diff --git a/.github/workflows/docs.yml b/.github/workflows/docs.yml
index 5d39e3d..57ea95d 100644
--- a/.github/workflows/docs.yml
+++ b/.github/workflows/docs.yml
@@ -23,7 +23,7 @@ jobs:
 
       - uses: actions/setup-python@v5
         with:
-          python-version: "3.11"
+          python-version: "3.10"
 
       - name: Install documentation dependencies
         run: uv sync --group docs
diff --git a/.github/workflows/tests.yml b/.github/workflows/tests.yml
index 6808c2a..572644a 100644
--- a/.github/workflows/tests.yml
+++ b/.github/workflows/tests.yml
@@ -38,7 +38,7 @@ jobs:
 
       - uses: actions/setup-python@v5
         with:
-          python-version: "3.11"
+          python-version: "3.10"
 
       - name: Install core + sr + md17 + dev extras
         run: uv sync --extra sr --extra md17 --extra dev
@@ -62,7 +62,7 @@ jobs:
 
       - uses: actions/setup-python@v5
         with:
-          python-version: "3.11"
+          python-version: "3.10"
 
       - name: Install all task extras + dev
         run: uv sync --extra sr --extra md17 --extra lqa --extra dev
@@ -88,7 +88,7 @@ jobs:
 
       - uses: actions/setup-python@v5
         with:
-          python-version: "3.11"
+          python-version: "3.10"
 
       - name: Install all task extras + dev
         run: uv sync --extra sr --extra md17 --extra lqa --extra dev
@@ -114,7 +114,7 @@ jobs:
 
       - uses: actions/setup-python@v5
         with:
-          python-version: "3.11"
+          python-version: "3.10"
 
       - name: Install all task extras + dev
         run: uv sync --extra sr --extra md17 --extra lqa --extra dev
diff --git a/README.md b/README.md
index e38e3eb..147bc34 100644
--- a/README.md
+++ b/README.md
@@ -1,6 +1,6 @@
 # Versor: A PyTorch Framework for Geometric Algebra Deep Learning
 
-[![License](https://img.shields.io/badge/License-Apache%202.0-blue.svg)](LICENSE) [![Python](https://img.shields.io/badge/python-3.9+-blue.svg)](https://www.python.org/downloads/) [![PyTorch](https://img.shields.io/badge/PyTorch-2.0+-ee4c2c.svg)](https://pytorch.org/) [![Docs](https://img.shields.io/badge/docs-MkDocs-brightgreen)](https://concode0.github.io/Versor/) [![DOI](https://zenodo.org/badge/1149480519.svg)](https://doi.org/10.5281/zenodo.18939518)
+[![License](https://img.shields.io/badge/License-Apache%202.0-blue.svg)](LICENSE) [![Python](https://img.shields.io/badge/python-3.10+-blue.svg)](https://www.python.org/downloads/) [![PyTorch](https://img.shields.io/badge/PyTorch-2.0+-ee4c2c.svg)](https://pytorch.org/) [![Docs](https://img.shields.io/badge/docs-MkDocs-brightgreen)](https://concode0.github.io/Versor/) [![DOI](https://zenodo.org/badge/1149480519.svg)](https://doi.org/10.5281/zenodo.18939518)
 
 > **"There is a ceiling above standard Deep Learning that no one saw. Versor opens the door above it."**
 
@@ -59,7 +59,7 @@ For code examples of each innovation, see [docs/innovations.md](docs/innovations
 
 ## Installation
 
-Versor requires Python 3.9+ and PyTorch.
+Versor requires Python 3.10+ and PyTorch.
 
 ```bash
 # Clone the repository
@@ -83,7 +83,7 @@ uv sync --extra all           # everything
 
 ```python
 import torch
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers.primitives.rotor import RotorLayer
 from layers.linear import CliffordLinear
 from functional.activation import GeometricGELU
@@ -361,4 +361,4 @@ By releasing this under Apache 2.0, we provide a **perpetual, royalty-free paten
 ```
 
 ## Reference:
- * Pence, T., Yamada, D., & Singh, V. (2025). "Composing Linear Layers from Irreducibles." arXiv:2507.11688v1 [cs.LG]
\ No newline at end of file
+ * Pence, T., Yamada, D., & Singh, V. (2025). "Composing Linear Layers from Irreducibles." arXiv:2507.11688v1 [cs.LG]
diff --git a/__init__.py b/__init__.py
index d520a36..abe7e8b 100644
--- a/__init__.py
+++ b/__init__.py
@@ -2,14 +2,20 @@
 
 __version__ = "1.0.0"
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.config import AlgebraConfig, make_algebra, make_algebra_from_config
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.context import AlgebraContext
 from layers import CliffordLinear, RotorLayer
 
 __all__ = [
     "__version__",
+    "AlgebraConfig",
+    "AlgebraContext",
     "CliffordAlgebra",
     "CliffordModule",
+    "make_algebra",
+    "make_algebra_from_config",
     "RotorLayer",
     "CliffordLinear",
 ]
diff --git a/benchmarks/benchmark_core.py b/benchmarks/benchmark_core.py
index eb00aa1..07f059b 100644
--- a/benchmarks/benchmark_core.py
+++ b/benchmarks/benchmark_core.py
@@ -55,16 +55,13 @@
 if str(_REPO_ROOT) not in sys.path:
     sys.path.insert(0, str(_REPO_ROOT))
 
-from core.algebra import CliffordAlgebra
-from core.decomposition import ExpPolicy, compiled_safe_decomposed_exp  # noqa: E402
-from core.device import resolve_device
+from core.config import make_algebra
+from core.runtime.decomposition import ExpPolicy, compiled_safe_decomposed_exp  # noqa: E402
+from core.foundation.device import FLOAT_DTYPES, resolve_device
+from core.foundation.device import dtype_name as _format_dtype_name
+from core.foundation.module import AlgebraLike
 
-DTYPES: dict[str, torch.dtype] = {
-    "float64": torch.float64,
-    "float32": torch.float32,
-    "bfloat16": torch.bfloat16,
-    "float16": torch.float16,
-}
+DTYPES: dict[str, torch.dtype] = FLOAT_DTYPES
 
 
 @dataclass(frozen=True)
@@ -97,7 +94,7 @@ def label(self) -> str:
 class CoreOpModule(nn.Module):
     """Small wrapper so core operators can be passed to torch.compile."""
 
-    def __init__(self, algebra: CliffordAlgebra, op: str):
+    def __init__(self, algebra: AlgebraLike, op: str):
         super().__init__()
         self.algebra = algebra
         self.op = op
@@ -160,20 +157,42 @@ def _parse_signature_csv(value: str) -> list[SignatureSpec]:
             raise ValueError(
                 f"invalid signature {raw!r}; use n, p:q, or p:q:r entries"
             )
-        if p < 0 or q < 0 or r < 0 or p + q + r < 1 or p + q + r > 12:
+        if p < 0 or q < 0 or r < 0 or p + q + r < 1 or p + q + r > 16:
             raise ValueError(
                 f"invalid signature {raw!r}; dimensions must be non-negative "
-                "and sum to 1..12"
+                "and sum to 1..16"
             )
         specs.append(SignatureSpec(p, q, r))
     return specs
 
 
 def _dtype_name(dtype: torch.dtype) -> str:
-    for name, candidate in DTYPES.items():
-        if candidate == dtype:
-            return name
-    return str(dtype).replace("torch.", "")
+    return _format_dtype_name(dtype)
+
+
+def setup_algebra(
+    p: int,
+    q: int = 0,
+    r: int = 0,
+    *,
+    device: str,
+    dtype: torch.dtype,
+    exp_policy: str | ExpPolicy = "balanced",
+    fixed_iterations: int | None = None,
+    args: argparse.Namespace | None = None,
+) -> AlgebraLike:
+    """Construct benchmark algebras through the shared core factory."""
+    return make_algebra(
+        p=p,
+        q=q,
+        r=r,
+        kernel=getattr(args, "algebra_kernel", "auto"),
+        dense_threshold=getattr(args, "dense_threshold", 8),
+        device=device,
+        dtype=dtype,
+        exp_policy=exp_policy,
+        fixed_iterations=fixed_iterations,
+    )
 
 
 def _ordered_modes(value: str) -> list[str]:
@@ -364,13 +383,13 @@ def _seed_all(seed: int, device: str) -> None:
 
 
 def _supported_dtypes(
-    device: str,
+    args: argparse.Namespace,
     requested: str,
     probe_n_values: Iterable[int],
 ) -> list[torch.dtype]:
     if requested == "auto":
         candidates = ["float64", "float32"]
-        if device.startswith("cuda") or device == "mps":
+        if args.device.startswith("cuda") or args.device == "mps":
             candidates += ["bfloat16", "float16"]
     else:
         candidates = _parse_csv(requested)
@@ -383,15 +402,15 @@ def _supported_dtypes(
         if name not in DTYPES:
             raise ValueError(f"unknown dtype {name!r}; valid: {sorted(DTYPES)}")
         dtype = DTYPES[name]
-        algebra: CliffordAlgebra | None = None
+        algebra: AlgebraLike | None = None
         x: torch.Tensor | None = None
         y: torch.Tensor | None = None
         try:
             for n in probe_ns:
-                algebra = CliffordAlgebra(n, 0, device=device, dtype=dtype)
-                x = torch.randn(2, algebra.dim, device=device, dtype=dtype)
+                algebra = setup_algebra(n, 0, device=args.device, dtype=dtype, args=args)
+                x = torch.randn(2, algebra.dim, device=args.device, dtype=dtype)
                 y = algebra.geometric_product(x, x)
-                _sync(device)
+                _sync(args.device)
                 if not torch.isfinite(y.float()).all().item():
                     raise RuntimeError(f"n={n} probe produced non-finite values")
                 algebra = None
@@ -399,14 +418,14 @@ def _supported_dtypes(
                 y = None
             supported.append(dtype)
         except Exception as exc:
-            print(f"Skipping dtype {name} on {device}: {exc}")
+            print(f"Skipping dtype {name} on {args.device}: {exc}")
         finally:
             algebra = None
             x = None
             y = None
-            _release_memory(device)
+            _release_memory(args.device)
     if not supported:
-        raise SystemExit(f"No requested dtypes are usable on {device}.")
+        raise SystemExit(f"No requested dtypes are usable on {args.device}.")
     return supported
 
 
@@ -532,7 +551,7 @@ def _nonzero_terms(tensor: torch.Tensor) -> int:
 
 
 def _analytic_forward_metrics(
-    algebra: CliffordAlgebra,
+    algebra: AlgebraLike,
     op: str,
     batch: int,
     channels: int,
@@ -544,16 +563,20 @@ def _analytic_forward_metrics(
     io_elements = 0
 
     if op == "gp":
-        flops = 2.0 * batch * _nonzero_terms(algebra.gp_signs)
+        signs = getattr(algebra, "gp_signs", None)
+        flops = 2.0 * batch * _nonzero_terms(signs) if signs is not None else float("nan")
         io_elements = 3 * batch * dim
     elif op == "wedge":
-        flops = 2.0 * batch * _nonzero_terms(algebra.wedge_gp_signs)
+        signs = getattr(algebra, "wedge_gp_signs", None)
+        flops = 2.0 * batch * _nonzero_terms(signs) if signs is not None else float("nan")
         io_elements = 3 * batch * dim
     elif op == "inner":
-        flops = 2.0 * batch * _nonzero_terms(algebra.inner_gp_signs)
+        signs = getattr(algebra, "inner_gp_signs", None)
+        flops = 2.0 * batch * _nonzero_terms(signs) if signs is not None else float("nan")
         io_elements = 3 * batch * dim
     elif op == "commutator":
-        flops = 2.0 * batch * _nonzero_terms(algebra.comm_gp_signs)
+        signs = getattr(algebra, "comm_gp_signs", None)
+        flops = 2.0 * batch * _nonzero_terms(signs) if signs is not None else float("nan")
         io_elements = 3 * batch * dim
     elif op in {"grade2", "reverse"}:
         flops = float(batch * dim)
@@ -1044,7 +1067,7 @@ def _commuting_pairs(n: int) -> list[tuple[int, int]]:
 
 
 def _make_commuting_nonsimple_bivector(
-    algebra: CliffordAlgebra,
+    algebra: AlgebraLike,
     batch: int,
     scale: float,
 ) -> torch.Tensor:
@@ -1065,7 +1088,7 @@ def _make_commuting_nonsimple_bivector(
     return b
 
 
-def _exact_commuting_exp(algebra: CliffordAlgebra, b: torch.Tensor) -> torch.Tensor:
+def _exact_commuting_exp(algebra: AlgebraLike, b: torch.Tensor) -> torch.Tensor:
     """Exact exp for the controlled non-simple bivector used here.
 
     The bivector is a sum of disjoint coordinate-plane bivectors. Those
@@ -1081,7 +1104,7 @@ def _exact_commuting_exp(algebra: CliffordAlgebra, b: torch.Tensor) -> torch.Ten
     return result
 
 
-def _grade_leak(algebra: CliffordAlgebra, x: torch.Tensor, grade: int) -> float:
+def _grade_leak(algebra: AlgebraLike, x: torch.Tensor, grade: int) -> float:
     mask = algebra.grade_masks_float[grade]
     if mask.dtype != x.dtype:
         mask = mask.to(dtype=x.dtype)
@@ -1090,7 +1113,7 @@ def _grade_leak(algebra: CliffordAlgebra, x: torch.Tensor, grade: int) -> float:
 
 
 def _make_speed_inputs(
-    algebra: CliffordAlgebra,
+    algebra: AlgebraLike,
     op: str,
     batch: int,
     channels: int,
@@ -1120,7 +1143,7 @@ def _make_speed_inputs(
 
 
 def _random_multivector(
-    algebra: CliffordAlgebra,
+    algebra: AlgebraLike,
     shape: tuple[int, ...],
     scale: float,
 ) -> torch.Tensor:
@@ -1131,7 +1154,7 @@ def _random_multivector(
 
 
 def _random_grade(
-    algebra: CliffordAlgebra,
+    algebra: AlgebraLike,
     shape: tuple[int, ...],
     grade: int,
     scale: float,
@@ -1164,13 +1187,14 @@ def run_stability_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) ->
         for sig_index, signature in enumerate(signatures):
             _release_memory(args.device)
             try:
-                algebra = CliffordAlgebra(
+                algebra = setup_algebra(
                     signature.p,
                     signature.q,
                     signature.r,
                     device=args.device,
                     dtype=dtype,
                     exp_policy=ExpPolicy.PRECISE,
+                    args=args,
                 )
             except Exception as exc:
                 _record_stability_failure(rows, args, dtype_name, signature, "setup", exc)
@@ -1368,13 +1392,14 @@ def per_channel_sandwich() -> tuple[float, float, str]:
             def exp_policy_consistency() -> tuple[float, float, str]:
                 if algebra.n < 2:
                     return 0.0, 0.0, "skipped: no bivectors"
-                balanced = CliffordAlgebra(
+                balanced = setup_algebra(
                     signature.p,
                     signature.q,
                     signature.r,
                     device=args.device,
                     dtype=dtype,
                     exp_policy=ExpPolicy.BALANCED,
+                    args=args,
                 )
                 with torch.no_grad():
                     b = torch.zeros(
@@ -1569,12 +1594,13 @@ def run_speed_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) -> list
                                 try:
                                     with _tf32_context(args.device, tf32_mode):
                                         row.update(_current_tf32_flags(args.device, tf32_mode))
-                                        algebra = CliffordAlgebra(
+                                        algebra = setup_algebra(
                                             n,
                                             0,
                                             device=args.device,
                                             dtype=dtype,
                                             exp_policy=_op_exp_policy(op),
+                                            args=args,
                                         )
                                         row.update(
                                             _analytic_forward_metrics(
@@ -1740,12 +1766,13 @@ def run_backward_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) -> l
                                 try:
                                     with _tf32_context(args.device, tf32_mode):
                                         row.update(_current_tf32_flags(args.device, tf32_mode))
-                                        algebra = CliffordAlgebra(
+                                        algebra = setup_algebra(
                                             n,
                                             0,
                                             device=args.device,
                                             dtype=dtype,
                                             exp_policy=_op_exp_policy(op),
+                                            args=args,
                                         )
                                         row.update(
                                             _analytic_forward_metrics(
@@ -1920,12 +1947,13 @@ def run_fusion_suite(
                                 try:
                                     with _tf32_context(args.device, tf32_mode):
                                         row.update(_current_tf32_flags(args.device, tf32_mode))
-                                        algebra = CliffordAlgebra(
+                                        algebra = setup_algebra(
                                             n,
                                             0,
                                             device=args.device,
                                             dtype=dtype,
                                             exp_policy=_op_exp_policy(op),
+                                            args=args,
                                         )
                                         row.update(
                                             _analytic_forward_metrics(
@@ -2070,7 +2098,7 @@ def run_nonsimple_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) ->
     for n in args.nonsimple_n_values:
         if n < 4:
             continue
-        ref_alg = CliffordAlgebra(n, 0, device="cpu", dtype=torch.float64, exp_policy="precise")
+        ref_alg = setup_algebra(n, 0, device="cpu", dtype=torch.float64, exp_policy="precise", args=args)
         b_ref = _make_commuting_nonsimple_bivector(ref_alg, args.error_batch, args.bivector_scale)
         r_ref = _exact_commuting_exp(ref_alg, b_ref)
         bb_ref = ref_alg.geometric_product(b_ref, b_ref)
@@ -2094,12 +2122,13 @@ def run_nonsimple_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) ->
                     "nonscalar_BB_norm": nonscalar_norm,
                 }
                 try:
-                    algebra = CliffordAlgebra(
+                    algebra = setup_algebra(
                         n,
                         0,
                         device=args.device,
                         dtype=dtype,
                         exp_policy=policy,
+                        args=args,
                     )
                     b = _make_commuting_nonsimple_bivector(
                         algebra,
@@ -2169,7 +2198,7 @@ def run_cumulative_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) ->
         if n < 4:
             continue
         for batch in args.cumulative_batch_sizes:
-            ref_alg = CliffordAlgebra(n, 0, device="cpu", dtype=torch.float64, exp_policy="precise")
+            ref_alg = setup_algebra(n, 0, device="cpu", dtype=torch.float64, exp_policy="precise", args=args)
             b_ref = _make_commuting_nonsimple_bivector(
                 ref_alg,
                 batch,
@@ -2182,7 +2211,14 @@ def run_cumulative_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) ->
                 dtype_name = _dtype_name(dtype)
                 _release_memory(args.device)
                 try:
-                    algebra = CliffordAlgebra(n, 0, device=args.device, dtype=dtype, exp_policy="balanced")
+                    algebra = setup_algebra(
+                        n,
+                        0,
+                        device=args.device,
+                        dtype=dtype,
+                        exp_policy="balanced",
+                        args=args,
+                    )
                     b = _make_commuting_nonsimple_bivector(
                         algebra,
                         batch,
@@ -2276,7 +2312,7 @@ def run_convergence_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) -
     for n in args.convergence_n_values:
         if n < 4:
             continue
-        ref_alg = CliffordAlgebra(n, 0, device="cpu", dtype=torch.float64, exp_policy="precise")
+        ref_alg = setup_algebra(n, 0, device="cpu", dtype=torch.float64, exp_policy="precise", args=args)
         b_ref = _make_commuting_nonsimple_bivector(ref_alg, args.error_batch, args.bivector_scale)
         r_ref = _exact_commuting_exp(ref_alg, b_ref)
 
@@ -2296,7 +2332,14 @@ def run_convergence_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) -
                     "error": "",
                 }
                 try:
-                    algebra = CliffordAlgebra(n, 0, device=args.device, dtype=dtype, exp_policy="precise")
+                    algebra = setup_algebra(
+                        n,
+                        0,
+                        device=args.device,
+                        dtype=dtype,
+                        exp_policy="precise",
+                        args=args,
+                    )
                     b = _make_commuting_nonsimple_bivector(
                         algebra,
                         args.error_batch,
@@ -2307,7 +2350,7 @@ def run_convergence_suite(args: argparse.Namespace, dtypes: list[torch.dtype]) -
 
                     def exp_fixed(
                         inp: torch.Tensor,
-                        alg: CliffordAlgebra = algebra,
+                        alg: AlgebraLike = algebra,
                         iterations: int = fixed_iterations,
                     ) -> torch.Tensor:
                         return compiled_safe_decomposed_exp(
@@ -3441,6 +3484,8 @@ def _collect_runtime_metadata(device: str) -> dict[str, Any]:
 def make_argparser() -> argparse.ArgumentParser:
     parser = argparse.ArgumentParser(description="Benchmark the Versor core package.")
     parser.add_argument("--device", default="auto", help="cpu, cuda, mps, or auto")
+    parser.add_argument("--algebra-kernel", default="auto", choices=("auto", "dense", "context"))
+    parser.add_argument("--dense-threshold", type=int, default=8)
     parser.add_argument("--out", default="benchmarks/results", help="artifact root")
     parser.add_argument("--sections", default="speed,backward,fusion,nonsimple,cumulative,convergence,stability")
     parser.add_argument("--n-values", type=_parse_int_csv, default=_parse_int_csv("2,3,4,5,6"))
@@ -3594,7 +3639,7 @@ def main() -> None:
     else:
         args.dtype_probe_n_values = sorted(set(args.dtype_probe_n_values))
 
-    dtypes = _supported_dtypes(args.device, args.dtypes, args.dtype_probe_n_values)
+    dtypes = _supported_dtypes(args, args.dtypes, args.dtype_probe_n_values)
     args.dtypes_resolved = dtypes
 
     timestamp = datetime.now().strftime("%Y%m%d_%H%M%S")
diff --git a/benchmarks/benchmark_framework_pipeline.py b/benchmarks/benchmark_framework_pipeline.py
new file mode 100644
index 0000000..3b54e26
--- /dev/null
+++ b/benchmarks/benchmark_framework_pipeline.py
@@ -0,0 +1,348 @@
+#!/usr/bin/env python3
+"""Benchmark framework-level dense, compact, pairwise, and layer pipeline paths.
+
+The core benchmark suite measures many algebra kernels in detail. This script is
+smaller and pipeline-oriented: it times the execution paths users hit when they
+compose product layers, functional products, compact layouts, and planned
+contexts.
+
+Examples:
+    uv run python benchmarks/benchmark_framework_pipeline.py --quick
+    uv run python benchmarks/benchmark_framework_pipeline.py --device cpu --n 8 --batch-size 1024
+"""
+
+from __future__ import annotations
+
+import argparse
+import csv
+import statistics
+import sys
+import time
+from dataclasses import dataclass
+from datetime import datetime
+from pathlib import Path
+from typing import Callable
+
+import torch
+import torch.nn as nn
+
+_REPO_ROOT = Path(__file__).resolve().parent.parent
+if str(_REPO_ROOT) not in sys.path:
+    sys.path.insert(0, str(_REPO_ROOT))
+
+from core.foundation.device import FLOAT_DTYPES, resolve_device  # noqa: E402
+from core.runtime.algebra import CliffordAlgebra  # noqa: E402
+from core.runtime.context import AlgebraContext  # noqa: E402
+from layers import ProductLayer, WedgeLayer  # noqa: E402
+
+
+@dataclass(frozen=True)
+class BenchmarkCase:
+    name: str
+    host: str
+    n: int
+    op: str
+    storage: str
+    batch_size: int
+    left_lanes: int
+    right_lanes: int
+    output_lanes: int
+    pairwise: bool
+    fn: Callable[[], torch.Tensor]
+
+
+def _sync(device: str) -> None:
+    if device == "mps":
+        torch.mps.synchronize()
+    elif device.startswith("cuda"):
+        torch.cuda.synchronize()
+
+
+def _time_case(case: BenchmarkCase, *, warmups: int, repeats: int, device: str) -> dict:
+    with torch.no_grad():
+        start = time.perf_counter()
+        first_output = case.fn()
+        _sync(device)
+        first_ms = (time.perf_counter() - start) * 1000.0
+
+        for _ in range(warmups):
+            case.fn()
+        _sync(device)
+
+        samples = []
+        for _ in range(repeats):
+            start = time.perf_counter()
+            output = case.fn()
+            _sync(device)
+            samples.append((time.perf_counter() - start) * 1000.0)
+
+    output_lanes = int(first_output.shape[-1])
+    if output_lanes != case.output_lanes:
+        raise RuntimeError(f"{case.name} expected {case.output_lanes} output lanes, got {output_lanes}")
+
+    median_ms = statistics.median(samples)
+    mean_ms = statistics.fmean(samples)
+    return {
+        "name": case.name,
+        "host": case.host,
+        "n": case.n,
+        "op": case.op,
+        "storage": case.storage,
+        "batch_size": case.batch_size,
+        "left_lanes": case.left_lanes,
+        "right_lanes": case.right_lanes,
+        "output_lanes": case.output_lanes,
+        "pairwise": case.pairwise,
+        "first_call_ms": first_ms,
+        "median_ms": median_ms,
+        "mean_ms": mean_ms,
+        "min_ms": min(samples),
+        "max_ms": max(samples),
+        "repeats": repeats,
+        "items_per_sec": case.batch_size * 1000.0 / median_ms if median_ms > 0 else float("inf"),
+    }
+
+
+def _dense_product_case(args, dtype: torch.dtype, device: str) -> BenchmarkCase:
+    algebra = CliffordAlgebra(args.n, 0, device=device, dtype=dtype)
+    left = algebra.embed_vector(torch.randn(args.batch_size, algebra.n, device=device, dtype=dtype))
+    right = algebra.embed_vector(torch.randn(args.batch_size, algebra.n, device=device, dtype=dtype))
+
+    return BenchmarkCase(
+        name="dense_full_gp_1x1",
+        host="CliffordAlgebra",
+        n=algebra.n,
+        op="gp",
+        storage="dense",
+        batch_size=args.batch_size,
+        left_lanes=algebra.dim,
+        right_lanes=algebra.dim,
+        output_lanes=algebra.dim,
+        pairwise=False,
+        fn=lambda: algebra.geometric_product(left, right),
+    )
+
+
+def _compact_product_case(args, dtype: torch.dtype, device: str) -> BenchmarkCase:
+    algebra = CliffordAlgebra(args.n, 0, device=device, dtype=dtype)
+    layout_1 = algebra.layout((1,))
+    layout_02 = algebra.layout((0, 2))
+    left = layout_1.compact(algebra.embed_vector(torch.randn(args.batch_size, algebra.n, device=device, dtype=dtype)))
+    right = layout_1.compact(algebra.embed_vector(torch.randn(args.batch_size, algebra.n, device=device, dtype=dtype)))
+
+    layer = ProductLayer(
+        algebra,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        left_compact=True,
+        right_compact=True,
+        compact_output=True,
+    )
+
+    return BenchmarkCase(
+        name="compact_layer_gp_1x1_to_02",
+        host="CliffordAlgebra",
+        n=algebra.n,
+        op="gp",
+        storage="compact",
+        batch_size=args.batch_size,
+        left_lanes=layout_1.dim,
+        right_lanes=layout_1.dim,
+        output_lanes=layout_02.dim,
+        pairwise=False,
+        fn=lambda: layer(left, right),
+    )
+
+
+def _context_compact_case(args, dtype: torch.dtype, device: str) -> BenchmarkCase:
+    context = AlgebraContext(args.context_n, 0, device=device, dtype=dtype)
+    layout_1 = context.layout((1,))
+    layout_02 = context.layout((0, 2))
+    left = torch.randn(args.batch_size, layout_1.dim, device=device, dtype=dtype)
+    right = torch.randn(args.batch_size, layout_1.dim, device=device, dtype=dtype)
+
+    return BenchmarkCase(
+        name="context_compact_gp_1x1_to_02",
+        host="AlgebraContext",
+        n=context.n,
+        op="gp",
+        storage="compact",
+        batch_size=args.batch_size,
+        left_lanes=layout_1.dim,
+        right_lanes=layout_1.dim,
+        output_lanes=layout_02.dim,
+        pairwise=False,
+        fn=lambda: context.geometric_product(
+            left,
+            right,
+            left_grades=(1,),
+            right_grades=(1,),
+            output_grades=(0, 2),
+            left_compact=True,
+            right_compact=True,
+            compact_output=True,
+        ),
+    )
+
+
+def _pairwise_context_case(args, dtype: torch.dtype, device: str) -> BenchmarkCase:
+    context = AlgebraContext(args.context_n, 0, device=device, dtype=dtype)
+    layout_2 = context.layout((2,))
+    layout_1 = context.layout((1,))
+    layout_3 = context.layout((3,))
+    left = torch.randn(args.batch_size, args.left_items, layout_2.dim, device=device, dtype=dtype)
+    right = torch.randn(args.batch_size, args.right_items, layout_1.dim, device=device, dtype=dtype)
+
+    return BenchmarkCase(
+        name="context_pairwise_wedge_2x1_to_3",
+        host="AlgebraContext",
+        n=context.n,
+        op="wedge",
+        storage="compact_pairwise",
+        batch_size=args.batch_size * args.left_items * args.right_items,
+        left_lanes=layout_2.dim,
+        right_lanes=layout_1.dim,
+        output_lanes=layout_3.dim,
+        pairwise=True,
+        fn=lambda: context.wedge(
+            left,
+            right,
+            left_grades=(2,),
+            right_grades=(1,),
+            output_grades=(3,),
+            left_compact=True,
+            right_compact=True,
+            compact_output=True,
+            pairwise=True,
+        ),
+    )
+
+
+def _layer_pipeline_case(args, dtype: torch.dtype, device: str) -> BenchmarkCase:
+    context = AlgebraContext(args.context_n, 0, device=device, dtype=dtype)
+    layout_1 = context.layout((1,))
+    layout_3 = context.layout((3,))
+
+    class Pipeline(nn.Module):
+        def __init__(self):
+            super().__init__()
+            self.wedge_vectors = WedgeLayer(
+                context,
+                left_grades=(1,),
+                right_grades=(1,),
+                output_grades=(2,),
+                compact_output=True,
+            )
+            self.wedge_trivector = WedgeLayer(
+                context,
+                left_grades=(2,),
+                right_grades=(1,),
+                output_grades=(3,),
+                left_compact=True,
+                right_compact=True,
+                compact_output=True,
+            )
+
+        def forward(self, left, right, third):
+            bivector = self.wedge_vectors(left, right)
+            return self.wedge_trivector(bivector, third)
+
+    model = Pipeline().to(device=device)
+    left = context.embed_vector(torch.randn(args.batch_size, context.n, device=device, dtype=dtype))
+    right = context.embed_vector(torch.randn(args.batch_size, context.n, device=device, dtype=dtype))
+    third = layout_1.compact(context.embed_vector(torch.randn(args.batch_size, context.n, device=device, dtype=dtype)))
+
+    return BenchmarkCase(
+        name="layer_pipeline_wedge_wedge",
+        host="AlgebraContext",
+        n=context.n,
+        op="wedge_pipeline",
+        storage="dense_to_compact",
+        batch_size=args.batch_size,
+        left_lanes=context.dim,
+        right_lanes=layout_1.dim,
+        output_lanes=layout_3.dim,
+        pairwise=False,
+        fn=lambda: model(left, right, third),
+    )
+
+
+def _write_results(rows: list[dict], output_dir: Path) -> None:
+    output_dir.mkdir(parents=True, exist_ok=True)
+    csv_path = output_dir / "framework_pipeline.csv"
+    with csv_path.open("w", newline="") as handle:
+        writer = csv.DictWriter(handle, fieldnames=list(rows[0]))
+        writer.writeheader()
+        writer.writerows(rows)
+
+    summary_path = output_dir / "summary.md"
+    with summary_path.open("w") as handle:
+        handle.write("# Framework Pipeline Benchmark\n\n")
+        handle.write("| case | host | storage | n | median ms | items/sec |\n")
+        handle.write("| --- | --- | --- | ---: | ---: | ---: |\n")
+        for row in rows:
+            handle.write(
+                f"| {row['name']} | {row['host']} | {row['storage']} | {row['n']} | "
+                f"{row['median_ms']:.4f} | {row['items_per_sec']:.2f} |\n"
+            )
+
+
+def parse_args() -> argparse.Namespace:
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--device", default="cpu", help="cpu, cuda, cuda:0, mps, or auto")
+    parser.add_argument("--dtype", default="float32", choices=sorted(FLOAT_DTYPES))
+    parser.add_argument("--n", type=int, default=6, help="Dense algebra dimension for dense-vs-compact comparison")
+    parser.add_argument("--context-n", type=int, default=12, help="Planned context dimension")
+    parser.add_argument("--batch-size", type=int, default=512)
+    parser.add_argument("--left-items", type=int, default=16)
+    parser.add_argument("--right-items", type=int, default=16)
+    parser.add_argument("--warmups", type=int, default=5)
+    parser.add_argument("--repeats", type=int, default=20)
+    parser.add_argument("--quick", action="store_true", help="Use smaller dimensions and repeat counts")
+    parser.add_argument("--no-save", action="store_true", help="Print rows without writing benchmark artifacts")
+    parser.add_argument("--output-dir", type=Path, default=None)
+    return parser.parse_args()
+
+
+def main() -> None:
+    args = parse_args()
+    if args.quick:
+        args.n = min(args.n, 5)
+        args.context_n = min(args.context_n, 8)
+        args.batch_size = min(args.batch_size, 64)
+        args.left_items = min(args.left_items, 8)
+        args.right_items = min(args.right_items, 8)
+        args.warmups = min(args.warmups, 2)
+        args.repeats = min(args.repeats, 5)
+
+    device = resolve_device(args.device) if args.device == "auto" else args.device
+    dtype = FLOAT_DTYPES[args.dtype]
+    torch.manual_seed(2026)
+
+    cases = [
+        _dense_product_case(args, dtype, device),
+        _compact_product_case(args, dtype, device),
+        _context_compact_case(args, dtype, device),
+        _pairwise_context_case(args, dtype, device),
+        _layer_pipeline_case(args, dtype, device),
+    ]
+    rows = [_time_case(case, warmups=args.warmups, repeats=args.repeats, device=device) for case in cases]
+
+    for row in rows:
+        print(
+            f"{row['name']}: median={row['median_ms']:.4f} ms, "
+            f"first={row['first_call_ms']:.4f} ms, items/sec={row['items_per_sec']:.2f}"
+        )
+
+    if not args.no_save:
+        output_dir = args.output_dir
+        if output_dir is None:
+            stamp = datetime.now().strftime("%Y%m%d_%H%M%S")
+            output_dir = _REPO_ROOT / "benchmarks" / "results" / f"framework_pipeline_{stamp}"
+        _write_results(rows, output_dir)
+        print(f"Wrote benchmark artifacts to {output_dir}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/conf/config.yaml b/conf/config.yaml
index 6bf8d7d..919f14c 100644
--- a/conf/config.yaml
+++ b/conf/config.yaml
@@ -6,7 +6,12 @@ algebra:
   p: 3
   q: 0
   r: 0
+  kernel: "auto"            # auto: dense for n<=dense_threshold, AlgebraContext above it
+  dense_threshold: 8
   device: "auto"  # Auto-detect: cuda > mps > cpu
+  dtype: "float32"
+  exp_policy: "balanced"
+  default_grades: null       # null = full layout fallback when no specific layout is declared
 
 training:
   epochs: 100
@@ -17,11 +22,15 @@ training:
   momentum: 0.9
   betas: [0.9, 0.999]
   max_bivector_norm: 10.0
+  scheduler:
+    factor: 0.5
+    patience: 10
   num_workers: 2  # DataLoader workers (0 to disable multiprocessing)
   pin_memory: null        # null = auto (true for CUDA)
   compile: false          # torch.compile() wrapper
   compile_backend: null   # null=auto (aot_eager on MPS, inductor on CUDA/CPU)
   amp: false              # Automatic mixed precision (CUDA only)
+  amp_dtype: null         # null=PyTorch autocast default; or "float16"/"bfloat16"
   cudnn_benchmark: null   # null = auto (true for CUDA)
 
 checkpoint: null
diff --git a/conf/task/deap_eeg.yaml b/conf/task/deap_eeg.yaml
index a7b10e2..80e6079 100644
--- a/conf/task/deap_eeg.yaml
+++ b/conf/task/deap_eeg.yaml
@@ -5,7 +5,6 @@ algebra:
   p: 3
   q: 1
   r: 0
-  device: "auto"
 
 dataset:
   name: "deap"
@@ -29,5 +28,3 @@ training:
   epochs: 100
   lr: 0.0005
   batch_size: 32
-  optimizer_type: "riemannian_adam"
-  max_bivector_norm: 10.0
diff --git a/conf/task/lqa.yaml b/conf/task/lqa.yaml
index 6f2d057..a4c83ff 100644
--- a/conf/task/lqa.yaml
+++ b/conf/task/lqa.yaml
@@ -8,7 +8,6 @@ algebra:
   p: 4
   q: 1
   r: 0
-  device: auto
 
 model:
   channels: 16
@@ -33,7 +32,6 @@ training:
   epochs: 50
   lr: 0.001
   batch_size: 64
-  optimizer_type: riemannian_adam
   # Auxiliary loss weights
   isometry_weight: 0.01
   asymmetry_weight: 0.1
diff --git a/conf/task/md17.yaml b/conf/task/md17.yaml
index c68d29b..f65dc6e 100644
--- a/conf/task/md17.yaml
+++ b/conf/task/md17.yaml
@@ -4,7 +4,6 @@ algebra:
   p: 3
   q: 0
   r: 1
-  device: "auto"
 dataset:
   name: "rmd17"
   molecule: "ethanol"
@@ -20,7 +19,6 @@ model:
   max_z: 100
   num_rbf: 20
   rbf_cutoff: 5.0
-  exp_policy: "balanced"
   use_rotor_backend: true
   use_geo_square: true
   use_checkpoint: false
@@ -28,7 +26,6 @@ training:
   epochs: 500
   lr: 0.0015
   batch_size: 96
-  optimizer_type: "riemannian_adam"
   loss_weights:
     energy: 0.1
     force: 100.0
@@ -36,4 +33,3 @@ training:
     conservative: 0.0
     grade_reg: 0.01
   target_spectrum: [0.35, 0.30, 0.20, 0.10, 0.05]
-  max_bivector_norm: 10.0
diff --git a/conf/task/sr.yaml b/conf/task/sr.yaml
index 514ded9..d3cc5a1 100644
--- a/conf/task/sr.yaml
+++ b/conf/task/sr.yaml
@@ -6,7 +6,7 @@ algebra:
   q: 0
   r: 0
   device: "cpu"
-  auto: true
+  metric_search: true
 
 dataset:
   name: "sr"
@@ -21,16 +21,12 @@ dataset:
 
 model:
   auto_capacity: true
-  exp_policy: "balanced"
 
 training:
   epochs: 200
   lr: 0.003
   batch_size: 128
-  optimizer_type: "riemannian_adam"
-  max_bivector_norm: 10.0
   sparsity_weight: 0.01
-  scheduler_patience: 10
 
 iterative:
   max_stages: 5
@@ -92,4 +88,4 @@ basis:
   corr_threshold: 0.05
   max_expansion_factor: 3
   dynamic_range_threshold: 100.0
-  exp_max_input: 700.0
\ No newline at end of file
+  exp_max_input: 700.0
diff --git a/core/__init__.py b/core/__init__.py
index 21934de..faf12f0 100644
--- a/core/__init__.py
+++ b/core/__init__.py
@@ -11,16 +11,48 @@
 until first access, keeping ``import core`` lightweight.
 """
 
-from .algebra import CliffordAlgebra
-from .decomposition import (
+from .config import AlgebraConfig, make_algebra, make_algebra_from_config
+from .foundation.basis import (
+    GradeProductOp,
+    basis_indices_for_grades,
+    basis_product,
+    expand_output_grades,
+    geometric_product_output_grades,
+    normalize_grades,
+    operation_coefficient,
+    operation_may_be_nonzero,
+    product_output_grades,
+    reverse_sign,
+)
+from .foundation.device import DeviceConfig, dtype_name, optional_dtype, resolve_device, resolve_dtype
+from .foundation.layout import AlgebraSpec, GradeLayout
+from .foundation.module import AlgebraLike, CliffordModule
+from .foundation.validation import check_channels, check_multivector
+from .planning.flow import GradeFlow
+from .planning.layouts import ProductRequest, build_product_request
+from .planning.planner import GradePlanner
+from .planning.policy import DEFAULT_PLANNING_LIMITS, PlanCost, PlanningLimits
+from .planning.product import GradeProductExecutor, GradeProductPlan, build_grade_product_plan
+from .planning.tree import GradePathNode, GradePlanTree, build_grade_plan_tree
+from .planning.unary import GradeUnaryExecutor, GradeUnaryOp, GradeUnaryPlan, UnaryRequest, build_unary_request
+from .runtime.accessors import (
+    as_multivector,
+    compact_values,
+    grade_indices,
+    hermitian_signs,
+    materialize_dense,
+    resolve_layout,
+)
+from .runtime.algebra import CliffordAlgebra
+from .runtime.context import AlgebraContext
+from .runtime.decomposition import (
     ExpPolicy,
     compiled_safe_decomposed_exp,
     differentiable_invariant_decomposition,
     exp_simple_bivector,
     ga_power_iteration,
 )
-from .device import DeviceConfig, resolve_device
-from .metric import (
+from .runtime.metric import (
     clifford_conjugate,
     geometric_distance,
     grade_hermitian_norm,
@@ -36,18 +68,30 @@
     signature_norm_squared,
     signature_trace_form,
 )
-from .module import CliffordModule
-from .multivector import Multivector
-from .validation import check_channels, check_multivector
+from .runtime.multivector import Multivector
 
 __all__ = [
     # algebra
+    "AlgebraContext",
     "CliffordAlgebra",
+    "AlgebraConfig",
+    "AlgebraLike",
     "CliffordModule",
     "Multivector",
+    "AlgebraSpec",
+    "GradeLayout",
+    "GradePlanner",
+    "PlanningLimits",
+    "PlanCost",
+    "DEFAULT_PLANNING_LIMITS",
+    "make_algebra",
+    "make_algebra_from_config",
     # device / validation
     "DeviceConfig",
+    "dtype_name",
+    "optional_dtype",
     "resolve_device",
+    "resolve_dtype",
     "check_multivector",
     "check_channels",
     # metric
@@ -65,12 +109,43 @@
     "hermitian_grade_spectrum",
     "signature_trace_form",
     "signature_norm_squared",
+    "as_multivector",
+    "compact_values",
+    "grade_indices",
+    "hermitian_signs",
+    "materialize_dense",
+    "resolve_layout",
     # decomposition
     "ExpPolicy",
     "ga_power_iteration",
     "differentiable_invariant_decomposition",
     "exp_simple_bivector",
     "compiled_safe_decomposed_exp",
+    # static sparse grade planning
+    "GradeProductOp",
+    "GradeProductExecutor",
+    "GradeProductPlan",
+    "GradePathNode",
+    "GradePlanTree",
+    "GradeFlow",
+    "ProductRequest",
+    "GradeUnaryExecutor",
+    "GradeUnaryOp",
+    "GradeUnaryPlan",
+    "UnaryRequest",
+    "basis_indices_for_grades",
+    "basis_product",
+    "build_grade_product_plan",
+    "build_grade_plan_tree",
+    "build_product_request",
+    "build_unary_request",
+    "expand_output_grades",
+    "geometric_product_output_grades",
+    "normalize_grades",
+    "operation_coefficient",
+    "operation_may_be_nonzero",
+    "product_output_grades",
+    "reverse_sign",
     # analysis (lazy)
     "MetricSearch",
     "GeodesicFlow",
diff --git a/core/analysis/commutator.py b/core/analysis/commutator.py
index 4c46dd3..24e0119 100644
--- a/core/analysis/commutator.py
+++ b/core/analysis/commutator.py
@@ -16,7 +16,7 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.foundation.module import AlgebraLike
 
 from ._types import CONSTANTS, CommutatorResult
 
@@ -29,14 +29,14 @@ class CommutatorAnalyzer:
     elements (bivectors), directly related to rotation planes.
 
     Args:
-        algebra: :class:`CliffordAlgebra` instance.
+        algebra: algebra kernel or planning context.
         max_bivectors: Maximum number of bivectors for Lie-bracket
             closure analysis (guards combinatorial cost).
     """
 
     def __init__(
         self,
-        algebra: CliffordAlgebra,
+        algebra: AlgebraLike,
         max_bivectors: int = 15,
     ):
         self.algebra = algebra
@@ -86,28 +86,34 @@ def commutativity_matrix(self, mv_data: torch.Tensor) -> torch.Tensor:
             ``[n, n]`` symmetric matrix of commutativity indices.
         """
         n = self.algebra.n
-        dim = self.algebra.dim
         device = mv_data.device
         dtype = mv_data.dtype
 
-        # Build per-direction projections: keep only the e_i component
-        g1_idx = (1 << torch.arange(n, device=device)).long()
-
+        g1_idx = self.algebra.grade_indices((1,), device=device)
         N = mv_data.shape[0]
 
-        # Build all n projected multivectors: [n, N, dim]
-        xi_all = torch.zeros(n, N, dim, device=device, dtype=dtype)
-        coeffs = mv_data[:, g1_idx]  # [N, n]
-        row_idx = torch.arange(n, device=device).unsqueeze(1).expand(n, N)
-        batch_idx = torch.arange(N, device=device).unsqueeze(0).expand(n, N)
-        col_idx = g1_idx.unsqueeze(1).expand(n, N)
-        xi_all[row_idx, batch_idx, col_idx] = coeffs.T
-
         # All (i, j) pairs with i < j
         i_idx, j_idx = torch.triu_indices(n, n, offset=1, device=device)
+        if i_idx.numel() == 0:
+            return torch.zeros(n, n, device=device, dtype=dtype)
 
-        # Batched commutator: [n_pairs, N, dim]
-        comm = self.algebra.commutator(xi_all[i_idx], xi_all[j_idx])
+        coeffs = mv_data[:, g1_idx]  # [N, n]
+        left = torch.zeros(i_idx.numel(), N, n, device=device, dtype=dtype)
+        right = torch.zeros_like(left)
+        left.scatter_(-1, i_idx.view(-1, 1, 1).expand(-1, N, 1), coeffs[:, i_idx].T.unsqueeze(-1))
+        right.scatter_(-1, j_idx.view(-1, 1, 1).expand(-1, N, 1), coeffs[:, j_idx].T.unsqueeze(-1))
+
+        # Batched compact commutator: [n_pairs, N, grade2_dim]
+        comm = self.algebra.commutator(
+            left,
+            right,
+            left_grades=(1,),
+            right_grades=(1,),
+            output_grades=(2,),
+            left_compact=True,
+            right_compact=True,
+            compact_output=True,
+        )
         vals = comm.norm(dim=-1).mean(dim=-1)  # [n_pairs]
 
         matrix = torch.zeros(n, n, device=device, dtype=dtype)
@@ -178,7 +184,6 @@ def lie_bracket_closure(self, mv_data: torch.Tensor) -> Dict:
             ``"closure_error"`` (scalar), and ``"basis_indices"`` (list
             of multivector-coefficient indices of the chosen bivectors).
         """
-        dim = self.algebra.dim
         n = self.algebra.n
         device = mv_data.device
         dtype = mv_data.dtype
@@ -190,15 +195,13 @@ def lie_bracket_closure(self, mv_data: torch.Tensor) -> Dict:
                 "basis_indices": [],
             }
 
-        # Extract grade-2 part of mean per-sample
-        bv_data = self.algebra.grade_projection(mv_data, 2)  # [N, dim]
-        mean_bv = bv_data.mean(dim=0)  # [dim]
+        # Extract compact grade-2 part of mean per-sample
+        bv_data = self.algebra.grade_projection(mv_data, 2, compact_output=True)  # [N, grade2_dim]
+        mean_bv = bv_data.mean(dim=0)  # [grade2_dim]
 
-        # Identify bivector basis indices
-        ii, jj = torch.triu_indices(n, n, offset=1)
-        bv_blade_indices = ((1 << ii) | (1 << jj)).tolist()
+        bv_blade_indices = self.algebra.grade_indices((2,), device=device)
 
-        if not bv_blade_indices:
+        if bv_blade_indices.numel() == 0:
             return {
                 "structure_constants": torch.zeros(0, 0, 0, device=device),
                 "closure_error": 0.0,
@@ -206,24 +209,29 @@ def lie_bracket_closure(self, mv_data: torch.Tensor) -> Dict:
             }
 
         # Pick top-k by energy in the mean bivector
-        bv_idx_tensor = torch.tensor(bv_blade_indices, dtype=torch.long, device=device)
-        energies = mean_bv[bv_idx_tensor].abs()
-        k = min(self.max_bivectors, len(bv_blade_indices))
-        topk_pos = energies.topk(k).indices.tolist()
+        energies = mean_bv.abs()
+        k = min(self.max_bivectors, int(bv_blade_indices.numel()))
+        topk_pos = energies.topk(k).indices
+        selected_indices = bv_blade_indices[topk_pos].tolist()
 
-        selected_indices = [bv_blade_indices[p] for p in topk_pos]
-
-        # Build basis bivector multivectors
-        B = torch.zeros(k, dim, device=device, dtype=dtype)
-        sel_tensor = torch.tensor(selected_indices, dtype=torch.long, device=device)
-        B[torch.arange(k, device=device), sel_tensor] = 1.0
+        # Build compact basis bivectors.
+        B = torch.zeros(k, bv_blade_indices.numel(), device=device, dtype=dtype)
+        B[torch.arange(k, device=device), topk_pos] = 1.0
 
         # Compute structure constants c_{a,b,c} such that [B_a, B_b] ~= Sum_c c_{abc} B_c
         a_idx, b_idx = torch.triu_indices(k, k, offset=1, device=device)
 
-        # Batched commutator and grade-2 projection
-        brackets = self.algebra.commutator(B[a_idx], B[b_idx])  # [n_pairs, dim]
-        brackets_bv = self.algebra.grade_projection(brackets, 2)  # [n_pairs, dim]
+        # Batched compact commutator and grade-2 projection.
+        brackets_bv = self.algebra.commutator(
+            B[a_idx],
+            B[b_idx],
+            left_grades=(2,),
+            right_grades=(2,),
+            output_grades=(2,),
+            left_compact=True,
+            right_compact=True,
+            compact_output=True,
+        )  # [n_pairs, grade2_dim]
 
         # Project onto basis: coeffs[p, c] = <bracket_bv_p, B_c>
         coeffs = brackets_bv @ B.T  # [n_pairs, k]
@@ -233,7 +241,7 @@ def lie_bracket_closure(self, mv_data: torch.Tensor) -> Dict:
         structure[b_idx, a_idx, :] = -coeffs  # antisymmetry
 
         # Closure errors: residual norm / bracket norm
-        projected = coeffs @ B  # [n_pairs, dim]
+        projected = coeffs @ B  # [n_pairs, grade2_dim]
         residuals = brackets_bv - projected
         res_norms = residuals.norm(dim=-1)  # [n_pairs]
         bracket_norms = brackets_bv.norm(dim=-1)  # [n_pairs]
@@ -251,14 +259,14 @@ def lie_bracket_closure(self, mv_data: torch.Tensor) -> Dict:
         }
 
 
-def compute_uncertainty_and_alignment(algebra: CliffordAlgebra, data_tensor: torch.Tensor):
+def compute_uncertainty_and_alignment(algebra: AlgebraLike, data_tensor: torch.Tensor):
     """Compute Geometric Uncertainty Index (U) and Procrustes Alignment (V).
 
     Used by :class:`~layers.adapters.mother.MotherEmbedding` to initialise
     per-group / per-subject alignment rotors.
 
     Args:
-        algebra: CliffordAlgebra instance.
+        algebra: algebra kernel or planning context.
         data_tensor: ``[N, D]`` tensor of raw features.
 
     Returns:
@@ -275,11 +283,18 @@ def compute_uncertainty_and_alignment(algebra: CliffordAlgebra, data_tensor: tor
     else:
         x_n = data_tensor[:, :n]
 
-    x = algebra.embed_vector(x_n)  # [N, dim]
-
-    # 2. Mean multivector and commutator [x_i, mu]
-    mu = x.mean(dim=0, keepdim=True)  # [1, dim]
-    comm = algebra.commutator(x, mu.expand_as(x))
+    # 2. Mean grade-1 vector and compact commutator [x_i, mu]
+    mu = x_n.mean(dim=0, keepdim=True)  # [1, n]
+    comm = algebra.commutator(
+        x_n,
+        mu.expand_as(x_n),
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(2,),
+        left_compact=True,
+        right_compact=True,
+        compact_output=True,
+    )
 
     U = torch.norm(comm, p=2, dim=-1).mean().item()
 
diff --git a/core/analysis/dimension.py b/core/analysis/dimension.py
index 958e33b..628d9b3 100644
--- a/core/analysis/dimension.py
+++ b/core/analysis/dimension.py
@@ -17,7 +17,8 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra
+from core.foundation.module import AlgebraLike
 
 from ._types import CONSTANTS, DimensionResult
 
@@ -196,7 +197,7 @@ def __init__(self, device: str = "cpu"):
     def lift(
         self,
         data: torch.Tensor,
-        target_algebra: CliffordAlgebra,
+        target_algebra: AlgebraLike,
         fill: float = 1.0,
     ) -> torch.Tensor:
         """Lifts data into the grade-1 subspace of a higher-dimensional algebra.
@@ -260,7 +261,7 @@ def test(
         data = data.to(self.device)
         results: Dict = {}
 
-        def _measure(alg: CliffordAlgebra, mv: torch.Tensor) -> Dict:
+        def _measure(alg: AlgebraLike, mv: torch.Tensor) -> Dict:
             gf = GeodesicFlow(alg, k=k)
             coh = gf.coherence(mv)
             curv = gf.curvature(mv)
@@ -273,15 +274,15 @@ def _measure(alg: CliffordAlgebra, mv: torch.Tensor) -> Dict:
                 "causal": (coh > coh_threshold) and (curv < CONSTANTS.curvature_causal_threshold),
             }
 
-        alg_orig = CliffordAlgebra(p, q, device=self.device)
+        alg_orig = make_algebra(p, q, device=self.device)
         mv_orig = alg_orig.embed_vector(data[..., : alg_orig.n])
         results["original"] = _measure(alg_orig, mv_orig)
 
-        alg_pos = CliffordAlgebra(p + 1, q, device=self.device)
+        alg_pos = make_algebra(p + 1, q, device=self.device)
         mv_pos = self.lift(data, alg_pos, fill=1.0)
         results["lift_positive"] = _measure(alg_pos, mv_pos)
 
-        alg_null = CliffordAlgebra(p, q + 1, device=self.device)
+        alg_null = make_algebra(p, q + 1, device=self.device)
         mv_null = self.lift(data, alg_null, fill=0.0)
         results["lift_null"] = _measure(alg_null, mv_null)
 
diff --git a/core/analysis/geodesic.py b/core/analysis/geodesic.py
index 6785a43..599e916 100644
--- a/core/analysis/geodesic.py
+++ b/core/analysis/geodesic.py
@@ -16,7 +16,7 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.foundation.module import AlgebraLike
 
 from ._types import CONSTANTS
 
@@ -44,7 +44,7 @@ class GeodesicFlow:
       collides with itself.  The signal is dominated by noise.
     """
 
-    def __init__(self, algebra: CliffordAlgebra, k: int = 8):
+    def __init__(self, algebra: AlgebraLike, k: int = 8):
         """Initialize Geodesic Flow.
 
         Args:
@@ -86,8 +86,7 @@ def _knn(self, mv: torch.Tensor) -> torch.Tensor:
         """
         N = mv.shape[0]
         k = min(self.k, N - 1)
-        diff = mv.unsqueeze(1) - mv.unsqueeze(0)  # [N, N, dim]
-        dists = diff.norm(dim=-1)  # [N, N]
+        dists = torch.cdist(mv, mv)  # [N, N]
         dists.fill_diagonal_(float("inf"))
         _, idx = dists.topk(k, dim=-1, largest=False)
         return idx  # [N, k]
@@ -112,10 +111,16 @@ def _connection_bivectors(self, mv: torch.Tensor) -> torch.Tensor:
 
         neighbors = mv[nn_idx]  # [N, k, dim]
         xi = mv.unsqueeze(1).expand(N, k, D).reshape(N * k, D)
-        xj_rev = self.algebra.reverse(neighbors.reshape(N * k, D))
+        xj_rev = self.algebra.reverse(neighbors.reshape(N * k, D), input_grades=(1,))
 
         # For grade-1 inputs, <xi * ~xj>_2 = wedge(xi, xj_rev) -- single pass
-        bv_raw = self.algebra.wedge(xi, xj_rev)  # [N*k, dim]
+        bv_raw = self.algebra.wedge(
+            xi,
+            xj_rev,
+            left_grades=(1,),
+            right_grades=(1,),
+            output_grades=(2,),
+        )  # [N*k, dim]
         bv_norm = bv_raw.norm(dim=-1, keepdim=True).clamp(min=self.algebra.eps)
         return (bv_raw / bv_norm).reshape(N, k, D)  # [N, k, dim]
 
diff --git a/core/analysis/pipeline.py b/core/analysis/pipeline.py
index 3f53b87..f7ed812 100644
--- a/core/analysis/pipeline.py
+++ b/core/analysis/pipeline.py
@@ -13,7 +13,8 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra
+from core.foundation.module import AlgebraLike
 
 from ._types import AnalysisConfig, AnalysisReport, SamplingConfig
 from .commutator import CommutatorAnalyzer
@@ -50,7 +51,7 @@ def __init__(self, config: Optional[AnalysisConfig] = None):
     def analyze(
         self,
         data: torch.Tensor,
-        algebra: Optional[CliffordAlgebra] = None,
+        algebra: Optional[AlgebraLike] = None,
     ) -> AnalysisReport:
         """Run the full geometric analysis pipeline.
 
@@ -124,7 +125,7 @@ def _run_full_pipeline(self, data: torch.Tensor, report: AnalysisReport) -> Anal
         if p + q + r < 2:
             p = max(p, 2 - q - r)
 
-        algebra = CliffordAlgebra(p, q, r, device=cfg.device)
+        algebra = make_algebra(p, q, r, device=cfg.device)
         mv_data = self._embed_raw(sampled, algebra)
 
         # 5. GA analyses (parallel)
@@ -135,7 +136,7 @@ def _run_full_pipeline(self, data: torch.Tensor, report: AnalysisReport) -> Anal
     def _run_ga_analyses(
         self,
         mv_data: torch.Tensor,
-        algebra: CliffordAlgebra,
+        algebra: AlgebraLike,
         report: AnalysisReport,
     ) -> AnalysisReport:
         cfg = self.config
@@ -173,7 +174,7 @@ def _run_ga_analyses(
         return report
 
     @staticmethod
-    def _embed_raw(data: torch.Tensor, algebra: CliffordAlgebra) -> torch.Tensor:
+    def _embed_raw(data: torch.Tensor, algebra: AlgebraLike) -> torch.Tensor:
         """Embed raw ``[N, D]`` data as grade-1 multivectors ``[N, 1, dim]``."""
         n = algebra.n
         D = data.shape[1]
diff --git a/core/analysis/sampler.py b/core/analysis/sampler.py
index 20d1665..a9804de 100644
--- a/core/analysis/sampler.py
+++ b/core/analysis/sampler.py
@@ -91,7 +91,7 @@ def _stratified(data: torch.Tensor, config: SamplingConfig) -> Tuple[torch.Tenso
         coherence) and unstructured (low coherence) regions of the data
         without assuming a specific metric signature.
         """
-        from core.algebra import CliffordAlgebra
+        from core.config import make_algebra
 
         from .geodesic import GeodesicFlow
 
@@ -102,7 +102,7 @@ def _stratified(data: torch.Tensor, config: SamplingConfig) -> Tuple[torch.Tenso
 
         # Cap algebra dimension for tractability
         alg_dim = min(D, 6)
-        algebra = CliffordAlgebra(alg_dim, 0, device=data.device)
+        algebra = make_algebra(alg_dim, 0, device=data.device)
 
         # Embed into algebra (truncate to alg_dim if needed)
         raw = data[:, :alg_dim].float()
diff --git a/core/analysis/signature.py b/core/analysis/signature.py
index 608cc25..60acc85 100644
--- a/core/analysis/signature.py
+++ b/core/analysis/signature.py
@@ -21,28 +21,85 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from layers import BladeSelector, CliffordLinear, RotorLayer
+from core.config import make_algebra
+from core.foundation.module import AlgebraLike, CliffordModule
 
 from ._types import CONSTANTS, DimensionResult, SamplingConfig, SignatureResult
 from .geodesic import GeodesicFlow
 
 
+class _ProbeLinear(CliffordModule):
+    """Core-local channel mixer used by metric-search probes."""
+
+    def __init__(self, algebra: AlgebraLike, in_channels: int, out_channels: int):
+        super().__init__(algebra)
+        self.weight = nn.Parameter(torch.empty(out_channels, in_channels))
+        self.bias = nn.Parameter(torch.empty(out_channels, algebra.dim))
+        self.reset_parameters()
+
+    def reset_parameters(self) -> None:
+        nn.init.xavier_uniform_(self.weight)
+        nn.init.zeros_(self.bias)
+
+    def forward(self, x: torch.Tensor) -> torch.Tensor:
+        return torch.einsum("oi,...id->...od", self.weight, x) + self.bias
+
+
+class _ProbeRotor(CliffordModule):
+    """Core-local dense rotor used only for signature search analysis."""
+
+    def __init__(self, algebra: AlgebraLike, channels: int):
+        super().__init__(algebra)
+        self.channels = channels
+        self.register_buffer("bivector_indices", algebra.grade_indices((2,), device=algebra.device))
+        self.bivector_weights = nn.Parameter(torch.empty(channels, self.bivector_indices.numel()))
+        self.bivector_weights._manifold = "spin"
+        self.reset_parameters()
+
+    def reset_parameters(self) -> None:
+        nn.init.normal_(self.bivector_weights, std=0.01)
+
+    def forward(self, x: torch.Tensor) -> torch.Tensor:
+        if not hasattr(self.algebra, "exp") or not hasattr(self.algebra, "per_channel_sandwich"):
+            raise ValueError(
+                "Signature probes require dense rotor execution; reduce dimensions or request dense kernel."
+            )
+        bivectors = x.new_zeros(self.channels, self.algebra.dim)
+        indices = self.bivector_indices.unsqueeze(0).expand(self.channels, -1)
+        bivectors.scatter_(1, indices, self.bivector_weights.to(dtype=x.dtype))
+        rotor = self.algebra.exp(-0.5 * bivectors)
+        return self.algebra.per_channel_sandwich(rotor, x, self.algebra.reverse(rotor))
+
+    def sparsity_loss(self) -> torch.Tensor:
+        return torch.norm(self.bivector_weights, p=1)
+
+
+class _ProbeBladeSelector(CliffordModule):
+    """Core-local blade gate for metric-search probes."""
+
+    def __init__(self, algebra: AlgebraLike, channels: int):
+        super().__init__(algebra)
+        self.weights = nn.Parameter(torch.ones(channels, algebra.dim))
+
+    def forward(self, x: torch.Tensor) -> torch.Tensor:
+        return x * torch.sigmoid(self.weights).unsqueeze(0)
+
+
 class _SignatureProbe(nn.Module):
     """Minimal single-rotor probe for bivector energy analysis.
 
-    Architecture: CliffordLinear(1, C) -> RotorLayer(C) -> BladeSelector(C).
+    Architecture: channel mixer -> rotor -> channel mixer -> blade selector.
     Only one linear layer for channel expansion; the rotor bivector energy
     is the primary signal for signature discovery.
     """
 
-    def __init__(self, algebra: CliffordAlgebra, channels: int = 4):
+    def __init__(self, algebra: AlgebraLike, channels: int = 4):
         super().__init__()
         self.algebra = algebra
-        self.linear_in = CliffordLinear(algebra, 1, channels)
-        self.rotor = RotorLayer(algebra, channels)
-        self.linear_out = CliffordLinear(algebra, channels, 1)
-        self.selector = BladeSelector(algebra, 1)
+        self.linear_in = _ProbeLinear(algebra, 1, channels)
+        self.rotor = _ProbeRotor(algebra, channels)
+        self.linear_out = _ProbeLinear(algebra, channels, 1)
+        self.selector = _ProbeBladeSelector(algebra, 1)
 
     def forward(self, x: torch.Tensor) -> torch.Tensor:
         x = self.linear_in(x)
@@ -51,35 +108,35 @@ def forward(self, x: torch.Tensor) -> torch.Tensor:
         x = self.selector(x)
         return x
 
-    def get_rotor_layers(self) -> List[RotorLayer]:
-        return [m for m in self.modules() if isinstance(m, RotorLayer)]
+    def get_rotor_layers(self) -> List[_ProbeRotor]:
+        return [m for m in self.modules() if isinstance(m, _ProbeRotor)]
 
 
 def _apply_biased_init(
     probe: _SignatureProbe,
-    algebra: CliffordAlgebra,
+    algebra: AlgebraLike,
     bias_type: str = "random",
 ) -> None:
     """Biases RotorLayer bivector weights based on signature type.
 
-    Uses ``algebra.bv_sq_scalar`` to classify each basis bivector:
+    Uses ``algebra.bivector_squared_signs()`` to classify each basis bivector:
     - bv_sq = -1: elliptic (positive-signature base vectors)
     - bv_sq = +1: hyperbolic (mixed-signature base vectors)
     - bv_sq =  0: null (degenerate base vectors)
     """
-    bv_sq = algebra.bv_sq_scalar
+    bv_sq = algebra.bivector_squared_signs(device=algebra.device, dtype=algebra.dtype)
     ell = CONSTANTS.bv_sq_elliptic_bound
     hyp = CONSTANTS.bv_sq_hyperbolic_bound
     for rotor in probe.get_rotor_layers():
         with torch.no_grad():
             if bias_type == "euclidean":
-                weights = torch.where(bv_sq < ell, torch.tensor(1.0), torch.tensor(0.1))
+                weights = torch.where(bv_sq < ell, torch.ones_like(bv_sq), torch.full_like(bv_sq, 0.1))
                 rotor.bivector_weights.copy_(
                     weights.unsqueeze(0).expand_as(rotor.bivector_weights)
                     + torch.randn_like(rotor.bivector_weights) * 0.05
                 )
             elif bias_type == "minkowski":
-                weights = torch.where(bv_sq.abs() > hyp, torch.tensor(1.0), torch.tensor(0.1))
+                weights = torch.where(bv_sq.abs() > hyp, torch.ones_like(bv_sq), torch.full_like(bv_sq, 0.1))
                 rotor.bivector_weights.copy_(
                     weights.unsqueeze(0).expand_as(rotor.bivector_weights)
                     + torch.randn_like(rotor.bivector_weights) * 0.05
@@ -129,7 +186,7 @@ def __init__(
         self.micro_batch_size = micro_batch_size
         self.early_stop_patience = early_stop_patience
 
-    def _lift_data(self, data: torch.Tensor) -> Tuple[torch.Tensor, CliffordAlgebra]:
+    def _lift_data(self, data: torch.Tensor) -> Tuple[torch.Tensor, AlgebraLike]:
         """Lifts [N, X] data into Cl(X+1, 1, 0) via CGA-style embedding."""
         data = data.to(self.device)
         N, X = data.shape
@@ -137,14 +194,16 @@ def _lift_data(self, data: torch.Tensor) -> Tuple[torch.Tensor, CliffordAlgebra]
         if X + 2 > 12:
             warnings.warn(
                 f"Data dimension {X} yields algebra dim 2^{X + 2}={2 ** (X + 2)}. "
-                f"Consider PCA pre-reduction to X <= 10 for tractable computation."
+                "MetricSearch probes require dense rotor execution; use SignatureSearchAnalyzer PCA "
+                "pre-reduction to X <= 10."
             )
+            raise ValueError("MetricSearch requires X <= 10 so the conformal probe algebra stays within Cl12.")
 
         norm_sq = 0.5 * (data**2).sum(dim=-1, keepdim=True)
         ones = torch.ones(N, 1, device=self.device, dtype=data.dtype)
         lifted = torch.cat([data, norm_sq, ones], dim=-1)
 
-        algebra = CliffordAlgebra(X + 1, 1, 0, device=self.device)
+        algebra = make_algebra(X + 1, 1, 0, kernel="dense", device=self.device)
         mv = algebra.embed_vector(lifted)
         mv = mv.unsqueeze(1)
         return mv, algebra
@@ -152,7 +211,7 @@ def _lift_data(self, data: torch.Tensor) -> Tuple[torch.Tensor, CliffordAlgebra]
     def _train_probe(
         self,
         mv_data: torch.Tensor,
-        algebra: CliffordAlgebra,
+        algebra: AlgebraLike,
         bias_type: str = "random",
     ) -> Dict:
         """Trains a single probe and returns results."""
@@ -217,13 +276,12 @@ def _train_probe(
     def _analyze_bivector_energy(
         self,
         probe: _SignatureProbe,
-        algebra: CliffordAlgebra,
+        algebra: AlgebraLike,
         original_dim: int,
     ) -> Tuple[Tuple[int, int, int], Dict]:
         """Maps learned bivector energy to (p, q, r) signature."""
-        bv_sq = algebra.bv_sq_scalar
-        bv_mask = algebra.grade_masks[2]
-        bv_indices = bv_mask.nonzero(as_tuple=False).squeeze(-1)
+        bv_sq = algebra.bivector_squared_signs(device=self.device, dtype=algebra.dtype)
+        bv_indices = algebra.grade_indices((2,), device=self.device)
 
         total_energy = torch.zeros(len(bv_indices), device=self.device)
         n_layers = 0
diff --git a/core/analysis/spectral.py b/core/analysis/spectral.py
index 4065c37..bc8d45b 100644
--- a/core/analysis/spectral.py
+++ b/core/analysis/spectral.py
@@ -15,9 +15,9 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
-from core.decomposition import differentiable_invariant_decomposition
-from core.metric import hermitian_grade_spectrum
+from core.foundation.module import AlgebraLike
+from core.runtime.decomposition import differentiable_invariant_decomposition
+from core.runtime.metric import hermitian_grade_spectrum
 
 from ._types import CONSTANTS, SpectralResult
 
@@ -35,14 +35,14 @@ class SpectralAnalyzer:
        operator :math:`L_x(y) = x \\cdot y` (only for small algebras).
 
     Args:
-        algebra: :class:`CliffordAlgebra` instance.
+        algebra: algebra kernel or planning context.
         max_simple_components: Maximum number of simple components to
             extract from the mean bivector.
     """
 
     def __init__(
         self,
-        algebra: CliffordAlgebra,
+        algebra: AlgebraLike,
         max_simple_components: int = 5,
     ):
         self.algebra = algebra
diff --git a/core/analysis/symmetry.py b/core/analysis/symmetry.py
index df27776..578ce78 100644
--- a/core/analysis/symmetry.py
+++ b/core/analysis/symmetry.py
@@ -15,7 +15,7 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.foundation.module import AlgebraLike
 
 from ._types import CONSTANTS, CommutatorResult, SymmetryResult
 
@@ -24,14 +24,14 @@ class SymmetryDetector:
     """Detect symmetries, null directions, and invariances.
 
     Args:
-        algebra: :class:`CliffordAlgebra` instance.
+        algebra: algebra kernel or planning context.
         null_threshold: Energy threshold below which a direction is
             considered effectively null.
     """
 
     def __init__(
         self,
-        algebra: CliffordAlgebra,
+        algebra: AlgebraLike,
         null_threshold: float = 0.01,
     ):
         self.algebra = algebra
@@ -90,7 +90,7 @@ def detect_null_directions(self, mv_data: torch.Tensor) -> Tuple[List[int], torc
             *null_indices* lists those with score < threshold.
         """
         n = self.algebra.n
-        g1_idx = (1 << torch.arange(n, device=mv_data.device)).long()
+        g1_idx = self.algebra.grade_indices((1,), device=mv_data.device)
 
         # Energy on each grade-1 component
         g1_coeffs = mv_data[:, g1_idx]  # [N, n]
@@ -141,7 +141,7 @@ def detect_reflection_symmetries(self, mv_data: torch.Tensor) -> List[Dict]:
 
         # Build all n basis vectors: [n, dim]
         basis_vecs = torch.zeros(n, dim, device=mv_data.device, dtype=mv_data.dtype)
-        blade_indices = (1 << torch.arange(n, device=mv_data.device)).long()
+        blade_indices = self.algebra.grade_indices((1,), device=mv_data.device)
         basis_vecs[torch.arange(n, device=mv_data.device), blade_indices] = 1.0
 
         # Batch reflect: [n, N, dim]
@@ -198,16 +198,14 @@ def detect_continuous_symmetries(
         dim = self.algebra.dim
         N = mv_data.shape[0]
 
-        ii, jj = torch.triu_indices(n, n, offset=1)
-        bv_indices = ((1 << ii) | (1 << jj)).tolist()
+        bv_idx_tensor = self.algebra.grade_indices((2,), device=mv_data.device)
 
-        if not bv_indices:
+        if bv_idx_tensor.numel() == 0:
             return 0
 
-        n_bv = len(bv_indices)
+        n_bv = int(bv_idx_tensor.numel())
         # Build all bivector bases: [n_bv, dim]
         bv_bases = torch.zeros(n_bv, dim, device=mv_data.device, dtype=mv_data.dtype)
-        bv_idx_tensor = torch.tensor(bv_indices, dtype=torch.long, device=mv_data.device)
         bv_bases[torch.arange(n_bv, device=mv_data.device), bv_idx_tensor] = 1.0
 
         # Batch commutator: [n_bv, N, dim]
diff --git a/core/config.py b/core/config.py
new file mode 100644
index 0000000..b1d121f
--- /dev/null
+++ b/core/config.py
@@ -0,0 +1,175 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Algebra construction config and dense/planned context dispatch."""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+from typing import Any, Iterable, Literal, Mapping, Optional
+
+import torch
+
+from core.foundation.device import resolve_device, resolve_dtype
+from core.foundation.module import AlgebraLike
+from core.planning.policy import PlanningLimits
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.context import AlgebraContext
+
+AlgebraKernel = Literal["auto", "dense", "context"]
+
+
+@dataclass(frozen=True)
+class AlgebraConfig:
+    """Dense/context algebra declaration."""
+
+    p: int
+    q: int = 0
+    r: int = 0
+    kernel: AlgebraKernel = "auto"
+    dense_threshold: int = 8
+    device: str = "cuda"
+    dtype: torch.dtype = torch.float32
+    exp_policy: str = "balanced"
+    fixed_iterations: Optional[int] = None
+    default_grades: Optional[tuple[int, ...]] = None
+    allow_full_layout_products: Optional[bool] = None
+    planning_limits: Optional[PlanningLimits] = None
+
+    @classmethod
+    def from_mapping(cls, config: Mapping[str, Any], **overrides) -> "AlgebraConfig":
+        """Build an algebra declaration from Hydra/OmegaConf config."""
+        values = {
+            "p": int(_mapping_get(config, "p", 0)),
+            "q": int(_mapping_get(config, "q", 0)),
+            "r": int(_mapping_get(config, "r", 0)),
+            "kernel": _mapping_get(config, "kernel", "auto"),
+            "dense_threshold": int(_mapping_get(config, "dense_threshold", 8)),
+            "device": _mapping_get(config, "device", "cuda"),
+            "dtype": resolve_dtype(_mapping_get(config, "dtype", torch.float32)),
+            "exp_policy": _mapping_get(config, "exp_policy", "balanced"),
+            "fixed_iterations": _optional_int(_mapping_get(config, "fixed_iterations", None)),
+            "default_grades": _optional_grades(_mapping_get(config, "default_grades", None)),
+            "allow_full_layout_products": _optional_bool(_mapping_get(config, "allow_full_layout_products", None)),
+        }
+        values.update({key: value for key, value in overrides.items() if value is not None})
+        values["dtype"] = resolve_dtype(values["dtype"])
+        return cls(**values)
+
+
+def make_algebra(
+    p: int,
+    q: int = 0,
+    r: int = 0,
+    *,
+    kernel: AlgebraKernel = "auto",
+    dense_threshold: int = 8,
+    device="cuda",
+    dtype: torch.dtype = torch.float32,
+    exp_policy: str = "balanced",
+    fixed_iterations: Optional[int] = None,
+    default_grades: Optional[Iterable[int]] = None,
+    allow_full_layout_products: Optional[bool] = None,
+    planning_limits: Optional[PlanningLimits] = None,
+    **deprecated_options,
+) -> AlgebraLike:
+    """Construct a dense low-dimensional algebra or high-dimensional planning context."""
+    kernel = _normalize_kernel(kernel)
+    n = p + q + r
+    if deprecated_options:
+        unknown = ", ".join(sorted(deprecated_options))
+        raise TypeError(f"Unsupported algebra construction option(s): {unknown}")
+
+    selected_kernel = "context" if kernel == "auto" and n > dense_threshold else kernel
+    if selected_kernel == "auto":
+        selected_kernel = "dense"
+
+    resolved_device = resolve_device(device) if str(device) == "auto" else device
+    resolved_dtype = resolve_dtype(dtype)
+
+    if selected_kernel == "dense":
+        return CliffordAlgebra(
+            p,
+            q,
+            r,
+            device=resolved_device,
+            dtype=resolved_dtype,
+            exp_policy=exp_policy,
+            fixed_iterations=fixed_iterations,
+            allow_large_dense=kernel == "dense",
+            planning_limits=planning_limits,
+        )
+
+    return AlgebraContext(
+        p,
+        q,
+        r,
+        device=resolved_device,
+        dtype=resolved_dtype,
+        default_grades=default_grades,
+        allow_full_layout_products=allow_full_layout_products,
+        planning_limits=planning_limits,
+    )
+
+
+def make_algebra_from_config(config: Mapping[str, Any], **overrides) -> AlgebraLike:
+    """Construct an algebra from a Hydra/OmegaConf-compatible config mapping."""
+    algebra_config = AlgebraConfig.from_mapping(config, **overrides)
+    return make_algebra(
+        algebra_config.p,
+        algebra_config.q,
+        algebra_config.r,
+        kernel=algebra_config.kernel,
+        dense_threshold=algebra_config.dense_threshold,
+        device=algebra_config.device,
+        dtype=algebra_config.dtype,
+        exp_policy=algebra_config.exp_policy,
+        fixed_iterations=algebra_config.fixed_iterations,
+        default_grades=algebra_config.default_grades,
+        allow_full_layout_products=algebra_config.allow_full_layout_products,
+        planning_limits=algebra_config.planning_limits,
+    )
+
+
+def _mapping_get(config: Mapping[str, Any], key: str, default):
+    """Return a value from plain mappings or OmegaConf DictConfig objects."""
+    if config is None:
+        return default
+    return config.get(key, default)
+
+
+def _normalize_kernel(kernel: str) -> AlgebraKernel:
+    """Validate and normalize algebra kernel names."""
+    normalized = str(kernel).lower()
+    if normalized not in {"auto", "dense", "context"}:
+        raise ValueError(f"Unknown algebra kernel {kernel!r}; expected 'auto', 'dense', or 'context'")
+    return normalized  # type: ignore[return-value]
+
+
+def _optional_int(value) -> Optional[int]:
+    if value is None:
+        return None
+    return int(value)
+
+
+def _optional_grades(value) -> Optional[tuple[int, ...]]:
+    if value is None:
+        return None
+    return tuple(int(grade) for grade in value)
+
+
+def _optional_bool(value) -> Optional[bool]:
+    if value is None:
+        return None
+    if isinstance(value, str):
+        lowered = value.strip().lower()
+        if lowered in {"true", "1", "yes", "on"}:
+            return True
+        if lowered in {"false", "0", "no", "off"}:
+            return False
+        raise ValueError(f"Cannot parse boolean value {value!r}")
+    return bool(value)
diff --git a/core/foundation/__init__.py b/core/foundation/__init__.py
new file mode 100644
index 0000000..5af000d
--- /dev/null
+++ b/core/foundation/__init__.py
@@ -0,0 +1,71 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Foundation value objects, basis utilities, validation, and device helpers."""
+
+from .basis import (
+    GradeProductOp,
+    basis_count_for_grades,
+    basis_index_tuple_for_grades,
+    basis_indices_for_grades,
+    basis_product,
+    expand_output_grades,
+    geometric_product_output_grades,
+    normalize_grades,
+    operation_coefficient,
+    operation_may_be_nonzero,
+    product_output_grades,
+    reverse_sign,
+)
+from .device import DeviceConfig, dtype_name, optional_dtype, resolve_device, resolve_dtype
+from .layout import AlgebraSpec, GradeLayout
+from .manifold import (
+    MANIFOLD_EUCLIDEAN,
+    MANIFOLD_ORDER,
+    MANIFOLD_SPHERE,
+    MANIFOLD_SPIN,
+    VALID_MANIFOLDS,
+    format_valid_manifolds,
+    tag_manifold,
+    validate_manifold,
+)
+from .module import AlgebraLike, CliffordModule
+from .validation import check_channels, check_multivector
+
+__all__ = [
+    "AlgebraLike",
+    "AlgebraSpec",
+    "CliffordModule",
+    "DeviceConfig",
+    "GradeLayout",
+    "GradeProductOp",
+    "MANIFOLD_EUCLIDEAN",
+    "MANIFOLD_ORDER",
+    "MANIFOLD_SPHERE",
+    "MANIFOLD_SPIN",
+    "VALID_MANIFOLDS",
+    "basis_count_for_grades",
+    "basis_index_tuple_for_grades",
+    "basis_indices_for_grades",
+    "basis_product",
+    "check_channels",
+    "check_multivector",
+    "dtype_name",
+    "expand_output_grades",
+    "format_valid_manifolds",
+    "geometric_product_output_grades",
+    "normalize_grades",
+    "operation_coefficient",
+    "operation_may_be_nonzero",
+    "optional_dtype",
+    "product_output_grades",
+    "reverse_sign",
+    "resolve_device",
+    "resolve_dtype",
+    "tag_manifold",
+    "validate_manifold",
+]
diff --git a/core/foundation/basis.py b/core/foundation/basis.py
new file mode 100644
index 0000000..71a691e
--- /dev/null
+++ b/core/foundation/basis.py
@@ -0,0 +1,223 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Canonical bitmask-basis utilities for Clifford algebra planning."""
+
+from __future__ import annotations
+
+from itertools import combinations
+from math import comb
+from typing import Iterable, Literal, Optional
+
+import torch
+
+GradeProductOp = Literal["gp", "wedge", "inner", "commutator", "anti_commutator"]
+
+# NOTE: Torch-backed executors currently store canonical basis blades as signed
+# int64 bitmasks. That makes n=63 the largest supported dimension: the highest
+# usable basis bit is 1 << 62, while n=64 would require 1 << 63, which is outside
+# torch.long's positive range. Supporting n>=64 requires kernel-level storage
+# engineering, for example compact-position-only kernels, declared blade objects,
+# multi-limb or variable-length bitsets, or another non-int64 basis identifier.
+TORCH_LONG_BASIS_MAX_N = 63
+_TORCH_LONG_MAX = (1 << TORCH_LONG_BASIS_MAX_N) - 1
+
+
+def normalize_grades(grades: Iterable[int], n: int, *, name: str = "grades") -> tuple[int, ...]:
+    """Return sorted unique grades validated against ``0 <= grade <= n``."""
+    normalized = tuple(sorted({int(grade) for grade in grades}))
+    if not normalized:
+        raise ValueError(f"{name} must contain at least one grade")
+    invalid = [grade for grade in normalized if grade < 0 or grade > n]
+    if invalid:
+        raise ValueError(f"{name} contains invalid grades for n={n}: {invalid}")
+    return normalized
+
+
+def basis_index_tuple_for_grades(n: int, grades: Iterable[int]) -> tuple[int, ...]:
+    """Return canonical bitmask basis indices whose popcount is in ``grades``."""
+    indices: list[int] = []
+    for grade in normalize_grades(grades, n):
+        indices.extend(_basis_indices_for_grade(n, grade))
+    return tuple(sorted(indices))
+
+
+def basis_count_for_grades(n: int, grades: Iterable[int]) -> int:
+    """Return the number of basis blades represented by ``grades``."""
+    return sum(comb(n, grade) for grade in normalize_grades(grades, n))
+
+
+def basis_indices_for_grades(n: int, grades: Iterable[int], *, device=None) -> torch.Tensor:
+    """Return canonical bitmask basis indices as a tensor."""
+    return basis_indices_tensor(basis_index_tuple_for_grades(n, grades), n=n, device=device)
+
+
+def basis_indices_tensor(
+    indices: Iterable[int],
+    *,
+    n: Optional[int] = None,
+    role: str = "basis indices",
+    device=None,
+) -> torch.Tensor:
+    """Tensorize canonical basis bitmasks with a clear signed-int64 boundary."""
+    values = tuple(int(index) for index in indices)
+    _validate_torch_long_basis_indices(values, n=n, role=role)
+    return torch.tensor(values, dtype=torch.long, device=device)
+
+
+def _basis_indices_for_grade(n: int, grade: int) -> tuple[int, ...]:
+    if grade == 0:
+        return (0,)
+    if grade == n:
+        return ((1 << n) - 1,)
+    return tuple(sum(1 << bit for bit in bits) for bits in combinations(range(n), grade))
+
+
+def _validate_torch_long_basis_indices(indices: tuple[int, ...], *, n: Optional[int], role: str) -> None:
+    if not indices:
+        return
+    if max(indices) <= _TORCH_LONG_MAX:
+        return
+    dimension = "" if n is None else f" for n={n}"
+    raise ValueError(
+        f"{role}{dimension} cannot be represented as torch.long basis bitmasks. "
+        f"Current Torch-backed executors support bitmask tensorization up to n={TORCH_LONG_BASIS_MAX_N}."
+    )
+
+
+def geometric_product_output_grades(left_grade: int, right_grade: int, n: int) -> tuple[int, ...]:
+    """Return the possible output grades of a homogeneous geometric product."""
+    low = abs(int(left_grade) - int(right_grade))
+    high = min(int(left_grade) + int(right_grade), 2 * n - int(left_grade) - int(right_grade))
+    return tuple(range(low, high + 1, 2))
+
+
+def expand_output_grades(
+    left_grades: Iterable[int],
+    right_grades: Iterable[int],
+    n: int,
+    *,
+    op: GradeProductOp = "gp",
+    project_grades: Optional[Iterable[int]] = None,
+) -> tuple[int, ...]:
+    """Expand input grade sets into output grades required by ``op``."""
+    left = normalize_grades(left_grades, n, name="left_grades")
+    right = normalize_grades(right_grades, n, name="right_grades")
+    if op not in {"gp", "wedge", "inner", "commutator", "anti_commutator"}:
+        raise ValueError(f"Unsupported grade product op {op!r}")
+
+    outputs: set[int] = set()
+    for left_grade in left:
+        for right_grade in right:
+            outputs.update(product_output_grades(left_grade, right_grade, n, op=op))
+
+    if project_grades is not None:
+        outputs &= set(normalize_grades(project_grades, n, name="project_grades"))
+    if not outputs:
+        raise ValueError(
+            f"Grade expansion is empty for op={op!r}, left_grades={left}, right_grades={right}, "
+            f"project_grades={None if project_grades is None else tuple(project_grades)}"
+        )
+    return tuple(sorted(outputs))
+
+
+def product_output_grades(left_grade: int, right_grade: int, n: int, *, op: GradeProductOp = "gp") -> tuple[int, ...]:
+    """Return possible output grades for one homogeneous product route."""
+    left_grade = int(left_grade)
+    right_grade = int(right_grade)
+    if op == "wedge":
+        grade = left_grade + right_grade
+        return (grade,) if grade <= n else ()
+
+    outputs = geometric_product_output_grades(left_grade, right_grade, n)
+    if op == "gp":
+        return outputs
+    if op not in {"inner", "commutator", "anti_commutator"}:
+        raise ValueError(f"Unsupported grade product op {op!r}")
+
+    filtered = []
+    for output_grade in outputs:
+        overlap = (left_grade + right_grade - output_grade) // 2
+        parity_odd = ((left_grade * right_grade - overlap) % 2) == 1
+        if op == "commutator":
+            keep = parity_odd
+        else:
+            keep = not parity_odd
+        if keep:
+            filtered.append(output_grade)
+    return tuple(filtered)
+
+
+def basis_product(index_a: int, index_b: int, p: int, q: int, r: int) -> tuple[int, float]:
+    """Return ``(index, sign)`` for two canonical basis blade products."""
+    n = p + q + r
+    swap_count = 0
+    for bit in range(n):
+        if index_a & (1 << bit):
+            swap_count += (index_b & ((1 << bit) - 1)).bit_count()
+
+    sign = -1.0 if swap_count % 2 else 1.0
+
+    negative_mask = sum(1 << bit for bit in range(p, p + q))
+    if ((index_a & index_b & negative_mask).bit_count() % 2) == 1:
+        sign = -sign
+
+    null_mask = sum(1 << bit for bit in range(p + q, n))
+    if (index_a & index_b & null_mask) != 0:
+        sign = 0.0
+
+    return index_a ^ index_b, sign
+
+
+def reverse_sign(index: int) -> float:
+    """Return the reversion sign for a canonical basis blade."""
+    grade = int(index).bit_count()
+    return -1.0 if ((grade * (grade - 1) // 2) % 2) else 1.0
+
+
+def operation_coefficient(index_a: int, index_b: int, p: int, q: int, r: int, op: GradeProductOp) -> float:
+    """Return the scalar coefficient multiplying ``A_i * B_j`` for ``op``."""
+    if not operation_may_be_nonzero(index_a, index_b, p, q, r, op):
+        return 0.0
+
+    _, sign_ab = basis_product(index_a, index_b, p, q, r)
+    if op in {"gp", "wedge", "inner"}:
+        return sign_ab
+
+    if op == "commutator":
+        return 2.0 * sign_ab
+    if op == "anti_commutator":
+        return 2.0 * sign_ab
+    raise ValueError(f"Unsupported grade product op {op!r}")
+
+
+def operation_may_be_nonzero(index_a: int, index_b: int, p: int, q: int, r: int, op: GradeProductOp) -> bool:
+    """Return whether an operator can have a non-zero coefficient for a basis pair."""
+    overlap_mask = int(index_a) & int(index_b)
+    if op == "wedge":
+        return overlap_mask == 0
+
+    if r > 0:
+        null_mask = ((1 << r) - 1) << (p + q)
+        if overlap_mask & null_mask:
+            return False
+    if op == "gp":
+        return True
+
+    parity_odd = _swap_parity_between_orders(index_a, index_b) == 1
+    if op == "commutator":
+        return parity_odd
+    if op in {"inner", "anti_commutator"}:
+        return not parity_odd
+    raise ValueError(f"Unsupported grade product op {op!r}")
+
+
+def _swap_parity_between_orders(index_a: int, index_b: int) -> int:
+    left_grade = int(index_a).bit_count()
+    right_grade = int(index_b).bit_count()
+    overlap = (int(index_a) & int(index_b)).bit_count()
+    return (left_grade * right_grade - overlap) % 2
diff --git a/core/device.py b/core/foundation/device.py
similarity index 68%
rename from core/device.py
rename to core/foundation/device.py
index bf96f69..b53961b 100644
--- a/core/device.py
+++ b/core/foundation/device.py
@@ -12,11 +12,55 @@
 
 from contextlib import nullcontext
 from dataclasses import dataclass
-from typing import ContextManager
+from typing import Any, ContextManager, Optional
 
 import torch
 import torch.nn as nn
 
+FLOAT_DTYPES: dict[str, torch.dtype] = {
+    "float64": torch.float64,
+    "float32": torch.float32,
+    "bfloat16": torch.bfloat16,
+    "float16": torch.float16,
+}
+
+_DTYPE_ALIASES: dict[str, torch.dtype] = {
+    **FLOAT_DTYPES,
+    "fp64": torch.float64,
+    "double": torch.float64,
+    "fp32": torch.float32,
+    "float": torch.float32,
+    "bf16": torch.bfloat16,
+    "fp16": torch.float16,
+    "half": torch.float16,
+}
+
+
+def optional_dtype(value: Any) -> Optional[torch.dtype]:
+    """Parse a torch dtype declaration, preserving ``None`` as unset."""
+    if value is None or isinstance(value, torch.dtype):
+        return value
+    if isinstance(value, str):
+        normalized = value.strip().lower()
+        if not normalized:
+            return None
+        if normalized in _DTYPE_ALIASES:
+            return _DTYPE_ALIASES[normalized]
+    raise ValueError(f"Unsupported torch dtype declaration: {value!r}")
+
+
+def resolve_dtype(value: Any, default: torch.dtype = torch.float32) -> torch.dtype:
+    """Parse a torch dtype declaration and fall back to ``default`` when unset."""
+    return optional_dtype(value) or default
+
+
+def dtype_name(dtype: torch.dtype) -> str:
+    """Return the canonical short name for a torch dtype."""
+    for name, candidate in FLOAT_DTYPES.items():
+        if candidate == dtype:
+            return name
+    return str(dtype).replace("torch.", "")
+
 
 def resolve_device(device: str = "auto") -> str:
     """Resolve ``'auto'`` to the best available accelerator.
@@ -47,6 +91,8 @@ class DeviceConfig:
             (``aot_eager`` for MPS, ``inductor`` for CUDA/CPU).
             MPS does not fully support the inductor backend.
         amp: Enable automatic mixed precision (CUDA only).
+        amp_dtype: Optional autocast dtype.  ``None`` uses PyTorch's autocast
+            default; explicit values should be ``float16`` or ``bfloat16``.
         cudnn_benchmark: Set :attr:`torch.backends.cudnn.benchmark`.
             ``None`` -> auto (``True`` for CUDA).
     """
@@ -57,10 +103,12 @@ class DeviceConfig:
     compile_model: bool = False
     compile_backend: str | None = None
     amp: bool = False
+    amp_dtype: torch.dtype | str | None = None
     cudnn_benchmark: bool | None = None
 
     def __post_init__(self) -> None:
         self.device = resolve_device(self.device)
+        self.amp_dtype = optional_dtype(self.amp_dtype)
 
         is_cuda = self.device.startswith("cuda")
 
@@ -74,6 +122,8 @@ def __post_init__(self) -> None:
         # AMP only makes sense on CUDA
         if self.amp and not is_cuda:
             self.amp = False
+        if self.amp_dtype is not None and self.amp_dtype not in {torch.float16, torch.bfloat16}:
+            raise ValueError("amp_dtype must be 'float16', 'bfloat16', or null")
 
     # Public helpers
 
@@ -115,23 +165,6 @@ def maybe_compile(self, model: nn.Module) -> nn.Module:
             )
             return model
 
-    @property
-    def dtype(self) -> torch.dtype:
-        """Working dtype for algebra tables and model parameters.
-
-        Returns ``torch.bfloat16`` when AMP is active on CUDA (bfloat16 is
-        native on Ampere+ GPUs and avoids fp16 overflow).  All other
-        backends keep ``torch.float32``.
-
-        Pass this to :class:`~core.algebra.CliffordAlgebra` as ``dtype`` and
-        to :meth:`~tasks.base.BaseTask` model setup so that tables and
-        parameters are created in the correct precision from the start
-        rather than requiring a post-hoc ``.to()`` cast.
-        """
-        if self.amp and self.device.startswith("cuda"):
-            return torch.bfloat16
-        return torch.float32
-
     def get_scaler(self) -> torch.amp.GradScaler | None:
         """Return a :class:`GradScaler` when AMP is active, else ``None``."""
         if not self.amp:
@@ -142,4 +175,6 @@ def autocast_context(self) -> ContextManager:
         """Return an ``autocast`` context manager or :func:`nullcontext`."""
         if not self.amp:
             return nullcontext()
-        return torch.amp.autocast("cuda")
+        if self.amp_dtype is None:
+            return torch.amp.autocast("cuda")
+        return torch.amp.autocast("cuda", dtype=self.amp_dtype)
diff --git a/core/foundation/layout.py b/core/foundation/layout.py
new file mode 100644
index 0000000..254263e
--- /dev/null
+++ b/core/foundation/layout.py
@@ -0,0 +1,173 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Static algebra and compact grade-layout value objects."""
+
+from __future__ import annotations
+
+from dataclasses import dataclass, field
+from typing import Iterable
+
+import torch
+
+from core.foundation.basis import basis_index_tuple_for_grades, basis_indices_tensor, normalize_grades
+
+
+@dataclass(frozen=True)
+class AlgebraSpec:
+    """Immutable Clifford signature metadata used by grade planners."""
+
+    p: int
+    q: int = 0
+    r: int = 0
+
+    def __post_init__(self) -> None:
+        if self.p < 0 or self.q < 0 or self.r < 0:
+            raise ValueError(f"signature counts must be non-negative, got Cl({self.p},{self.q},{self.r})")
+
+    @classmethod
+    def from_algebra(cls, algebra) -> "AlgebraSpec":
+        """Build a spec from any algebra-like object with ``p``, ``q``, and ``r`` attributes."""
+        return cls(int(algebra.p), int(algebra.q), int(algebra.r))
+
+    @property
+    def n(self) -> int:
+        """Number of basis vectors."""
+        return self.p + self.q + self.r
+
+    @property
+    def dim(self) -> int:
+        """Number of canonical basis blades."""
+        return 1 << self.n
+
+    def layout(self, grades: Iterable[int]) -> "GradeLayout":
+        """Return a compact layout for ``grades``."""
+        return GradeLayout(self, normalize_grades(grades, self.n))
+
+
+@dataclass(frozen=True)
+class GradeLayout:
+    """Compact basis-lane layout for a fixed grade set."""
+
+    spec: AlgebraSpec
+    grades: tuple[int, ...]
+    _basis_indices: tuple[int, ...] = field(init=False, repr=False)
+    _indices_cache: dict[str, torch.Tensor] = field(default_factory=dict, init=False, repr=False, compare=False)
+    _grade_index_cache: dict[str, torch.Tensor] = field(default_factory=dict, init=False, repr=False, compare=False)
+    _conversion_cache: dict[tuple[AlgebraSpec, tuple[int, ...], str], tuple[torch.Tensor, torch.Tensor]] = field(
+        default_factory=dict,
+        init=False,
+        repr=False,
+        compare=False,
+    )
+
+    def __post_init__(self) -> None:
+        grades = normalize_grades(self.grades, self.spec.n)
+        object.__setattr__(self, "grades", grades)
+        object.__setattr__(self, "_basis_indices", basis_index_tuple_for_grades(self.spec.n, grades))
+
+    @property
+    def basis_indices(self) -> tuple[int, ...]:
+        """Canonical dense basis indices represented by this compact layout."""
+        return self._basis_indices
+
+    @property
+    def dim(self) -> int:
+        """Number of compact lanes."""
+        return len(self.basis_indices)
+
+    @property
+    def dense_dim(self) -> int:
+        """Full dense multivector dimension."""
+        return self.spec.dim
+
+    def contains_grade(self, grade: int) -> bool:
+        """Return whether ``grade`` is present in this layout."""
+        return int(grade) in self.grades
+
+    def indices_tensor(self, *, device=None) -> torch.Tensor:
+        """Return basis indices as a tensor on ``device``."""
+        key = _device_cache_key(device)
+        cached = self._indices_cache.get(key)
+        if cached is None:
+            cached = basis_indices_tensor(self.basis_indices, n=self.spec.n, role="layout basis indices", device=device)
+            self._indices_cache[key] = cached
+        return cached
+
+    def grade_indices_tensor(self, *, device=None) -> torch.Tensor:
+        """Return per-lane grade ids on ``device``."""
+        key = _device_cache_key(device)
+        cached = self._grade_index_cache.get(key)
+        if cached is None:
+            grades = tuple(index.bit_count() for index in self.basis_indices)
+            cached = torch.tensor(grades, dtype=torch.long, device=device)
+            self._grade_index_cache[key] = cached
+        return cached
+
+    def convert(self, values: torch.Tensor, source: "GradeLayout") -> torch.Tensor:
+        """Convert compact values from ``source`` into this layout.
+
+        Shared basis lanes are copied by canonical basis index. Lanes present in
+        this layout but absent from ``source`` are filled with zeros, which makes
+        the method usable for both projections and sparse layout unions without
+        materializing a full dense multivector.
+        """
+        if source.spec != self.spec:
+            raise ValueError(f"source layout signature {source.spec} does not match target spec {self.spec}")
+        if values.shape[-1] != source.dim:
+            raise ValueError(f"source values last dimension must be {source.dim}, got {values.shape[-1]}")
+        if source == self:
+            return values
+
+        output = values.new_zeros(*values.shape[:-1], self.dim)
+        gather, scatter = self._conversion_tensors(source, device=values.device)
+        if gather.numel() == 0:
+            return output
+
+        copied = torch.index_select(values, -1, gather)
+        return output.index_copy(-1, scatter, copied)
+
+    def compact(self, dense: torch.Tensor) -> torch.Tensor:
+        """Gather compact lanes from a dense multivector tensor."""
+        if dense.shape[-1] != self.dense_dim:
+            raise ValueError(f"dense last dimension must be {self.dense_dim}, got {dense.shape[-1]}")
+        return torch.index_select(dense, -1, self.indices_tensor(device=dense.device))
+
+    def dense(self, values: torch.Tensor) -> torch.Tensor:
+        """Materialize compact lane values into a dense multivector tensor."""
+        if values.shape[-1] != self.dim:
+            raise ValueError(f"values last dimension must be {self.dim}, got {values.shape[-1]}")
+        output = values.new_zeros(*values.shape[:-1], self.dense_dim)
+        return output.index_copy(-1, self.indices_tensor(device=values.device), values)
+
+    def _conversion_tensors(self, source: "GradeLayout", *, device=None) -> tuple[torch.Tensor, torch.Tensor]:
+        key = (source.spec, source.grades, _device_cache_key(device))
+        cached = self._conversion_cache.get(key)
+        if cached is not None:
+            return cached
+
+        source_positions = {index: position for position, index in enumerate(source.basis_indices)}
+        gather_positions: list[int] = []
+        scatter_positions: list[int] = []
+        for target_position, index in enumerate(self.basis_indices):
+            source_position = source_positions.get(index)
+            if source_position is None:
+                continue
+            gather_positions.append(source_position)
+            scatter_positions.append(target_position)
+
+        gather = torch.tensor(gather_positions, dtype=torch.long, device=device)
+        scatter = torch.tensor(scatter_positions, dtype=torch.long, device=device)
+        cached = (gather, scatter)
+        self._conversion_cache[key] = cached
+        return cached
+
+
+def _device_cache_key(device) -> str:
+    if device is None:
+        return "cpu"
+    return str(torch.device(device))
diff --git a/core/foundation/manifold.py b/core/foundation/manifold.py
new file mode 100644
index 0000000..efa0b3d
--- /dev/null
+++ b/core/foundation/manifold.py
@@ -0,0 +1,30 @@
+"""Shared manifold metadata for framework parameters."""
+
+from __future__ import annotations
+
+import torch.nn as nn
+
+MANIFOLD_SPIN = "spin"
+MANIFOLD_SPHERE = "sphere"
+MANIFOLD_EUCLIDEAN = "euclidean"
+
+MANIFOLD_ORDER = (MANIFOLD_SPIN, MANIFOLD_SPHERE, MANIFOLD_EUCLIDEAN)
+VALID_MANIFOLDS = frozenset(MANIFOLD_ORDER)
+
+
+def format_valid_manifolds() -> str:
+    """Return a stable, user-facing list of supported manifold names."""
+    return ", ".join(repr(name) for name in MANIFOLD_ORDER)
+
+
+def validate_manifold(manifold: str) -> str:
+    """Validate and return a manifold tag."""
+    if manifold not in VALID_MANIFOLDS:
+        raise ValueError(f"Unknown manifold {manifold!r}. Must be one of {format_valid_manifolds()}")
+    return manifold
+
+
+def tag_manifold(param: nn.Parameter, manifold: str) -> nn.Parameter:
+    """Tag a parameter with its manifold type and return the parameter."""
+    param._manifold = validate_manifold(manifold)
+    return param
diff --git a/core/foundation/module.py b/core/foundation/module.py
new file mode 100644
index 0000000..8139fc7
--- /dev/null
+++ b/core/foundation/module.py
@@ -0,0 +1,169 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Base PyTorch module for components that share a Clifford algebra."""
+
+from typing import Iterable, Optional, Protocol, runtime_checkable
+
+import torch
+import torch.nn as nn
+
+from core.foundation.layout import GradeLayout
+
+
+@runtime_checkable
+class AlgebraLike(Protocol):
+    """Protocol implemented by dense kernels and planned algebra contexts."""
+
+    p: int
+    q: int
+    r: int
+    n: int
+    dim: int
+    num_grades: int
+    eps: float
+    eps_sq: float
+    planner: object
+    planning_limits: object
+    allow_full_layout_products: bool
+
+    @property
+    def device(self):
+        """Return the device of algebra-owned buffers."""
+        ...
+
+    @property
+    def dtype(self) -> torch.dtype:
+        """Return the dtype of algebra-owned floating-point buffers."""
+        ...
+
+    def _apply(self, fn):
+        """Move/cast algebra-owned buffers."""
+        ...
+
+    def layout(self, grades: Optional[Iterable[int]] = None) -> GradeLayout:
+        """Return a compact grade layout or the algebra's default layout."""
+        ...
+
+    def default_layout(self) -> GradeLayout:
+        """Return the algebra default layout."""
+        ...
+
+    def resolve_layout(
+        self,
+        *,
+        layout: Optional[GradeLayout] = None,
+        grades: Optional[Iterable[int]] = None,
+        mv=None,
+        allow_full: bool = True,
+        warn_full: bool = True,
+    ) -> GradeLayout:
+        """Resolve static layout metadata for tensors or multivectors."""
+        ...
+
+    def grade_indices(self, grades: Iterable[int], *, device=None) -> torch.Tensor:
+        """Return canonical dense basis indices for ``grades``."""
+        ...
+
+    def hermitian_signs(
+        self,
+        layout: Optional[GradeLayout] = None,
+        *,
+        grades: Optional[Iterable[int]] = None,
+        device=None,
+        dtype: Optional[torch.dtype] = None,
+    ) -> torch.Tensor:
+        """Return Hermitian signs for a dense or compact layout."""
+        ...
+
+    def projected_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply a declared grade-restricted binary product."""
+        ...
+
+    def geometric_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply the geometric product."""
+        ...
+
+    def wedge(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply the exterior product."""
+        ...
+
+    def inner_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply the inner product."""
+        ...
+
+    def commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply the commutator product."""
+        ...
+
+    def anti_commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply the anti-commutator product."""
+        ...
+
+    def grade_projection(self, mv: torch.Tensor, grade: int, **kwargs) -> torch.Tensor:
+        """Project to a single grade."""
+        ...
+
+    def reverse(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply reversion."""
+        ...
+
+    def grade_involution(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply grade involution."""
+        ...
+
+    def clifford_conjugation(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply Clifford conjugation."""
+        ...
+
+
+class CliffordModule(nn.Module):
+    """Base module for Clifford algebra-aware components.
+
+    ``CliffordModule`` belongs to :mod:`core` because it is shared by layers,
+    functional losses/activations, models, examples, and experiments. Keeping it
+    out of :mod:`layers` prevents functional code from importing the eager layer
+    package just to subclass this base type.
+
+    The module stores a shared algebra reference without registering it as a
+    PyTorch submodule. In Versor, one algebra instance often owns the
+    precomputed geometric tensors used by many modules.
+    """
+
+    def __init__(self, algebra: AlgebraLike):
+        """Set up the module with a shared algebra instance."""
+        super().__init__()
+        # Bypass nn.Module.__setattr__ to avoid registering algebra as a child.
+        object.__setattr__(self, "_algebra", algebra)
+
+    @property
+    def algebra(self) -> AlgebraLike:
+        """Return the shared algebra instance."""
+        return self._algebra
+
+    @property
+    def p(self):
+        return self._algebra.p
+
+    @property
+    def q(self):
+        return self._algebra.q
+
+    @property
+    def r(self):
+        return self._algebra.r
+
+    def _apply(self, fn):
+        """Apply device/dtype moves to this module and its shared algebra."""
+        result = super()._apply(fn)
+        if self._algebra is not None:
+            self._algebra._apply(fn)
+        return result
+
+    def forward(self, x):
+        """Perform the forward pass computation."""
+        raise NotImplementedError
diff --git a/core/validation.py b/core/foundation/validation.py
similarity index 100%
rename from core/validation.py
rename to core/foundation/validation.py
diff --git a/core/metric.py b/core/metric.py
deleted file mode 100644
index c9c23f9..0000000
--- a/core/metric.py
+++ /dev/null
@@ -1,351 +0,0 @@
-# Versor: Universal Geometric Algebra Neural Network
-# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
-#
-# Licensed under the Apache License, Version 2.0 (the "License");
-# you may not use this file except in compliance with the License.
-#
-
-"""Metric definitions for Clifford algebras.
-
-Provides distances, norms, and inner products that respect
-the metric signature.
-"""
-
-import torch
-
-from core.algebra import CliffordAlgebra
-
-
-def _hermitian_signs(algebra: CliffordAlgebra) -> torch.Tensor:
-    """Return the precomputed Hermitian sign tensor from the algebra.
-
-    The Hermitian inner product on Cl(p,q) is:
-        <A, B>_H = sum_I (conj_sign_I * metric_sign_I) * a_I * b_I
-
-    This is precomputed as ``conj_signs * diagonal(cayley_signs)``
-    and registered as a buffer on the algebra in ``_init_derived_tables()``.
-
-    Returns:
-        Sign tensor [Dim] with values +1, -1, or 0 (null blades).
-    """
-    return algebra._hermitian_signs
-
-
-def inner_product(algebra: CliffordAlgebra, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
-    """Compute the scalar product via projection onto grade 0.
-
-    Computes <A B>_0.
-
-    Args:
-        algebra (CliffordAlgebra): The algebra instance.
-        A (torch.Tensor): First multivector [Batch, Dim].
-        B (torch.Tensor): Second multivector [Batch, Dim].
-
-    Returns:
-        torch.Tensor: Scalar part [Batch, 1].
-    """
-    # Optimized: A * B then extract grade 0
-    prod = algebra.geometric_product(A, B)
-    scalar_part = prod[..., 0:1]  # Grade 0
-    return scalar_part
-
-
-def induced_norm(algebra: CliffordAlgebra, A: torch.Tensor) -> torch.Tensor:
-    """Compute the induced norm respecting the metric signature.
-
-    Computes ||A|| = sqrt(|<A ~A>_0|).
-
-    Args:
-        algebra (CliffordAlgebra): The algebra instance.
-        A (torch.Tensor): Multivector [Batch, Dim].
-
-    Returns:
-        torch.Tensor: Norm [Batch, 1].
-    """
-    A_rev = algebra.reverse(A)
-    # Scalar product <A A~>_0
-    sq_norm = inner_product(algebra, A, A_rev)
-
-    # In mixed signatures, sq_norm can be negative.
-    # We return sqrt(|sq_norm|)
-    return torch.sqrt(torch.abs(sq_norm))
-
-
-def geometric_distance(algebra: CliffordAlgebra, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
-    """Computes geometric distance.
-
-    dist(A, B) = ||A - B||.
-
-    Args:
-        algebra (CliffordAlgebra): The algebra instance.
-        A (torch.Tensor): First multivector.
-        B (torch.Tensor): Second multivector.
-
-    Returns:
-        torch.Tensor: Distance.
-    """
-    diff = A - B
-    return induced_norm(algebra, diff)
-
-
-def grade_purity(algebra: CliffordAlgebra, A: torch.Tensor, grade: int) -> torch.Tensor:
-    """Checks the purity of the grade by examining coefficient energy.
-
-    Purity = ||<A>_k||^2 / ||A||^2.
-
-    Args:
-        algebra (CliffordAlgebra): The algebra instance.
-        A (torch.Tensor): Multivector [..., Dim].
-        grade (int): Target grade.
-
-    Returns:
-        torch.Tensor: Purity score [0, 1].
-    """
-    # Project to grade
-    A_k = algebra.grade_projection(A, grade)
-
-    # Compute energies (using standard squared norm of coefficients for stability)
-    energy_k = (A_k**2).sum(dim=-1)
-    energy_total = (A**2).sum(dim=-1).clamp(min=algebra.eps)
-
-    return energy_k / energy_total
-
-
-def mean_active_grade(algebra: CliffordAlgebra, A: torch.Tensor) -> torch.Tensor:
-    """Average grade. Identifies the grade where the majority of the energy resides.
-
-    Mean Grade = Sum(k * ||<A>_k||^2) / ||A||^2.
-
-    Args:
-        algebra (CliffordAlgebra): The algebra instance.
-        A (torch.Tensor): Multivector.
-
-    Returns:
-        torch.Tensor: Average grade index.
-    """
-    energy_total = (A**2).sum(dim=-1).clamp(min=algebra.eps)
-    weighted_sum = torch.zeros_like(energy_total)
-
-    for k in range(algebra.n + 1):
-        A_k = algebra.grade_projection(A, k)
-        energy_k = (A_k**2).sum(dim=-1)
-        weighted_sum += k * energy_k
-
-    return weighted_sum / energy_total
-
-
-# Hermitian Metrics for Mixed-Signature Algebras
-#
-# In Cl(p,q) with q > 0, the standard norm <A~A>_0 can be negative
-# because basis blades involving negative-signature dimensions square
-# to -1. This breaks gradient-based optimization.
-#
-# The Hermitian inner product uses the algebraically proper formula:
-#
-#   <A, B>_H = <bar{A} B>_0 = Sum_I (conj_sign_I * metric_sign_I) * a_I * b_I
-#
-# where conj_sign_I is the Clifford conjugation sign and metric_sign_I
-# is the basis blade self-product sign. We precompute these signs once
-# via _hermitian_signs(). For Euclidean algebras Cl(p,0), all signs are
-# +1 and this reduces to the simple coefficient inner product.
-#
-# Additionally, we provide the Clifford conjugate (bar involution)
-# and the signature-aware trace form for algebraic computations.
-
-
-def clifford_conjugate(algebra: CliffordAlgebra, mv: torch.Tensor) -> torch.Tensor:
-    """Clifford conjugation (bar involution).
-
-    Combines reversion with grade involution:
-        A_bar_k = (-1)^k * (-1)^{k(k-1)/2} * A_k
-
-    This is the natural *-involution on Cl(p,q). Useful for
-    algebraic computations (e.g., spinor norms, Lipschitz groups).
-
-    Args:
-        algebra: The algebra instance.
-        mv: Multivector [..., Dim].
-
-    Returns:
-        Conjugated multivector [..., Dim].
-    """
-    result = mv.clone()
-    for i in range(algebra.dim):
-        k = bin(i).count("1")
-        sign = ((-1) ** k) * ((-1) ** (k * (k - 1) // 2))
-        if sign == -1:
-            result[..., i] = -result[..., i]
-    return result
-
-
-def hermitian_inner_product(algebra: CliffordAlgebra, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
-    """Hermitian inner product on Cl(p,q): <bar{A} B>_0.
-
-    <A, B>_H = Sum_I (conj_sign_I * metric_sign_I) * a_I * b_I
-
-    Uses precomputed sign arrays so that the result equals the scalar
-    part of the geometric product of the Clifford conjugate of A with B.
-    For Euclidean algebras (q=0), all signs are +1 and this reduces to
-    the simple coefficient inner product Sum a_I b_I.
-
-    Args:
-        algebra: The algebra instance.
-        A: First multivector [..., Dim].
-        B: Second multivector [..., Dim].
-
-    Returns:
-        Scalar inner product [..., 1].
-    """
-    signs = _hermitian_signs(algebra)
-    if signs.dtype != A.dtype:
-        signs = signs.to(dtype=A.dtype)
-    return (signs * A * B).sum(dim=-1, keepdim=True)
-
-
-def hermitian_norm(algebra: CliffordAlgebra, A: torch.Tensor) -> torch.Tensor:
-    """Hermitian norm: ||A||_H = sqrt(|<A, A>_H|).
-
-    Always real and non-negative for any signature.
-    Uses abs() since the signed inner product can produce negative
-    self-products in mixed-signature algebras.
-
-    Args:
-        algebra: The algebra instance.
-        A: Multivector [..., Dim].
-
-    Returns:
-        Norm [..., 1]. Always >= 0.
-    """
-    sq = hermitian_inner_product(algebra, A, A)
-    return torch.sqrt(torch.abs(sq))
-
-
-def hermitian_distance(algebra: CliffordAlgebra, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
-    """Hermitian distance: d_H(A, B) = ||A - B||_H.
-
-    Positive-definite metric distance for any signature.
-    Satisfies: non-negativity, symmetry, triangle inequality, identity.
-
-    Args:
-        algebra: The algebra instance.
-        A: First multivector [..., Dim].
-        B: Second multivector [..., Dim].
-
-    Returns:
-        Distance [..., 1]. Always >= 0.
-    """
-    return hermitian_norm(algebra, A - B)
-
-
-def hermitian_angle(algebra: CliffordAlgebra, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
-    """Hermitian angle between multivectors.
-
-    cos(theta) = <A, B>_H / (||A||_H * ||B||_H)
-
-    Args:
-        algebra: The algebra instance.
-        A: First multivector [..., Dim].
-        B: Second multivector [..., Dim].
-
-    Returns:
-        Angle in radians [..., 1].
-    """
-    signs = _hermitian_signs(algebra)
-    if signs.dtype != A.dtype:
-        signs = signs.to(dtype=A.dtype)
-    ip = (signs * A * B).sum(dim=-1, keepdim=True)
-    sq_a = (signs * A * A).sum(dim=-1, keepdim=True)
-    sq_b = (signs * B * B).sum(dim=-1, keepdim=True)
-    # Use sqrt(sq_a * sq_b) instead of sqrt(sq_a)*sqrt(sq_b) to avoid
-    # float32 precision loss from two separate sqrt operations.
-    denom = torch.sqrt(torch.abs(sq_a) * torch.abs(sq_b)).clamp(min=algebra.eps)
-    cos_theta = ip / denom
-    cos_theta = torch.clamp(cos_theta, -1.0, 1.0)
-    return torch.acos(cos_theta)
-
-
-def grade_hermitian_norm(algebra: CliffordAlgebra, A: torch.Tensor, grade: int) -> torch.Tensor:
-    """Hermitian norm restricted to a single grade.
-
-    ||<A>_k||_H = sqrt(Sum_{I: |I|=k} a_I**2)
-
-    Measures the energy contribution of a specific grade
-    in a signature-independent way.
-
-    Args:
-        algebra: The algebra instance.
-        A: Multivector [..., Dim].
-        grade: Target grade.
-
-    Returns:
-        Grade-specific norm [..., 1].
-    """
-    A_k = algebra.grade_projection(A, grade)
-    return hermitian_norm(algebra, A_k)
-
-
-def hermitian_grade_spectrum(algebra: CliffordAlgebra, A: torch.Tensor) -> torch.Tensor:
-    """Full Hermitian grade spectrum.
-
-    Returns |<A_k, A_k>_H| for each grade k = 0, ..., n.
-    Uses abs() to ensure non-negative values in mixed signatures.
-
-    Args:
-        algebra: The algebra instance.
-        A: Multivector [..., Dim].
-
-    Returns:
-        Grade energies [..., n+1]. Each entry >= 0.
-    """
-    signs = _hermitian_signs(algebra)
-    if signs.dtype != A.dtype:
-        signs = signs.to(dtype=A.dtype)
-    spectrum = []
-    for k in range(algebra.n + 1):
-        A_k = algebra.grade_projection(A, k)
-        sq = (signs * A_k * A_k).sum(dim=-1, keepdim=True)
-        spectrum.append(torch.abs(sq))
-    return torch.cat(spectrum, dim=-1)
-
-
-def signature_trace_form(algebra: CliffordAlgebra, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
-    """Signature-aware trace form: <~A B>_0.
-
-    The standard Clifford algebra scalar product. NOT positive-definite
-    in mixed signatures. Use hermitian_inner_product for optimization.
-
-    This form is signature-aware and useful for:
-    - Rotor normalization (R~R = 1)
-    - Versor validation
-    - Spinor norm computation
-
-    Args:
-        algebra: The algebra instance.
-        A: First multivector [..., Dim].
-        B: Second multivector [..., Dim].
-
-    Returns:
-        Scalar trace form [..., 1]. Can be negative in mixed signatures.
-    """
-    A_rev = algebra.reverse(A)
-    prod = algebra.geometric_product(A_rev, B)
-    return prod[..., 0:1]
-
-
-def signature_norm_squared(algebra: CliffordAlgebra, A: torch.Tensor) -> torch.Tensor:
-    """Signature-aware squared norm: <A~A>_0.
-
-    Can be negative in mixed-signature algebras. Returns the raw value
-    without absolute value, preserving causal structure information.
-
-    For Cl(n,0): always non-negative.
-    For Cl(p,q) with q>0: sign encodes causal character.
-
-    Args:
-        algebra: The algebra instance.
-        A: Multivector [..., Dim].
-
-    Returns:
-        Signed squared norm [..., 1].
-    """
-    return signature_trace_form(algebra, A, A)
diff --git a/core/module.py b/core/module.py
deleted file mode 100644
index 4f28798..0000000
--- a/core/module.py
+++ /dev/null
@@ -1,60 +0,0 @@
-# Versor: Universal Geometric Algebra Neural Network
-# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
-#
-# Licensed under the Apache License, Version 2.0 (the "License");
-# you may not use this file except in compliance with the License.
-#
-
-"""Base PyTorch module for components that share a Clifford algebra."""
-
-import torch.nn as nn
-
-from .algebra import CliffordAlgebra
-
-
-class CliffordModule(nn.Module):
-    """Base module for Clifford algebra-aware components.
-
-    ``CliffordModule`` belongs to :mod:`core` because it is shared by layers,
-    functional losses/activations, models, examples, and experiments. Keeping it
-    out of :mod:`layers` prevents functional code from importing the eager layer
-    package just to subclass this base type.
-
-    The module stores a shared :class:`CliffordAlgebra` reference without
-    registering it as a PyTorch submodule. In Versor, one algebra instance often
-    owns the precomputed geometric tensors used by many modules.
-    """
-
-    def __init__(self, algebra: CliffordAlgebra):
-        """Set up the module with a shared algebra instance."""
-        super().__init__()
-        # Bypass nn.Module.__setattr__ to avoid registering algebra as a child.
-        object.__setattr__(self, "_algebra", algebra)
-
-    @property
-    def algebra(self) -> CliffordAlgebra:
-        """Return the shared algebra instance."""
-        return self._algebra
-
-    @property
-    def p(self):
-        return self._algebra.p
-
-    @property
-    def q(self):
-        return self._algebra.q
-
-    @property
-    def r(self):
-        return self._algebra.r
-
-    def _apply(self, fn):
-        """Apply device/dtype moves to this module and its shared algebra."""
-        result = super()._apply(fn)
-        if self._algebra is not None:
-            self._algebra._apply(fn)
-        return result
-
-    def forward(self, x):
-        """Perform the forward pass computation."""
-        raise NotImplementedError
diff --git a/core/multivector.py b/core/multivector.py
deleted file mode 100644
index 313bd98..0000000
--- a/core/multivector.py
+++ /dev/null
@@ -1,267 +0,0 @@
-# Versor: Universal Geometric Algebra Neural Network
-# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
-#
-# Licensed under the Apache License, Version 2.0 (the "License");
-# you may not use this file except in compliance with the License.
-#
-
-"""Object-oriented multivector wrapper with operator overloading."""
-
-from __future__ import annotations
-
-import torch
-
-from core.algebra import CliffordAlgebra
-
-
-class Multivector:
-    """Object-oriented multivector wrapper with operator overloading.
-
-    Wraps a raw coefficient tensor and its parent ``CliffordAlgebra``,
-    exposing every core algebra operation as a method or Python operator.
-
-    Attributes:
-        algebra (CliffordAlgebra): The backend.
-        tensor (torch.Tensor): The raw data [..., Dim].
-    """
-
-    __slots__ = ("algebra", "tensor")
-
-    def __init__(self, algebra: CliffordAlgebra, tensor: torch.Tensor):
-        self.algebra = algebra
-        self.tensor = tensor
-
-    @classmethod
-    def from_vectors(cls, algebra: CliffordAlgebra, vectors: torch.Tensor) -> Multivector:
-        """Promotes vectors to multivectors (Grade 1)."""
-        return cls(algebra, algebra.embed_vector(vectors))
-
-    @classmethod
-    def scalar(
-        cls, algebra: CliffordAlgebra, value: float | torch.Tensor, batch_shape: tuple[int, ...] = ()
-    ) -> Multivector:
-        """Creates a scalar multivector (grade 0 only)."""
-        dim = 2**algebra.n
-        t = torch.zeros(*batch_shape, dim, device=algebra.device, dtype=algebra.dtype)
-        t[..., 0] = value
-        return cls(algebra, t)
-
-    def __repr__(self):
-        return f"Multivector(shape={self.tensor.shape}, algebra=Cl({self.algebra.p},{self.algebra.q},{self.algebra.r}))"
-
-    def _check_algebra(self, other: Multivector) -> None:
-        s, o = self.algebra, other.algebra
-        if (s.p, s.q, s.r) != (o.p, o.q, o.r):
-            raise ValueError(f"Algebra mismatch: Cl({s.p},{s.q},{s.r}) vs Cl({o.p},{o.q},{o.r})")
-
-    def _wrap(self, tensor: torch.Tensor) -> Multivector:
-        return Multivector(self.algebra, tensor)
-
-    def __add__(self, other):
-        if isinstance(other, Multivector):
-            self._check_algebra(other)
-            return self._wrap(self.tensor + other.tensor)
-        if isinstance(other, (int, float, torch.Tensor)):
-            return self._wrap(self.tensor + other)
-        return NotImplemented
-
-    def __radd__(self, other):
-        return self.__add__(other)
-
-    def __sub__(self, other):
-        if isinstance(other, Multivector):
-            self._check_algebra(other)
-            return self._wrap(self.tensor - other.tensor)
-        if isinstance(other, (int, float, torch.Tensor)):
-            return self._wrap(self.tensor - other)
-        return NotImplemented
-
-    def __rsub__(self, other):
-        if isinstance(other, (int, float, torch.Tensor)):
-            return self._wrap(other - self.tensor)
-        return NotImplemented
-
-    def __neg__(self):
-        return self._wrap(-self.tensor)
-
-    def __mul__(self, other):
-        """Geometric product ``A * B``, or scalar scaling."""
-        if isinstance(other, Multivector):
-            self._check_algebra(other)
-            return self._wrap(self.algebra.geometric_product(self.tensor, other.tensor))
-        if isinstance(other, (int, float)):
-            return self._wrap(self.tensor * other)
-        if isinstance(other, torch.Tensor):
-            return self._wrap(self.tensor * other)
-        return NotImplemented
-
-    def __rmul__(self, other):
-        if isinstance(other, (int, float, torch.Tensor)):
-            return self._wrap(self.tensor * other)
-        return NotImplemented
-
-    def __truediv__(self, other):
-        if isinstance(other, (int, float)):
-            return self._wrap(self.tensor / other)
-        if isinstance(other, torch.Tensor):
-            return self._wrap(self.tensor / other)
-        return NotImplemented
-
-    def __xor__(self, other):
-        """Wedge (outer) product ``A ^ B``."""
-        if isinstance(other, Multivector):
-            self._check_algebra(other)
-            return self._wrap(self.algebra.wedge(self.tensor, other.tensor))
-        return NotImplemented
-
-    def __or__(self, other):
-        """Inner product ``A | B``."""
-        if isinstance(other, Multivector):
-            self._check_algebra(other)
-            return self._wrap(self.algebra.inner_product(self.tensor, other.tensor))
-        return NotImplemented
-
-    def __invert__(self):
-        """Reversion ``~A``."""
-        return self._wrap(self.algebra.reverse(self.tensor))
-
-    def grade(self, k: int) -> Multivector:
-        """Extract the grade-k component."""
-        return self._wrap(self.algebra.grade_projection(self.tensor, k))
-
-    def reverse(self) -> Multivector:
-        """Reversion (same as ``~self``)."""
-        return self._wrap(self.algebra.reverse(self.tensor))
-
-    def grade_involution(self) -> Multivector:
-        """Grade involution (main involution): flips odd-grade signs."""
-        return self._wrap(self.algebra.grade_involution(self.tensor))
-
-    def clifford_conjugation(self) -> Multivector:
-        """Clifford conjugation: grade_involution(reverse(x))."""
-        return self._wrap(self.algebra.clifford_conjugation(self.tensor))
-
-    def dual(self) -> Multivector:
-        """Hodge dual: maps grade-k to grade-(n-k)."""
-        return self._wrap(self.algebra.dual(self.tensor))
-
-    def inverse(self) -> Multivector:
-        """Blade inverse: B^{-1} = ~B / <B~B>_0."""
-        return self._wrap(self.algebra.blade_inverse(self.tensor))
-
-    def geometric_product(self, other: Multivector) -> Multivector:
-        """Explicit geometric product (same as ``self * other``)."""
-        self._check_algebra(other)
-        return self._wrap(self.algebra.geometric_product(self.tensor, other.tensor))
-
-    def wedge(self, other: Multivector) -> Multivector:
-        """Wedge (outer) product (same as ``self ^ other``)."""
-        self._check_algebra(other)
-        return self._wrap(self.algebra.wedge(self.tensor, other.tensor))
-
-    def inner(self, other: Multivector) -> Multivector:
-        """Inner product (same as ``self | other``)."""
-        self._check_algebra(other)
-        return self._wrap(self.algebra.inner_product(self.tensor, other.tensor))
-
-    def left_contraction(self, other: Multivector) -> Multivector:
-        """Left contraction: ``self _| other``."""
-        self._check_algebra(other)
-        return self._wrap(self.algebra.left_contraction(self.tensor, other.tensor))
-
-    def right_contraction(self, other: Multivector) -> Multivector:
-        """Right contraction: ``self |_ other``."""
-        self._check_algebra(other)
-        return self._wrap(self.algebra.right_contraction(self.tensor, other.tensor))
-
-    def commutator(self, other: Multivector) -> Multivector:
-        """Commutator (Lie bracket): ``[self, other] = self*other - other*self``."""
-        self._check_algebra(other)
-        return self._wrap(self.algebra.commutator(self.tensor, other.tensor))
-
-    def anti_commutator(self, other: Multivector) -> Multivector:
-        """Anti-commutator: ``{self, other} = self*other + other*self``."""
-        self._check_algebra(other)
-        return self._wrap(self.algebra.anti_commutator(self.tensor, other.tensor))
-
-    def norm(self) -> torch.Tensor:
-        """Induced metric norm (returns scalar tensor)."""
-        from core.metric import induced_norm
-
-        return induced_norm(self.algebra, self.tensor)
-
-    def norm_sq(self) -> torch.Tensor:
-        """Squared norm: <x * ~x>_0 (returns scalar tensor)."""
-        return self.algebra.norm_sq(self.tensor)
-
-    def get_grade_norms(self) -> torch.Tensor:
-        """Per-grade L2 norms."""
-        return self.algebra.get_grade_norms(self.tensor)
-
-    def exp(self) -> Multivector:
-        """Exponential map (bivector -> rotor)."""
-        return self._wrap(self.algebra.exp(self.tensor))
-
-    def sandwich(self, x: Multivector) -> Multivector:
-        """Sandwich product: ``self * x * ~self``.
-
-        Falls back to two geometric products when the tensor shapes
-        don't match the optimized [N, D] + [N, C, D] layout.
-        """
-        self._check_algebra(x)
-        R, xt = self.tensor, x.tensor
-        # Optimized path: R is [N, D], x is [N, C, D]
-        if R.dim() == 2 and xt.dim() == 3:
-            return self._wrap(self.algebra.sandwich_product(R, xt))
-        # General fallback: two GPs
-        R_rev = self.algebra.reverse(R)
-        return self._wrap(self.algebra.geometric_product(self.algebra.geometric_product(R, xt), R_rev))
-
-    def reflect(self, n: Multivector) -> Multivector:
-        """Reflect self through hyperplane orthogonal to vector n."""
-        self._check_algebra(n)
-        return self._wrap(self.algebra.reflect(self.tensor, n.tensor))
-
-    def versor_product(self, x: Multivector) -> Multivector:
-        """General versor action: ``hat(self) * x * self^{-1}``."""
-        self._check_algebra(x)
-        return self._wrap(self.algebra.versor_product(self.tensor, x.tensor))
-
-    def blade_project(self, blade: Multivector) -> Multivector:
-        """Project onto blade subspace: ``(self · B) B^{-1}``."""
-        self._check_algebra(blade)
-        return self._wrap(self.algebra.blade_project(self.tensor, blade.tensor))
-
-    def blade_reject(self, blade: Multivector) -> Multivector:
-        """Reject from blade subspace: ``self - proj_B(self)``."""
-        self._check_algebra(blade)
-        return self._wrap(self.algebra.blade_reject(self.tensor, blade.tensor))
-
-    def to(self, *args, **kwargs) -> Multivector:
-        """Move/cast the underlying tensor (same API as ``torch.Tensor.to``)."""
-        return self._wrap(self.tensor.to(*args, **kwargs))
-
-    def detach(self) -> Multivector:
-        """Detach from computation graph."""
-        return self._wrap(self.tensor.detach())
-
-    def clone(self) -> Multivector:
-        """Clone the underlying tensor."""
-        return self._wrap(self.tensor.clone())
-
-    def requires_grad_(self, requires_grad: bool = True) -> Multivector:
-        """Set requires_grad in-place."""
-        self.tensor.requires_grad_(requires_grad)
-        return self
-
-    @property
-    def shape(self) -> torch.Size:
-        return self.tensor.shape
-
-    @property
-    def device(self) -> torch.device:
-        return self.tensor.device
-
-    @property
-    def dtype(self) -> torch.dtype:
-        return self.tensor.dtype
diff --git a/core/planning/__init__.py b/core/planning/__init__.py
new file mode 100644
index 0000000..570aa81
--- /dev/null
+++ b/core/planning/__init__.py
@@ -0,0 +1,37 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Static grade planning and Torch executor lowering."""
+
+from .flow import GradeFlow
+from .layouts import ProductRequest, build_product_request
+from .planner import GradePlanner
+from .policy import DEFAULT_PLANNING_LIMITS, PlanCost, PlanningLimits
+from .product import GradeProductExecutor, GradeProductPlan, build_grade_product_plan
+from .tree import GradePathNode, GradePlanTree, build_grade_plan_tree
+from .unary import GradeUnaryExecutor, GradeUnaryOp, GradeUnaryPlan, UnaryRequest, build_unary_request
+
+__all__ = [
+    "GradeFlow",
+    "GradePathNode",
+    "GradeProductExecutor",
+    "GradeProductPlan",
+    "GradePlanTree",
+    "GradePlanner",
+    "PlanningLimits",
+    "PlanCost",
+    "DEFAULT_PLANNING_LIMITS",
+    "GradeUnaryExecutor",
+    "GradeUnaryOp",
+    "GradeUnaryPlan",
+    "ProductRequest",
+    "UnaryRequest",
+    "build_grade_product_plan",
+    "build_grade_plan_tree",
+    "build_product_request",
+    "build_unary_request",
+]
diff --git a/core/planning/flow.py b/core/planning/flow.py
new file mode 100644
index 0000000..e32c82b
--- /dev/null
+++ b/core/planning/flow.py
@@ -0,0 +1,97 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Grade-flow metadata for AoT layout propagation."""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+from typing import Iterable, Optional
+
+from core.foundation.basis import GradeProductOp, expand_output_grades, normalize_grades
+from core.foundation.layout import AlgebraSpec, GradeLayout
+
+
+@dataclass(frozen=True)
+class GradeFlow:
+    """Static grade/layout metadata passed between planned operations."""
+
+    spec: AlgebraSpec
+    layout: GradeLayout
+
+    @classmethod
+    def from_grades(cls, spec: AlgebraSpec, grades: Iterable[int]) -> "GradeFlow":
+        """Create a flow from active grades."""
+        return cls(spec=spec, layout=spec.layout(grades))
+
+    @classmethod
+    def full(cls, spec: AlgebraSpec) -> "GradeFlow":
+        """Create a full-layout flow."""
+        return cls.from_grades(spec, range(spec.n + 1))
+
+    @classmethod
+    def scalar(cls, spec: AlgebraSpec) -> "GradeFlow":
+        """Create a scalar-only flow."""
+        return cls.from_grades(spec, (0,))
+
+    @classmethod
+    def vector(cls, spec: AlgebraSpec) -> "GradeFlow":
+        """Create a vector-only flow."""
+        return cls.from_grades(spec, (1,))
+
+    @property
+    def grades(self) -> tuple[int, ...]:
+        """Active grades represented by this flow."""
+        return self.layout.grades
+
+    @property
+    def dim(self) -> int:
+        """Compact lane count represented by this flow."""
+        return self.layout.dim
+
+    def project(self, grades: Iterable[int]) -> "GradeFlow":
+        """Narrow the flow to a subset of active grades."""
+        projected = normalize_grades(grades, self.spec.n, name="grades")
+        missing = tuple(grade for grade in projected if grade not in self.grades)
+        if missing:
+            raise ValueError(f"Cannot project missing grades {missing} from active grades {self.grades}")
+        return GradeFlow.from_grades(self.spec, projected)
+
+    def unary(self, op: str, output_grades: Optional[Iterable[int]] = None) -> "GradeFlow":
+        """Propagate flow through a unary operation."""
+        if output_grades is not None:
+            return self.project(output_grades)
+        if op in {"identity", "reverse", "grade_involution", "clifford_conjugation"}:
+            return self
+        if op == "grade_projection":
+            raise ValueError("grade_projection requires output_grades")
+        raise ValueError(f"Unsupported unary flow op {op!r}")
+
+    def product(
+        self,
+        other: "GradeFlow",
+        *,
+        op: GradeProductOp = "gp",
+        output_grades: Optional[Iterable[int]] = None,
+    ) -> "GradeFlow":
+        """Propagate flow through a bilinear grade product."""
+        self._check_spec(other)
+        grades = (
+            expand_output_grades(self.grades, other.grades, self.spec.n, op=op)
+            if output_grades is None
+            else normalize_grades(output_grades, self.spec.n, name="output_grades")
+        )
+        return GradeFlow.from_grades(self.spec, grades)
+
+    def merge(self, other: "GradeFlow") -> "GradeFlow":
+        """Union two flows, e.g. for addition or concatenation."""
+        self._check_spec(other)
+        return GradeFlow.from_grades(self.spec, (*self.grades, *other.grades))
+
+    def _check_spec(self, other: "GradeFlow") -> None:
+        if self.spec != other.spec:
+            raise ValueError(f"GradeFlow spec mismatch: {self.spec} vs {other.spec}")
diff --git a/core/planning/layouts.py b/core/planning/layouts.py
new file mode 100644
index 0000000..bcfc364
--- /dev/null
+++ b/core/planning/layouts.py
@@ -0,0 +1,221 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Layout and request normalization for static grade planning."""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+from typing import Optional
+
+import torch
+
+from core.foundation.basis import GradeProductOp, expand_output_grades, normalize_grades
+from core.foundation.layout import AlgebraSpec, GradeLayout
+
+_VALID_PRODUCT_OPS = {"gp", "wedge", "inner", "commutator", "anti_commutator"}
+
+
+@dataclass(frozen=True)
+class ProductRequest:
+    """Fully resolved static request for one bilinear product.
+
+    The request is the planner's intermediate representation. It removes
+    ambiguity from caller input before any executor is built: layouts are
+    normalized, compact-vs-dense operand storage is known, and output grades are
+    inferred when callers do not explicitly project them.
+    """
+
+    spec: AlgebraSpec
+    op: GradeProductOp
+    left_layout: GradeLayout
+    right_layout: GradeLayout
+    output_layout: GradeLayout
+    left_compact: bool
+    right_compact: bool
+    dtype: torch.dtype
+    device: torch.device
+
+    @property
+    def left_grades(self) -> tuple[int, ...]:
+        return self.left_layout.grades
+
+    @property
+    def right_grades(self) -> tuple[int, ...]:
+        return self.right_layout.grades
+
+    @property
+    def output_grades(self) -> tuple[int, ...]:
+        return self.output_layout.grades
+
+    @property
+    def cache_key(self) -> tuple[object, ...]:
+        """Stable key for executor caching."""
+        return (
+            self.spec,
+            str(self.device),
+            str(self.dtype),
+            self.op,
+            self.left_grades,
+            self.right_grades,
+            self.output_grades,
+        )
+
+
+def build_product_request(
+    spec: AlgebraSpec,
+    left: torch.Tensor,
+    right: torch.Tensor,
+    *,
+    op: str = "gp",
+    left_grades=None,
+    right_grades=None,
+    output_grades=None,
+    left_layout: Optional[GradeLayout] = None,
+    right_layout: Optional[GradeLayout] = None,
+    output_layout: Optional[GradeLayout] = None,
+    left_compact: bool = False,
+    right_compact: bool = False,
+    full_layout_allowed: bool = True,
+) -> ProductRequest:
+    """Resolve caller input into a static product request."""
+    normalized_op = normalize_product_op(op)
+    left_layout = resolve_operand_layout(
+        spec,
+        left,
+        grades=left_grades,
+        layout=left_layout,
+        compact=left_compact,
+        side="left",
+        full_layout_allowed=full_layout_allowed,
+    )
+    right_layout = resolve_operand_layout(
+        spec,
+        right,
+        grades=right_grades,
+        layout=right_layout,
+        compact=right_compact,
+        side="right",
+        full_layout_allowed=full_layout_allowed,
+    )
+    output_layout = resolve_output_layout(
+        spec,
+        op=normalized_op,
+        left_layout=left_layout,
+        right_layout=right_layout,
+        output_grades=output_grades,
+        output_layout=output_layout,
+    )
+
+    left_compact = left_compact or is_compact_tensor(spec, left, left_layout)
+    right_compact = right_compact or is_compact_tensor(spec, right, right_layout)
+
+    return ProductRequest(
+        spec=spec,
+        op=normalized_op,
+        left_layout=left_layout,
+        right_layout=right_layout,
+        output_layout=output_layout,
+        left_compact=left_compact,
+        right_compact=right_compact,
+        dtype=torch.promote_types(left.dtype, right.dtype),
+        device=left.device,
+    )
+
+
+def normalize_product_op(op: str) -> GradeProductOp:
+    """Validate and normalize a product operation name."""
+    normalized = str(op)
+    if normalized not in _VALID_PRODUCT_OPS:
+        raise ValueError(f"Unsupported grade product op {op!r}")
+    return normalized  # type: ignore[return-value]
+
+
+def resolve_operand_layout(
+    spec: AlgebraSpec,
+    tensor: torch.Tensor,
+    *,
+    grades=None,
+    layout: Optional[GradeLayout] = None,
+    compact: bool = False,
+    side: str,
+    full_layout_allowed: bool = True,
+) -> GradeLayout:
+    """Resolve one operand's grade layout from explicit metadata or tensor shape."""
+    if layout is not None:
+        check_layout_spec(spec, layout, f"{side}_layout")
+        if grades is not None and layout.grades != normalize_grades(grades, spec.n, name=f"{side}_grades"):
+            raise ValueError(f"{side}_layout and {side}_grades disagree")
+        _check_operand_shape(spec, tensor, layout, compact=compact, side=side)
+        return layout
+
+    if grades is not None:
+        layout = spec.layout(grades)
+        _check_operand_shape(spec, tensor, layout, compact=compact, side=side)
+        return layout
+
+    if compact:
+        raise ValueError(f"{side}_layout or {side}_grades is required for compact {side} input")
+    if tensor.shape[-1] != spec.dim:
+        raise ValueError(
+            f"{side} input has last dimension {tensor.shape[-1]}; declare {side}_layout or "
+            f"{side}_grades for compact planned execution"
+        )
+    if not full_layout_allowed:
+        raise ValueError(
+            f"{side} input would require a full Cl({spec.p},{spec.q},{spec.r}) layout. "
+            "Declare active grades or enable an explicit low-dimensional full-layout fallback."
+        )
+    return spec.layout(range(spec.n + 1))
+
+
+def resolve_output_layout(
+    spec: AlgebraSpec,
+    *,
+    op: GradeProductOp,
+    left_layout: GradeLayout,
+    right_layout: GradeLayout,
+    output_grades=None,
+    output_layout: Optional[GradeLayout] = None,
+) -> GradeLayout:
+    """Resolve the output layout for a product request."""
+    if output_layout is not None:
+        check_layout_spec(spec, output_layout, "output_layout")
+        if output_grades is not None and output_layout.grades != normalize_grades(
+            output_grades, spec.n, name="output_grades"
+        ):
+            raise ValueError("output_layout and output_grades disagree")
+        return output_layout
+
+    if output_grades is None:
+        output_grades = expand_output_grades(left_layout.grades, right_layout.grades, spec.n, op=op)
+    return spec.layout(output_grades)
+
+
+def check_layout_spec(spec: AlgebraSpec, layout: GradeLayout, name: str) -> None:
+    """Validate that a layout belongs to ``spec``."""
+    if layout.spec != spec:
+        raise ValueError(f"{name} signature {layout.spec} does not match product spec {spec}")
+
+
+def is_compact_tensor(spec: AlgebraSpec, tensor: torch.Tensor, layout: GradeLayout) -> bool:
+    """Return whether ``tensor`` already uses ``layout``'s compact lane count."""
+    return layout.dim != spec.dim and tensor.shape[-1] == layout.dim
+
+
+def _check_operand_shape(
+    spec: AlgebraSpec,
+    tensor: torch.Tensor,
+    layout: GradeLayout,
+    *,
+    compact: bool,
+    side: str,
+) -> None:
+    expected = layout.dim if compact or is_compact_tensor(spec, tensor, layout) else spec.dim
+    if tensor.shape[-1] != expected:
+        storage = "compact" if expected == layout.dim else "dense"
+        raise ValueError(f"{side} {storage} last dimension must be {expected}, got {tensor.shape[-1]}")
diff --git a/core/planning/planner.py b/core/planning/planner.py
new file mode 100644
index 0000000..978111b
--- /dev/null
+++ b/core/planning/planner.py
@@ -0,0 +1,335 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Grade-aware planner from algebraic intent to static executors."""
+
+from __future__ import annotations
+
+import torch
+
+from core.foundation.basis import operation_coefficient
+from core.foundation.layout import AlgebraSpec, GradeLayout
+from core.planning.layouts import ProductRequest, build_product_request, normalize_product_op
+from core.planning.policy import (
+    full_layout_allowed,
+    validate_grades_cost,
+    validate_layout_cost,
+    validate_product_grades_cost,
+    validate_product_request,
+    validate_unary_request,
+    warn_full_layout_fallback,
+)
+from core.planning.product import GradeProductExecutor, build_grade_product_plan_from_request
+from core.planning.tree import build_grade_plan_tree
+from core.planning.unary import (
+    GradeUnaryExecutor,
+    UnaryRequest,
+    build_unary_plan_from_request,
+    build_unary_request,
+    normalize_unary_op,
+)
+
+
+class GradePlanner:
+    """Owns layout and product-plan lowering for one algebra instance.
+
+    The planner is deliberately not an ``nn.Module``. It builds static
+    executor modules keyed by signature, grades, dtype, and device, while the
+    algebra remains the source of truth for buffers and dense reference paths.
+    """
+
+    def __init__(self, algebra):
+        self.algebra = algebra
+        self.spec = AlgebraSpec.from_algebra(algebra)
+        self._product_executors = {}
+        self._unary_executors = {}
+
+    def layout(self, grades):
+        """Return the compact layout for ``grades``."""
+        return self.spec.layout(validate_grades_cost(self.algebra, self.spec, grades))
+
+    def full_layout(self) -> GradeLayout:
+        """Return the full dense basis layout."""
+        return self.layout(range(self.spec.n + 1))
+
+    def grade_indices(self, grades, *, device=None) -> torch.Tensor:
+        """Return canonical dense basis indices for ``grades``."""
+        if device is None:
+            device = getattr(self.algebra, "device", None)
+        return self.layout(grades).indices_tensor(device=device)
+
+    def convert_values(self, values: torch.Tensor, *, source_layout: GradeLayout, target_layout: GradeLayout):
+        """Convert compact values between layouts without full dense materialization."""
+        return target_layout.convert(values, source_layout)
+
+    def bivector_squared_signs(self, *, device=None, dtype: torch.dtype = None) -> torch.Tensor:
+        """Return ``(e_ab)^2`` signs in canonical grade-2 layout order."""
+        if device is None:
+            device = getattr(self.algebra, "device", None)
+        if dtype is None:
+            dtype = getattr(self.algebra, "dtype", torch.float32)
+        layout = self.layout((2,))
+        signs = [
+            operation_coefficient(index, index, self.spec.p, self.spec.q, self.spec.r, "gp")
+            for index in layout.basis_indices
+        ]
+        return torch.tensor(signs, dtype=dtype, device=device)
+
+    def clear_cache(self) -> None:
+        """Drop cached executor modules."""
+        self._product_executors.clear()
+        self._unary_executors.clear()
+
+    def _apply(self, fn):
+        """Apply a PyTorch module-style transform to cached executor buffers."""
+        product_executors = list(self._product_executors.values())
+        self._product_executors.clear()
+        for executor in product_executors:
+            executor._apply(fn)
+            self._product_executors[self._product_cache_key(executor)] = executor
+
+        unary_executors = list(self._unary_executors.values())
+        self._unary_executors.clear()
+        for executor in unary_executors:
+            executor._apply(fn)
+            self._unary_executors[self._unary_cache_key(executor)] = executor
+        return self
+
+    def product_executor(
+        self,
+        *,
+        op: str,
+        left_grades,
+        right_grades,
+        output_grades,
+        dtype,
+        device,
+        cache: bool = True,
+    ):
+        """Return a cached static executor for a projected bilinear product."""
+        left_grades, right_grades, output_grades = validate_product_grades_cost(
+            self.algebra,
+            self.spec,
+            op=op,
+            left_grades=left_grades,
+            right_grades=right_grades,
+            output_grades=output_grades,
+        )
+        request = ProductRequest(
+            spec=self.spec,
+            op=normalize_product_op(op),
+            left_layout=self.layout(left_grades),
+            right_layout=self.layout(right_grades),
+            output_layout=self.layout(output_grades),
+            left_compact=False,
+            right_compact=False,
+            dtype=dtype,
+            device=torch.device(device),
+        )
+        return self.product_executor_for_request(request, cache=cache)
+
+    def product_request(
+        self,
+        left: torch.Tensor,
+        right: torch.Tensor,
+        *,
+        op: str = "gp",
+        left_grades=None,
+        right_grades=None,
+        output_grades=None,
+        left_layout: GradeLayout = None,
+        right_layout: GradeLayout = None,
+        output_layout: GradeLayout = None,
+        left_compact: bool = False,
+        right_compact: bool = False,
+    ) -> ProductRequest:
+        """Normalize product intent into a static request without executing tensors."""
+        left_grades = self._default_operand_grades(left_grades, left_layout)
+        right_grades = self._default_operand_grades(right_grades, right_layout)
+        if self._implicit_full_operand(left, grades=left_grades, layout=left_layout, compact=left_compact) or (
+            self._implicit_full_operand(right, grades=right_grades, layout=right_layout, compact=right_compact)
+        ):
+            warn_full_layout_fallback(self.algebra)
+        self._validate_product_grade_cost_before_layouts(
+            op=op,
+            left_grades=left_grades,
+            right_grades=right_grades,
+            output_grades=output_grades,
+            left_layout=left_layout,
+            right_layout=right_layout,
+            output_layout=output_layout,
+        )
+        request = build_product_request(
+            self.spec,
+            left,
+            right,
+            op=op,
+            left_grades=left_grades,
+            right_grades=right_grades,
+            output_grades=output_grades,
+            left_layout=left_layout,
+            right_layout=right_layout,
+            output_layout=output_layout,
+            left_compact=left_compact,
+            right_compact=right_compact,
+            full_layout_allowed=self._full_layout_allowed(),
+        )
+        validate_product_request(self.algebra, request)
+        return request
+
+    def product_executor_for_request(self, request: ProductRequest, *, cache: bool = True) -> GradeProductExecutor:
+        """Return an executor for an already normalized product request."""
+        validate_product_request(self.algebra, request)
+        key = request.cache_key
+        executor = self._product_executors.get(key) if cache else None
+        if executor is None:
+            plan = build_grade_product_plan_from_request(request)
+            executor = GradeProductExecutor(plan)
+            if cache:
+                self._product_executors[key] = executor
+        return executor
+
+    def product_tree(self, *, op: str, left_grades, right_grades, output_grades=None):
+        """Return planner-only grade tree metadata for a product route."""
+        return build_grade_plan_tree(
+            self.spec,
+            op=op,
+            left_grades=left_grades,
+            right_grades=right_grades,
+            output_grades=output_grades,
+        )
+
+    def unary_request(
+        self,
+        values: torch.Tensor,
+        *,
+        op: str,
+        input_grades=None,
+        output_grades=None,
+        input_layout: GradeLayout = None,
+        output_layout: GradeLayout = None,
+        input_compact: bool = False,
+    ) -> UnaryRequest:
+        """Normalize unary intent into a static request without executing tensors."""
+        if not (op == "grade_projection" and output_grades is not None):
+            input_grades = self._default_operand_grades(input_grades, input_layout)
+        if self._implicit_full_operand(values, grades=input_grades, layout=input_layout, compact=input_compact):
+            warn_full_layout_fallback(self.algebra)
+        request = build_unary_request(
+            self.spec,
+            values,
+            op=op,
+            input_grades=input_grades,
+            output_grades=output_grades,
+            input_layout=input_layout,
+            output_layout=output_layout,
+            input_compact=input_compact,
+            full_layout_allowed=self._full_layout_allowed(),
+        )
+        validate_unary_request(self.algebra, request)
+        return request
+
+    def unary_executor(
+        self,
+        *,
+        op: str,
+        input_grades,
+        output_grades=None,
+        dtype,
+        device,
+        cache: bool = True,
+    ) -> GradeUnaryExecutor:
+        """Return a cached static executor for a unary operation."""
+        op = normalize_unary_op(op)
+        if op == "grade_projection" and output_grades is None:
+            raise ValueError("output_grades is required for grade_projection")
+        input_layout = self.layout(input_grades)
+        output_layout = input_layout if output_grades is None else self.layout(output_grades)
+        request = UnaryRequest(
+            spec=self.spec,
+            op=op,
+            input_layout=input_layout,
+            output_layout=output_layout,
+            input_compact=False,
+            dtype=dtype,
+            device=torch.device(device),
+        )
+        return self.unary_executor_for_request(request, cache=cache)
+
+    def unary_executor_for_request(self, request: UnaryRequest, *, cache: bool = True) -> GradeUnaryExecutor:
+        """Return an executor for an already normalized unary request."""
+        key = request.cache_key
+        executor = self._unary_executors.get(key) if cache else None
+        if executor is None:
+            plan = build_unary_plan_from_request(request)
+            executor = GradeUnaryExecutor(plan)
+            if cache:
+                self._unary_executors[key] = executor
+        return executor
+
+    def _product_cache_key(self, executor: GradeProductExecutor) -> tuple[object, ...]:
+        return (
+            self.spec,
+            str(executor.coefficients.device),
+            str(executor.coefficients.dtype),
+            executor.op,
+            executor.left_grades,
+            executor.right_grades,
+            executor.output_grades,
+        )
+
+    def _unary_cache_key(self, executor: GradeUnaryExecutor) -> tuple[object, ...]:
+        return (
+            self.spec,
+            str(executor.signs.device),
+            str(executor.signs.dtype),
+            executor.op,
+            executor.input_layout.grades,
+            executor.output_layout.grades,
+        )
+
+    def _full_layout_allowed(self) -> bool:
+        return full_layout_allowed(self.algebra, self.spec)
+
+    def _implicit_full_operand(self, tensor: torch.Tensor, *, grades, layout, compact: bool) -> bool:
+        return (
+            grades is None
+            and layout is None
+            and not compact
+            and self._full_layout_allowed()
+            and tensor.shape[-1] == self.spec.dim
+        )
+
+    def _default_operand_grades(self, grades, layout: GradeLayout = None):
+        if grades is not None or layout is not None:
+            return grades
+        return getattr(self.algebra, "_default_grades", None)
+
+    def _validate_product_grade_cost_before_layouts(
+        self,
+        *,
+        op: str,
+        left_grades,
+        right_grades,
+        output_grades,
+        left_layout: GradeLayout = None,
+        right_layout: GradeLayout = None,
+        output_layout: GradeLayout = None,
+    ) -> None:
+        left = left_layout.grades if left_layout is not None else left_grades
+        right = right_layout.grades if right_layout is not None else right_grades
+        if left is None or right is None:
+            return
+        output = output_layout.grades if output_layout is not None else output_grades
+        validate_product_grades_cost(
+            self.algebra,
+            self.spec,
+            op=op,
+            left_grades=left,
+            right_grades=right,
+            output_grades=output,
+        )
diff --git a/core/planning/policy.py b/core/planning/policy.py
new file mode 100644
index 0000000..fb49967
--- /dev/null
+++ b/core/planning/policy.py
@@ -0,0 +1,233 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Static planning policy for layout fallback and plan-cost diagnostics."""
+
+from __future__ import annotations
+
+import warnings
+from dataclasses import dataclass
+from typing import Optional
+
+from core.foundation.basis import basis_count_for_grades, expand_output_grades, normalize_grades
+from core.foundation.layout import AlgebraSpec, GradeLayout
+
+FULL_LAYOUT_WARN_N = 8
+FULL_LAYOUT_MAX_N = 12
+
+
+@dataclass(frozen=True)
+class PlanningLimits:
+    """Static cost limits for planned grade execution.
+
+    ``max_lanes`` protects compact tensor width. ``max_pairs`` protects the
+    gather/reduce interaction count generated by a bilinear product plan.
+    """
+
+    warn_lanes: int = 2048
+    max_lanes: int = 4096
+    warn_pairs: int = 1_000_000
+    max_pairs: int = 8_000_000
+
+
+@dataclass(frozen=True)
+class PlanCost:
+    """Static cost summary for a layout or planned operation."""
+
+    spec: AlgebraSpec
+    kind: str
+    op: str
+    left_lanes: int = 0
+    right_lanes: int = 0
+    output_lanes: int = 0
+    pair_count: int = 0
+    input_grades: tuple[int, ...] = ()
+    left_grades: tuple[int, ...] = ()
+    right_grades: tuple[int, ...] = ()
+    output_grades: tuple[int, ...] = ()
+
+    @property
+    def max_lanes(self) -> int:
+        return max(self.left_lanes, self.right_lanes, self.output_lanes)
+
+
+DEFAULT_PLANNING_LIMITS = PlanningLimits()
+_WARNED_FULL_LAYOUT_SIGNATURES: set[tuple[int, int, int]] = set()
+_WARNED_PLAN_COSTS: set[tuple[object, ...]] = set()
+
+
+def planning_limits_for(algebra) -> PlanningLimits:
+    """Return per-algebra planning limits, falling back to defaults."""
+    return getattr(algebra, "planning_limits", DEFAULT_PLANNING_LIMITS)
+
+
+def full_layout_allowed(algebra, spec: AlgebraSpec) -> bool:
+    """Return whether implicit full-layout planning is enabled for this algebra."""
+    return bool(getattr(algebra, "allow_full_layout_products", False)) and spec.n <= FULL_LAYOUT_MAX_N
+
+
+def warn_full_layout_fallback(algebra) -> None:
+    """Warn once per signature when implicit full layout is used at n>=8."""
+    if getattr(algebra, "n", 0) < FULL_LAYOUT_WARN_N:
+        return
+    signature = (int(algebra.p), int(algebra.q), int(algebra.r))
+    if signature in _WARNED_FULL_LAYOUT_SIGNATURES:
+        return
+    _WARNED_FULL_LAYOUT_SIGNATURES.add(signature)
+    warnings.warn(
+        f"Using implicit full Cl({algebra.p},{algebra.q},{algebra.r}) layout at n={algebra.n}. "
+        "Declare active grades or default_grades to avoid full-layout planning.",
+        RuntimeWarning,
+        stacklevel=3,
+    )
+
+
+def validate_layout_cost(algebra, layout: GradeLayout, *, role: str = "layout") -> GradeLayout:
+    """Validate one compact layout against static lane limits."""
+    cost = PlanCost(
+        spec=layout.spec,
+        kind="layout",
+        op=role,
+        output_lanes=layout.dim,
+        output_grades=layout.grades,
+    )
+    validate_plan_cost(algebra, cost)
+    return layout
+
+
+def validate_grades_cost(algebra, spec: AlgebraSpec, grades, *, role: str = "layout") -> tuple[int, ...]:
+    """Validate a grade set before materializing its basis indices."""
+    normalized = normalize_grades(grades, spec.n)
+    cost = PlanCost(
+        spec=spec,
+        kind="layout",
+        op=role,
+        output_lanes=basis_count_for_grades(spec.n, normalized),
+        output_grades=normalized,
+    )
+    validate_plan_cost(algebra, cost)
+    return normalized
+
+
+def validate_product_grades_cost(
+    algebra,
+    spec: AlgebraSpec,
+    *,
+    op: str,
+    left_grades,
+    right_grades,
+    output_grades=None,
+) -> tuple[tuple[int, ...], tuple[int, ...], tuple[int, ...]]:
+    """Validate product grade sets before materializing layouts."""
+    left = normalize_grades(left_grades, spec.n, name="left_grades")
+    right = normalize_grades(right_grades, spec.n, name="right_grades")
+    output = (
+        expand_output_grades(left, right, spec.n, op=op)
+        if output_grades is None
+        else normalize_grades(output_grades, spec.n, name="output_grades")
+    )
+    left_lanes = basis_count_for_grades(spec.n, left)
+    right_lanes = basis_count_for_grades(spec.n, right)
+    output_lanes = basis_count_for_grades(spec.n, output)
+    validate_plan_cost(
+        algebra,
+        PlanCost(
+            spec=spec,
+            kind="product",
+            op=str(op),
+            left_lanes=left_lanes,
+            right_lanes=right_lanes,
+            output_lanes=output_lanes,
+            pair_count=left_lanes * right_lanes,
+            left_grades=left,
+            right_grades=right,
+            output_grades=output,
+        ),
+    )
+    return left, right, output
+
+
+def product_plan_cost(request) -> PlanCost:
+    """Return static cost summary for a product request."""
+    return PlanCost(
+        spec=request.spec,
+        kind="product",
+        op=request.op,
+        left_lanes=request.left_layout.dim,
+        right_lanes=request.right_layout.dim,
+        output_lanes=request.output_layout.dim,
+        pair_count=request.left_layout.dim * request.right_layout.dim,
+        left_grades=request.left_grades,
+        right_grades=request.right_grades,
+        output_grades=request.output_grades,
+    )
+
+
+def unary_plan_cost(request) -> PlanCost:
+    """Return static cost summary for a unary request."""
+    return PlanCost(
+        spec=request.spec,
+        kind="unary",
+        op=request.op,
+        left_lanes=request.input_layout.dim,
+        output_lanes=request.output_layout.dim,
+        input_grades=request.input_grades,
+        output_grades=request.output_grades,
+    )
+
+
+def validate_product_request(algebra, request) -> None:
+    """Validate product request static cost."""
+    validate_plan_cost(algebra, product_plan_cost(request))
+
+
+def validate_unary_request(algebra, request) -> None:
+    """Validate unary request static cost."""
+    validate_plan_cost(algebra, unary_plan_cost(request))
+
+
+def validate_plan_cost(algebra, cost: PlanCost, *, limits: Optional[PlanningLimits] = None) -> None:
+    """Raise on excessive static cost and warn once near configured limits."""
+    limits = planning_limits_for(algebra) if limits is None else limits
+    errors = []
+    if cost.max_lanes > limits.max_lanes:
+        errors.append(f"active lanes {cost.max_lanes} exceed max_lanes={limits.max_lanes}")
+    if cost.pair_count > limits.max_pairs:
+        errors.append(f"basis interactions {cost.pair_count} exceed max_pairs={limits.max_pairs}")
+    if errors:
+        detail = "; ".join(errors)
+        raise ValueError(f"Static {cost.kind} plan for {cost.op} is too large: {detail}. Declare fewer grades.")
+
+    warnings_to_emit = []
+    if cost.max_lanes >= limits.warn_lanes:
+        warnings_to_emit.append(f"active lanes {cost.max_lanes} are near max_lanes={limits.max_lanes}")
+    if cost.pair_count >= limits.warn_pairs:
+        warnings_to_emit.append(f"basis interactions {cost.pair_count} are near max_pairs={limits.max_pairs}")
+    if warnings_to_emit:
+        _warn_plan_cost_once(cost, "; ".join(warnings_to_emit))
+
+
+def _warn_plan_cost_once(cost: PlanCost, detail: str) -> None:
+    key = (
+        cost.spec,
+        cost.kind,
+        cost.op,
+        cost.input_grades,
+        cost.left_grades,
+        cost.right_grades,
+        cost.output_grades,
+        cost.max_lanes,
+        cost.pair_count,
+    )
+    if key in _WARNED_PLAN_COSTS:
+        return
+    _WARNED_PLAN_COSTS.add(key)
+    warnings.warn(
+        f"Static {cost.kind} plan for {cost.op} is large: {detail}.",
+        RuntimeWarning,
+        stacklevel=3,
+    )
diff --git a/core/planning/product.py b/core/planning/product.py
new file mode 100644
index 0000000..e09afa6
--- /dev/null
+++ b/core/planning/product.py
@@ -0,0 +1,349 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Static grade-path plans for sparse high-dimensional Clifford products.
+
+This module describes the lower-level AoT shape needed by high-dimensional
+sparse execution: input grades are declared or inferred at construction time,
+all basis interactions are expanded once, and forward execution is only gather,
+multiply, and indexed reduction over the required output grade lanes.
+"""
+
+from __future__ import annotations
+
+from typing import Iterable, Optional
+
+import torch
+import torch.nn as nn
+
+from core.foundation.basis import (
+    GradeProductOp,
+    basis_index_tuple_for_grades,
+    basis_indices_tensor,
+    expand_output_grades,
+    normalize_grades,
+    operation_coefficient,
+)
+from core.foundation.layout import AlgebraSpec
+from core.planning.layouts import ProductRequest
+from core.planning.tree import GradePlanTree, build_grade_plan_tree
+
+
+class GradeProductPlan:
+    """AoT basis interaction plan for one grade-restricted bilinear product."""
+
+    def __init__(
+        self,
+        *,
+        p: int,
+        q: int,
+        r: int,
+        op: GradeProductOp,
+        left_grades: tuple[int, ...],
+        right_grades: tuple[int, ...],
+        output_grades: tuple[int, ...],
+        left_indices: torch.Tensor,
+        right_indices: torch.Tensor,
+        output_indices: torch.Tensor,
+        output_positions: torch.Tensor,
+        left_compact_positions: torch.Tensor,
+        right_compact_positions: torch.Tensor,
+        coefficients: torch.Tensor,
+        active_output_indices: torch.Tensor,
+        tree: GradePlanTree,
+    ):
+        self.spec = AlgebraSpec(p, q, r)
+        self.op = op
+        self.left_layout = self.spec.layout(left_grades)
+        self.right_layout = self.spec.layout(right_grades)
+        self.output_layout = self.spec.layout(output_grades)
+        self.left_indices = left_indices
+        self.right_indices = right_indices
+        self.output_indices = output_indices
+        self.output_positions = output_positions
+        self.left_compact_positions = left_compact_positions
+        self.right_compact_positions = right_compact_positions
+        self.coefficients = coefficients
+        self.active_output_indices = active_output_indices
+        self.tree = tree
+
+    @property
+    def p(self) -> int:
+        return self.spec.p
+
+    @property
+    def q(self) -> int:
+        return self.spec.q
+
+    @property
+    def r(self) -> int:
+        return self.spec.r
+
+    @property
+    def left_grades(self) -> tuple[int, ...]:
+        return self.left_layout.grades
+
+    @property
+    def right_grades(self) -> tuple[int, ...]:
+        return self.right_layout.grades
+
+    @property
+    def output_grades(self) -> tuple[int, ...]:
+        return self.output_layout.grades
+
+    @property
+    def n(self) -> int:
+        return self.p + self.q + self.r
+
+    @property
+    def dim(self) -> int:
+        return 1 << self.n
+
+    @property
+    def pair_count(self) -> int:
+        return int(self.left_indices.numel())
+
+    @property
+    def output_dim(self) -> int:
+        return int(self.active_output_indices.numel())
+
+    @property
+    def is_empty(self) -> bool:
+        return self.pair_count == 0
+
+    @property
+    def density(self) -> float:
+        if self.tree.estimated_pairs == 0:
+            return 0.0
+        return self.pair_count / self.tree.estimated_pairs
+
+
+def build_grade_product_plan(
+    p: int,
+    q: int = 0,
+    r: int = 0,
+    *,
+    left_grades: Iterable[int],
+    right_grades: Iterable[int],
+    output_grades: Optional[Iterable[int]] = None,
+    op: GradeProductOp = "gp",
+    device=None,
+    dtype: torch.dtype = torch.float32,
+) -> GradeProductPlan:
+    """Build an exact static basis-pair plan for a grade-restricted operation."""
+    spec = AlgebraSpec(int(p), int(q), int(r))
+    tree = build_grade_plan_tree(
+        spec,
+        left_grades=left_grades,
+        right_grades=right_grades,
+        output_grades=output_grades,
+        op=op,
+    )
+    return build_grade_product_plan_from_tree(tree, device=device, dtype=dtype)
+
+
+def build_grade_product_plan_from_request(
+    request: ProductRequest,
+    *,
+    device=None,
+    dtype: Optional[torch.dtype] = None,
+) -> GradeProductPlan:
+    """Build a plan from a normalized product request."""
+    tree = build_grade_plan_tree(
+        request.spec,
+        left_grades=request.left_grades,
+        right_grades=request.right_grades,
+        output_grades=request.output_grades,
+        op=request.op,
+    )
+    return build_grade_product_plan_from_tree(
+        tree,
+        device=request.device if device is None else device,
+        dtype=request.dtype if dtype is None else dtype,
+    )
+
+
+def build_grade_product_plan_from_tree(
+    tree: GradePlanTree,
+    *,
+    device=None,
+    dtype: torch.dtype = torch.float32,
+) -> GradeProductPlan:
+    """Lower a planner tree into flat Torch gather/reduce buffers."""
+    spec = tree.spec
+    p, q, r = spec.p, spec.q, spec.r
+    n = spec.n
+    left_grade_tuple = tree.left_grades
+    right_grade_tuple = tree.right_grades
+    output_grade_tuple = tree.output_grades
+
+    left_basis_by_grade = {grade: basis_index_tuple_for_grades(n, (grade,)) for grade in left_grade_tuple}
+    right_basis_by_grade = {grade: basis_index_tuple_for_grades(n, (grade,)) for grade in right_grade_tuple}
+    left_position_by_index = {
+        index: position for position, index in enumerate(spec.layout(left_grade_tuple).basis_indices)
+    }
+    right_position_by_index = {
+        index: position for position, index in enumerate(spec.layout(right_grade_tuple).basis_indices)
+    }
+    active_outputs = basis_index_tuple_for_grades(n, output_grade_tuple)
+    output_position_by_index = {index: position for position, index in enumerate(active_outputs)}
+
+    plan_left: list[int] = []
+    plan_right: list[int] = []
+    plan_output: list[int] = []
+    plan_positions: list[int] = []
+    plan_left_compact_positions: list[int] = []
+    plan_right_compact_positions: list[int] = []
+    plan_coefficients: list[float] = []
+
+    for path in tree.paths:
+        path_output_grades = set(path.output_grades)
+        for left_index in left_basis_by_grade[path.left_grade]:
+            for right_index in right_basis_by_grade[path.right_grade]:
+                output_index = left_index ^ right_index
+                if output_index.bit_count() not in path_output_grades:
+                    continue
+                output_position = output_position_by_index.get(output_index)
+                if output_position is None:
+                    continue
+                coefficient = operation_coefficient(left_index, right_index, p, q, r, tree.op)
+                if coefficient == 0.0:
+                    continue
+                plan_left.append(left_index)
+                plan_right.append(right_index)
+                plan_output.append(output_index)
+                plan_positions.append(output_position)
+                plan_left_compact_positions.append(left_position_by_index[left_index])
+                plan_right_compact_positions.append(right_position_by_index[right_index])
+                plan_coefficients.append(coefficient)
+
+    return GradeProductPlan(
+        p=p,
+        q=q,
+        r=r,
+        op=tree.op,
+        left_grades=left_grade_tuple,
+        right_grades=right_grade_tuple,
+        output_grades=output_grade_tuple,
+        left_indices=basis_indices_tensor(plan_left, n=n, role="left product basis indices", device=device),
+        right_indices=basis_indices_tensor(plan_right, n=n, role="right product basis indices", device=device),
+        output_indices=basis_indices_tensor(plan_output, n=n, role="output product basis indices", device=device),
+        output_positions=torch.tensor(plan_positions, dtype=torch.long, device=device),
+        left_compact_positions=torch.tensor(plan_left_compact_positions, dtype=torch.long, device=device),
+        right_compact_positions=torch.tensor(plan_right_compact_positions, dtype=torch.long, device=device),
+        coefficients=torch.tensor(plan_coefficients, dtype=dtype, device=device),
+        active_output_indices=basis_indices_tensor(active_outputs, n=n, role="active output basis indices", device=device),
+        tree=tree,
+    )
+
+
+class GradeProductExecutor(nn.Module):
+    """Compile-friendly grade-restricted product using a static interaction plan.
+
+    ``forward`` returns compact output lanes ordered by ``active_output_indices``.
+    ``forward_dense`` is an explicit materialization helper for parity checks and
+    dense callers.
+    """
+
+    def __init__(self, plan: GradeProductPlan):
+        super().__init__()
+        self.p = plan.p
+        self.q = plan.q
+        self.r = plan.r
+        self.n = plan.n
+        self.dim = plan.dim
+        self.op = plan.op
+        self.left_grades = plan.left_grades
+        self.right_grades = plan.right_grades
+        self.output_grades = plan.output_grades
+        self.left_layout = plan.left_layout
+        self.right_layout = plan.right_layout
+        self.output_layout = plan.output_layout
+        self._output_dim = plan.output_dim
+        self._pair_count = plan.pair_count
+        self.register_buffer("left_indices", plan.left_indices, persistent=False)
+        self.register_buffer("right_indices", plan.right_indices, persistent=False)
+        self.register_buffer("output_indices", plan.output_indices, persistent=False)
+        self.register_buffer("output_positions", plan.output_positions, persistent=False)
+        self.register_buffer("coefficients", plan.coefficients, persistent=False)
+        self.register_buffer("active_output_indices", plan.active_output_indices, persistent=False)
+        self.register_buffer("left_compact_positions", plan.left_compact_positions, persistent=False)
+        self.register_buffer("right_compact_positions", plan.right_compact_positions, persistent=False)
+
+    @property
+    def output_dim(self) -> int:
+        return self._output_dim
+
+    @property
+    def pair_count(self) -> int:
+        return self._pair_count
+
+    def forward(self, left: torch.Tensor, right: torch.Tensor) -> torch.Tensor:
+        """Return compact grade-lane output for full dense input tensors."""
+        if left.shape[-1] != self.dim:
+            raise ValueError(f"left last dimension must be {self.dim}, got {left.shape[-1]}")
+        if right.shape[-1] != self.dim:
+            raise ValueError(f"right last dimension must be {self.dim}, got {right.shape[-1]}")
+
+        left_terms = torch.index_select(left, -1, self.left_indices)
+        right_terms = torch.index_select(right, -1, self.right_indices)
+        left_terms, right_terms = torch.broadcast_tensors(left_terms, right_terms)
+        terms = left_terms * right_terms * self._coefficients_for(left_terms, right_terms)
+
+        output = terms.new_zeros(*terms.shape[:-1], self.output_dim)
+        return output.index_add(-1, self.output_positions, terms)
+
+    def forward_compact(self, left: torch.Tensor, right: torch.Tensor) -> torch.Tensor:
+        """Return compact output for inputs already stored in this plan's compact layouts."""
+        if left.shape[-1] != self.left_layout.dim:
+            raise ValueError(f"left compact dimension must be {self.left_layout.dim}, got {left.shape[-1]}")
+        if right.shape[-1] != self.right_layout.dim:
+            raise ValueError(f"right compact dimension must be {self.right_layout.dim}, got {right.shape[-1]}")
+
+        left_terms = torch.index_select(left, -1, self.left_compact_positions)
+        right_terms = torch.index_select(right, -1, self.right_compact_positions)
+        left_terms, right_terms = torch.broadcast_tensors(left_terms, right_terms)
+        terms = left_terms * right_terms * self._coefficients_for(left_terms, right_terms)
+
+        output = terms.new_zeros(*terms.shape[:-1], self.output_dim)
+        return output.index_add(-1, self.output_positions, terms)
+
+    def forward_pairwise_compact(self, left: torch.Tensor, right: torch.Tensor) -> torch.Tensor:
+        """Pairwise compact product for sequence-style bilinear scoring.
+
+        ``left`` is ``[..., left_items, left_layout.dim]`` and ``right`` is
+        ``[..., right_items, right_layout.dim]``. The result is
+        ``[..., left_items, right_items, output_layout.dim]``.
+        """
+        if left.shape[-1] != self.left_layout.dim:
+            raise ValueError(f"left compact dimension must be {self.left_layout.dim}, got {left.shape[-1]}")
+        if right.shape[-1] != self.right_layout.dim:
+            raise ValueError(f"right compact dimension must be {self.right_layout.dim}, got {right.shape[-1]}")
+
+        prefix = torch.broadcast_shapes(left.shape[:-2], right.shape[:-2])
+        left = left.expand(*prefix, *left.shape[-2:])
+        right = right.expand(*prefix, *right.shape[-2:])
+
+        left_terms = torch.index_select(left, -1, self.left_compact_positions)
+        right_terms = torch.index_select(right, -1, self.right_compact_positions)
+        terms = left_terms.unsqueeze(-2) * right_terms.unsqueeze(-3) * self._coefficients_for(left_terms, right_terms)
+
+        output = terms.new_zeros(*terms.shape[:-1], self.output_dim)
+        return output.index_add(-1, self.output_positions, terms)
+
+    def forward_dense(self, left: torch.Tensor, right: torch.Tensor) -> torch.Tensor:
+        """Return a full ``[..., 2**n]`` dense tensor for dense-kernel parity checks."""
+        compact = self.forward(left, right)
+        output = compact.new_zeros(*compact.shape[:-1], self.dim)
+        return output.index_copy(-1, self.active_output_indices, compact)
+
+    def _coefficients_for(self, left: torch.Tensor, right: torch.Tensor) -> torch.Tensor:
+        dtype = torch.promote_types(left.dtype, right.dtype)
+        coefficients = self.coefficients
+        if coefficients.dtype == dtype and coefficients.device == left.device:
+            return coefficients
+        return coefficients.to(device=left.device, dtype=dtype)
diff --git a/core/planning/tree.py b/core/planning/tree.py
new file mode 100644
index 0000000..c1312b9
--- /dev/null
+++ b/core/planning/tree.py
@@ -0,0 +1,136 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Planner-only grade tree metadata.
+
+The tree here is not a runtime backend. It groups declared grade routes before
+they are lowered into flat Torch executor buffers.
+"""
+
+from __future__ import annotations
+
+import math
+from dataclasses import dataclass
+from typing import Iterable, Optional
+
+from core.foundation.basis import GradeProductOp, expand_output_grades, normalize_grades, product_output_grades
+from core.foundation.layout import AlgebraSpec
+
+
+@dataclass(frozen=True)
+class GradePathNode:
+    """One homogeneous left-grade/right-grade route in a product plan."""
+
+    path_index: int
+    left_grade: int
+    right_grade: int
+    output_grades: tuple[int, ...]
+    left_dim: int
+    right_dim: int
+
+    @property
+    def estimated_pairs(self) -> int:
+        """Upper-bound number of basis pairs before metric zero pruning."""
+        return self.left_dim * self.right_dim
+
+
+@dataclass(frozen=True)
+class GradePlanTree:
+    """Planner-side grouping for a grade-restricted product."""
+
+    spec: AlgebraSpec
+    op: GradeProductOp
+    left_grades: tuple[int, ...]
+    right_grades: tuple[int, ...]
+    output_grades: tuple[int, ...]
+    paths: tuple[GradePathNode, ...]
+    chunk_pair_limit: Optional[int] = None
+
+    @property
+    def path_count(self) -> int:
+        """Number of active homogeneous product routes."""
+        return len(self.paths)
+
+    @property
+    def estimated_pairs(self) -> int:
+        """Upper-bound number of basis pairs across all paths."""
+        return sum(path.estimated_pairs for path in self.paths)
+
+    @property
+    def estimated_chunks(self) -> int:
+        """Number of planner chunks implied by ``chunk_pair_limit``."""
+        if self.chunk_pair_limit is None or self.chunk_pair_limit <= 0:
+            return 1 if self.paths else 0
+        return sum(math.ceil(path.estimated_pairs / self.chunk_pair_limit) for path in self.paths)
+
+    def path_for_grades(self, left_grade: int, right_grade: int) -> Optional[GradePathNode]:
+        """Return the active path for a homogeneous grade pair."""
+        for path in self.paths:
+            if path.left_grade == left_grade and path.right_grade == right_grade:
+                return path
+        return None
+
+
+def build_grade_plan_tree(
+    spec: AlgebraSpec,
+    *,
+    left_grades: Iterable[int],
+    right_grades: Iterable[int],
+    output_grades: Optional[Iterable[int]] = None,
+    op: GradeProductOp = "gp",
+    chunk_pair_limit: Optional[int] = None,
+) -> GradePlanTree:
+    """Build planner metadata for grade route grouping."""
+    left = normalize_grades(left_grades, spec.n, name="left_grades")
+    right = normalize_grades(right_grades, spec.n, name="right_grades")
+    output = (
+        expand_output_grades(left, right, spec.n, op=op)
+        if output_grades is None
+        else normalize_grades(output_grades, spec.n, name="output_grades")
+    )
+    output_set = set(output)
+
+    paths = []
+    for left_grade in left:
+        left_dim = _grade_dim(spec.n, left_grade)
+        for right_grade in right:
+            route_outputs = product_output_grades(left_grade, right_grade, spec.n, op=op)
+            route_outputs = tuple(grade for grade in route_outputs if grade in output_set)
+            if not route_outputs:
+                continue
+            paths.append(
+                GradePathNode(
+                    path_index=len(paths),
+                    left_grade=left_grade,
+                    right_grade=right_grade,
+                    output_grades=route_outputs,
+                    left_dim=left_dim,
+                    right_dim=_grade_dim(spec.n, right_grade),
+                )
+            )
+
+    return GradePlanTree(
+        spec=spec,
+        op=op,
+        left_grades=left,
+        right_grades=right,
+        output_grades=output,
+        paths=tuple(paths),
+        chunk_pair_limit=chunk_pair_limit,
+    )
+
+
+def _grade_dim(n: int, grade: int) -> int:
+    if grade < 0 or grade > n:
+        return 0
+    grade = min(grade, n - grade)
+    numerator = 1
+    denominator = 1
+    for i in range(1, grade + 1):
+        numerator *= n - grade + i
+        denominator *= i
+    return numerator // denominator
diff --git a/core/planning/unary.py b/core/planning/unary.py
new file mode 100644
index 0000000..5906b97
--- /dev/null
+++ b/core/planning/unary.py
@@ -0,0 +1,252 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Compile-friendly planned unary operators."""
+
+from __future__ import annotations
+
+from dataclasses import dataclass
+from typing import Literal, Optional
+
+import torch
+import torch.nn as nn
+
+from core.foundation.basis import normalize_grades, reverse_sign
+from core.foundation.layout import AlgebraSpec, GradeLayout
+from core.planning.layouts import check_layout_spec, is_compact_tensor, resolve_operand_layout
+
+GradeUnaryOp = Literal["identity", "reverse", "grade_involution", "clifford_conjugation", "grade_projection"]
+_VALID_UNARY_OPS = {"identity", "reverse", "grade_involution", "clifford_conjugation", "grade_projection"}
+
+
+@dataclass(frozen=True)
+class UnaryRequest:
+    """Fully resolved request for one unary planned operation."""
+
+    spec: AlgebraSpec
+    op: GradeUnaryOp
+    input_layout: GradeLayout
+    output_layout: GradeLayout
+    input_compact: bool
+    dtype: torch.dtype
+    device: torch.device
+
+    @property
+    def input_grades(self) -> tuple[int, ...]:
+        return self.input_layout.grades
+
+    @property
+    def output_grades(self) -> tuple[int, ...]:
+        return self.output_layout.grades
+
+    @property
+    def cache_key(self) -> tuple[object, ...]:
+        return (
+            self.spec,
+            str(self.device),
+            str(self.dtype),
+            self.op,
+            self.input_grades,
+            self.output_grades,
+        )
+
+
+class GradeUnaryPlan:
+    """Static gather/sign plan for one unary operation."""
+
+    def __init__(
+        self,
+        *,
+        spec: AlgebraSpec,
+        op: GradeUnaryOp,
+        input_grades: tuple[int, ...],
+        output_grades: tuple[int, ...],
+        input_positions: torch.Tensor,
+        output_indices: torch.Tensor,
+        signs: torch.Tensor,
+    ):
+        self.spec = spec
+        self.op = op
+        self.input_layout = spec.layout(input_grades)
+        self.output_layout = spec.layout(output_grades)
+        self.input_positions = input_positions
+        self.output_indices = output_indices
+        self.signs = signs
+
+    @property
+    def dim(self) -> int:
+        return self.spec.dim
+
+    @property
+    def output_dim(self) -> int:
+        return self.output_layout.dim
+
+
+class GradeUnaryExecutor(nn.Module):
+    """Torch module for planned unary gather/sign execution."""
+
+    def __init__(self, plan: GradeUnaryPlan):
+        super().__init__()
+        self.spec = plan.spec
+        self.op = plan.op
+        self.input_layout = plan.input_layout
+        self.output_layout = plan.output_layout
+        self.dim = plan.dim
+        self.register_buffer("input_positions", plan.input_positions, persistent=False)
+        self.register_buffer("output_indices", plan.output_indices, persistent=False)
+        self.register_buffer("signs", plan.signs, persistent=False)
+
+    @property
+    def output_dim(self) -> int:
+        return self.output_layout.dim
+
+    def forward(self, values: torch.Tensor) -> torch.Tensor:
+        """Return compact output lanes for dense input coefficients."""
+        if values.shape[-1] != self.dim:
+            raise ValueError(f"dense last dimension must be {self.dim}, got {values.shape[-1]}")
+        output = torch.index_select(values, -1, self.output_indices)
+        return output * self.signs.to(dtype=output.dtype)
+
+    def forward_compact(self, values: torch.Tensor) -> torch.Tensor:
+        """Return compact output lanes for compact input coefficients."""
+        if values.shape[-1] != self.input_layout.dim:
+            raise ValueError(f"compact last dimension must be {self.input_layout.dim}, got {values.shape[-1]}")
+        output = torch.index_select(values, -1, self.input_positions)
+        return output * self.signs.to(dtype=output.dtype)
+
+    def forward_dense(self, values: torch.Tensor) -> torch.Tensor:
+        """Return dense output coefficients for dense input coefficients."""
+        compact = self.forward(values)
+        return self.output_layout.dense(compact)
+
+
+def build_unary_request(
+    spec: AlgebraSpec,
+    values: torch.Tensor,
+    *,
+    op: str,
+    input_grades=None,
+    output_grades=None,
+    input_layout: Optional[GradeLayout] = None,
+    output_layout: Optional[GradeLayout] = None,
+    input_compact: bool = False,
+    full_layout_allowed: bool = True,
+) -> UnaryRequest:
+    """Resolve caller input into a static unary request."""
+    op = normalize_unary_op(op)
+    if op == "grade_projection" and input_grades is None and input_layout is None and not input_compact:
+        if output_layout is not None:
+            input_layout = output_layout
+        elif output_grades is not None:
+            input_grades = output_grades
+    input_layout = resolve_operand_layout(
+        spec,
+        values,
+        grades=input_grades,
+        layout=input_layout,
+        compact=input_compact,
+        side="input",
+        full_layout_allowed=full_layout_allowed,
+    )
+    output_layout = resolve_unary_output_layout(
+        spec,
+        op=op,
+        input_layout=input_layout,
+        output_grades=output_grades,
+        output_layout=output_layout,
+    )
+    input_compact = input_compact or is_compact_tensor(spec, values, input_layout)
+    return UnaryRequest(
+        spec=spec,
+        op=op,
+        input_layout=input_layout,
+        output_layout=output_layout,
+        input_compact=input_compact,
+        dtype=values.dtype,
+        device=values.device,
+    )
+
+
+def build_unary_plan_from_request(request: UnaryRequest) -> GradeUnaryPlan:
+    """Lower a unary request into static gather/sign buffers."""
+    input_position_by_index = {index: pos for pos, index in enumerate(request.input_layout.basis_indices)}
+    input_positions = []
+    signs = []
+    for index in request.output_layout.basis_indices:
+        position = input_position_by_index.get(index)
+        if position is None:
+            raise ValueError(
+                f"output basis index {index} is not available in input grades {request.input_layout.grades}"
+            )
+        input_positions.append(position)
+        signs.append(_unary_sign(request.op, index))
+
+    return GradeUnaryPlan(
+        spec=request.spec,
+        op=request.op,
+        input_grades=request.input_grades,
+        output_grades=request.output_grades,
+        input_positions=torch.tensor(input_positions, dtype=torch.long, device=request.device),
+        output_indices=torch.tensor(request.output_layout.basis_indices, dtype=torch.long, device=request.device),
+        signs=torch.tensor(signs, dtype=request.dtype, device=request.device),
+    )
+
+
+def normalize_unary_op(op: str) -> GradeUnaryOp:
+    """Validate and normalize a unary operation name."""
+    normalized = str(op)
+    if normalized not in _VALID_UNARY_OPS:
+        raise ValueError(f"Unsupported grade unary op {op!r}")
+    return normalized  # type: ignore[return-value]
+
+
+def resolve_unary_output_layout(
+    spec: AlgebraSpec,
+    *,
+    op: GradeUnaryOp,
+    input_layout: GradeLayout,
+    output_grades=None,
+    output_layout: Optional[GradeLayout] = None,
+) -> GradeLayout:
+    """Resolve output layout for a planned unary operation."""
+    if output_layout is not None:
+        check_layout_spec(spec, output_layout, "output_layout")
+        if output_grades is not None and output_layout.grades != normalize_grades(
+            output_grades, spec.n, name="output_grades"
+        ):
+            raise ValueError("output_layout and output_grades disagree")
+        return output_layout
+
+    if op == "grade_projection":
+        if output_grades is None:
+            raise ValueError("output_grades is required for grade_projection")
+        projected = normalize_grades(output_grades, spec.n, name="output_grades")
+        missing = tuple(grade for grade in projected if grade not in input_layout.grades)
+        if missing:
+            raise ValueError(f"Cannot project missing grades {missing} from input grades {input_layout.grades}")
+        return spec.layout(projected)
+
+    if output_grades is not None:
+        projected = normalize_grades(output_grades, spec.n, name="output_grades")
+        missing = tuple(grade for grade in projected if grade not in input_layout.grades)
+        if missing:
+            raise ValueError(f"Cannot project missing grades {missing} from input grades {input_layout.grades}")
+        return spec.layout(projected)
+    return input_layout
+
+
+def _unary_sign(op: GradeUnaryOp, index: int) -> float:
+    grade = int(index).bit_count()
+    if op in {"identity", "grade_projection"}:
+        return 1.0
+    if op == "reverse":
+        return reverse_sign(index)
+    if op == "grade_involution":
+        return -1.0 if grade % 2 else 1.0
+    if op == "clifford_conjugation":
+        return (-1.0 if grade % 2 else 1.0) * reverse_sign(index)
+    raise ValueError(f"Unsupported grade unary op {op!r}")
diff --git a/core/runtime/__init__.py b/core/runtime/__init__.py
new file mode 100644
index 0000000..0c108e6
--- /dev/null
+++ b/core/runtime/__init__.py
@@ -0,0 +1,27 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Runtime algebra hosts and dense reference operations."""
+
+from .accessors import as_multivector, compact_values, grade_indices, hermitian_signs, materialize_dense, resolve_layout
+from .algebra import CliffordAlgebra
+from .context import AlgebraContext
+from .multivector import Multivector
+from .projected import AlgebraRuntimeMixin
+
+__all__ = [
+    "AlgebraContext",
+    "AlgebraRuntimeMixin",
+    "CliffordAlgebra",
+    "Multivector",
+    "as_multivector",
+    "compact_values",
+    "grade_indices",
+    "hermitian_signs",
+    "materialize_dense",
+    "resolve_layout",
+]
diff --git a/core/runtime/accessors.py b/core/runtime/accessors.py
new file mode 100644
index 0000000..7cbd44d
--- /dev/null
+++ b/core/runtime/accessors.py
@@ -0,0 +1,220 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Functional accessors for algebra layout, grade, and storage metadata."""
+
+from __future__ import annotations
+
+from typing import Iterable, Optional
+
+import torch
+
+from core.foundation.basis import normalize_grades, operation_coefficient, reverse_sign
+from core.foundation.layout import AlgebraSpec, GradeLayout
+from core.planning.policy import (
+    FULL_LAYOUT_MAX_N,
+    validate_grades_cost,
+    validate_layout_cost,
+    warn_full_layout_fallback,
+)
+
+
+def resolve_layout(
+    algebra,
+    *,
+    layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+    mv=None,
+    allow_full: bool = True,
+    warn_full: bool = True,
+) -> GradeLayout:
+    """Resolve static grade layout metadata without inspecting tensor values."""
+    spec = AlgebraSpec.from_algebra(algebra)
+    if layout is not None:
+        _check_layout_spec(spec, layout, "layout")
+        if grades is not None and layout.grades != normalize_grades(grades, spec.n, name="grades"):
+            raise ValueError("layout and grades disagree")
+        return validate_layout_cost(algebra, layout)
+
+    if grades is not None:
+        return spec.layout(validate_grades_cost(algebra, spec, grades))
+
+    if _is_multivector(mv) and getattr(mv, "layout", None) is not None:
+        mv_layout = mv.layout
+        _check_layout_spec(spec, mv_layout, "mv.layout")
+        return validate_layout_cost(algebra, mv_layout)
+
+    default_grades = getattr(algebra, "_default_grades", None)
+    if default_grades is not None:
+        cached = getattr(algebra, "_default_layout", None)
+        if cached is not None:
+            _check_layout_spec(spec, cached, "default_layout")
+            return validate_layout_cost(algebra, cached, role="default_layout")
+        resolved = spec.layout(validate_grades_cost(algebra, spec, default_grades, role="default_layout"))
+        if hasattr(algebra, "_default_layout"):
+            algebra._default_layout = resolved
+        return validate_layout_cost(algebra, resolved, role="default_layout")
+
+    if not allow_full or not bool(getattr(algebra, "allow_full_layout_products", True)):
+        raise ValueError("No grade layout is available. Declare active grades or configure default_grades.")
+    if spec.n > FULL_LAYOUT_MAX_N:
+        raise ValueError(
+            f"Implicit full Cl({spec.p},{spec.q},{spec.r}) layout is disabled for n>{FULL_LAYOUT_MAX_N}. "
+            "Declare active grades or configure default_grades."
+        )
+    if warn_full:
+        warn_full_layout_fallback(algebra)
+    return spec.layout(validate_grades_cost(algebra, spec, range(spec.n + 1), role="full_layout"))
+
+
+def default_layout(algebra) -> GradeLayout:
+    """Return the algebra default layout using the central fallback policy."""
+    return resolve_layout(algebra)
+
+
+def grade_indices(algebra, grades: Iterable[int], *, device=None) -> torch.Tensor:
+    """Return canonical dense basis indices for ``grades``."""
+    if device is None:
+        device = getattr(algebra, "device", None)
+    return resolve_layout(algebra, grades=grades, warn_full=False).indices_tensor(device=device)
+
+
+def hermitian_signs(
+    algebra,
+    layout: Optional[GradeLayout] = None,
+    *,
+    grades: Optional[Iterable[int]] = None,
+    device=None,
+    dtype: Optional[torch.dtype] = None,
+) -> torch.Tensor:
+    """Return Hermitian metric signs for a dense or compact layout."""
+    resolved = resolve_layout(algebra, layout=layout, grades=grades)
+    if device is None:
+        device = getattr(algebra, "device", None)
+    if dtype is None:
+        dtype = getattr(algebra, "dtype", torch.float32)
+
+    dense_signs = getattr(algebra, "_hermitian_signs", None)
+    if dense_signs is not None:
+        indices = resolved.indices_tensor(device=dense_signs.device)
+        signs = torch.index_select(dense_signs, -1, indices)
+        return signs.to(device=device, dtype=dtype)
+
+    values = [_hermitian_sign_for_index(algebra, index) for index in resolved.basis_indices]
+    return torch.tensor(values, dtype=dtype, device=device)
+
+
+def compact_values(
+    algebra,
+    value,
+    *,
+    layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+) -> tuple[torch.Tensor, GradeLayout]:
+    """Return compact values plus layout for a tensor or ``Multivector``."""
+    resolved = resolve_layout(algebra, layout=layout, grades=grades, mv=value)
+    if _is_multivector(value):
+        _check_algebra(algebra, value.algebra)
+        if value.is_compact:
+            return resolved.convert(value.values, value.layout), resolved
+        return resolved.compact(value.coefficients), resolved
+
+    if not isinstance(value, torch.Tensor):
+        raise TypeError(f"Expected Tensor or Multivector-like value, got {type(value)!r}")
+    if value.shape[-1] == resolved.dim:
+        return value, resolved
+    if value.shape[-1] == resolved.dense_dim:
+        return resolved.compact(value), resolved
+    raise ValueError(f"value last dimension must be {resolved.dim} compact or {resolved.dense_dim} dense")
+
+
+def materialize_dense(
+    algebra,
+    value,
+    *,
+    layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+) -> torch.Tensor:
+    """Return dense coefficients subject to the central full-layout policy."""
+    if _is_multivector(value):
+        _check_algebra(algebra, value.algebra)
+        if not value.is_compact:
+            return value.coefficients
+        _check_dense_materialization_allowed(algebra)
+        return value.layout.dense(value.values)
+
+    if not isinstance(value, torch.Tensor):
+        raise TypeError(f"Expected Tensor or Multivector-like value, got {type(value)!r}")
+    if value.shape[-1] == getattr(algebra, "dim"):
+        return value
+    resolved = resolve_layout(algebra, layout=layout, grades=grades)
+    if value.shape[-1] != resolved.dim:
+        raise ValueError(f"value compact last dimension must be {resolved.dim}, got {value.shape[-1]}")
+    _check_dense_materialization_allowed(algebra)
+    return resolved.dense(value)
+
+
+def as_multivector(
+    algebra,
+    value,
+    *,
+    layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+):
+    """Wrap a tensor or return an existing ``Multivector``."""
+    from core.runtime.multivector import Multivector
+
+    if isinstance(value, Multivector):
+        _check_algebra(algebra, value.algebra)
+        if layout is None and grades is None:
+            return value
+        resolved = resolve_layout(algebra, layout=layout, grades=grades, mv=value)
+        return value.with_layout(resolved)
+
+    if layout is None and grades is None:
+        if not isinstance(value, torch.Tensor):
+            raise TypeError(f"Expected Tensor or Multivector-like value, got {type(value)!r}")
+        return Multivector(algebra, value)
+
+    resolved = resolve_layout(algebra, layout=layout, grades=grades)
+    if value.shape[-1] == resolved.dim:
+        return Multivector(algebra, values=value, layout=resolved)
+    return Multivector(algebra, tensor=value, layout=resolved)
+
+
+def _hermitian_sign_for_index(algebra, index: int) -> float:
+    grade = int(index).bit_count()
+    grade_sign = -1.0 if grade % 2 else 1.0
+    metric_sign = operation_coefficient(index, index, algebra.p, algebra.q, algebra.r, "gp")
+    return grade_sign * reverse_sign(index) * metric_sign
+
+
+def _check_layout_spec(spec: AlgebraSpec, layout: GradeLayout, name: str) -> None:
+    if layout.spec != spec:
+        raise ValueError(f"{name} signature {layout.spec} does not match algebra signature {spec}")
+
+
+def _check_algebra(expected, actual) -> None:
+    lhs = (expected.p, expected.q, expected.r)
+    rhs = (actual.p, actual.q, actual.r)
+    if lhs != rhs:
+        raise ValueError(f"Algebra mismatch: Cl{lhs} vs Cl{rhs}")
+
+
+def _check_dense_materialization_allowed(algebra) -> None:
+    if not bool(getattr(algebra, "allow_full_layout_products", True)):
+        raise ValueError("Dense materialization is disabled for this algebra. Keep compact values.")
+    if getattr(algebra, "n", 0) > FULL_LAYOUT_MAX_N:
+        raise ValueError(
+            f"Dense materialization is disabled for n>{FULL_LAYOUT_MAX_N}. "
+            "Keep compact values or declare a smaller active layout."
+        )
+    warn_full_layout_fallback(algebra)
+
+
+def _is_multivector(value) -> bool:
+    return hasattr(value, "algebra") and hasattr(value, "layout") and hasattr(value, "is_compact")
diff --git a/core/algebra.py b/core/runtime/algebra.py
similarity index 88%
rename from core/algebra.py
rename to core/runtime/algebra.py
index 56e749e..8dad30e 100644
--- a/core/algebra.py
+++ b/core/runtime/algebra.py
@@ -17,10 +17,12 @@
 import torch
 import torch.nn as nn
 
-from core.validation import check_multivector
+from core.foundation.validation import check_multivector
+from core.planning.policy import DEFAULT_PLANNING_LIMITS, PlanningLimits
+from core.runtime.projected import AlgebraRuntimeMixin
 
 
-class CliffordAlgebra(nn.Module):
+class CliffordAlgebra(AlgebraRuntimeMixin, nn.Module):
     """Differentiable Clifford algebra kernel with memory-optimized blocked accumulation.
 
     Extends ``nn.Module`` so that all Cayley tables are registered as
@@ -50,6 +52,8 @@ def __init__(
         dtype: torch.dtype = torch.float32,
         exp_policy: str = "balanced",
         fixed_iterations: Optional[int] = None,
+        allow_large_dense: bool = False,
+        planning_limits: Optional[PlanningLimits] = None,
     ):
         """Initialize the algebra and cache the Cayley table.
 
@@ -64,11 +68,11 @@ def __init__(
                 precision (e.g. AMP on CUDA bfloat16 mode).
             exp_policy (str or ExpPolicy, optional): Bivector exp policy.
                 ``'balanced'`` (default) or ``'precise'``.
-                See :class:`core.decomposition.ExpPolicy`.
+                See :class:`core.runtime.decomposition.ExpPolicy`.
             fixed_iterations (int, optional): Power-iteration step count for
                 the compiled-safe decomposed exp path (used when n>=4).
                 ``None`` (default) auto-derives from ``(exp_policy, dtype, n)``
-                via :func:`core.decomposition.resolve_fixed_iterations`,
+                via :func:`core.runtime.decomposition.resolve_fixed_iterations`,
                 pinned statically at init.
         """
         super().__init__()
@@ -76,11 +80,17 @@ def __init__(
         assert p >= 0, f"p must be non-negative, got {p}"
         assert q >= 0, f"q must be non-negative, got {q}"
         assert r >= 0, f"r must be non-negative, got {r}"
-        assert p + q + r <= 12, f"p + q + r must be <= 12, got {p + q + r}"
+        max_dense_n = 12 if allow_large_dense else 8
+        assert p + q + r <= max_dense_n, (
+            f"p + q + r must be <= {max_dense_n} for dense CliffordAlgebra, got {p + q + r}. "
+            "Use make_algebra(..., kernel='auto') for AlgebraContext or kernel='dense' to explicitly allow Cl9-Cl12."
+        )
 
         self.p, self.q, self.r = p, q, r
         self.n = p + q + r
         self.dim = 2**self.n
+        self.allow_full_layout_products = True
+        self.planning_limits = DEFAULT_PLANNING_LIMITS if planning_limits is None else planning_limits
 
         # Exp regime: dispatch at init
         if p == 0 or q == 0:
@@ -91,7 +101,7 @@ def __init__(
             self._exp_regime = "mixed"
 
         # Exp policy: controls decomposition iteration budget
-        from core.decomposition import ExpPolicy, resolve_fixed_iterations
+        from core.runtime.decomposition import ExpPolicy, resolve_fixed_iterations
 
         self._exp_policy = exp_policy if isinstance(exp_policy, ExpPolicy) else ExpPolicy(exp_policy)
 
@@ -146,6 +156,7 @@ def __init__(
         stacked = torch.stack(grade_masks_list)  # [n+1, dim]
         self.register_buffer("_grade_masks", stacked, persistent=False)
         self.register_buffer("_grade_masks_float", stacked.to(dtype=cayley_signs.dtype), persistent=False)
+        self.register_buffer("_g1_indices", stacked[1].nonzero(as_tuple=False).squeeze(-1), persistent=False)
 
         # Bivector indices
         if self.n >= 2:
@@ -163,6 +174,10 @@ def __init__(
         self.eps: float = float(_finfo.eps)
         self.eps_sq: float = float(_finfo.eps**2)
 
+        from core.planning.planner import GradePlanner
+
+        self.planner = GradePlanner(self)
+
     @property
     def device(self):
         """Return the device of the algebra tables."""
@@ -177,12 +192,23 @@ def dtype(self) -> torch.dtype:
         """
         return self.cayley_signs.dtype
 
+    def bivector_squared_signs(self, *, device=None, dtype: Optional[torch.dtype] = None) -> torch.Tensor:
+        """Return ``(e_ab)^2`` signs in canonical grade-2 layout order."""
+        signs = self.bv_sq_scalar
+        if device is not None or dtype is not None:
+            signs = signs.to(
+                device=self.device if device is None else device,
+                dtype=self.dtype if dtype is None else dtype,
+            )
+        return signs
+
     def _apply(self, fn):
         """Propagate device/dtype moves and keep eps tolerances in sync."""
         result = super()._apply(fn)
         _finfo = torch.finfo(self.cayley_signs.dtype)
         self.eps = float(_finfo.eps)
         self.eps_sq = float(_finfo.eps**2)
+        self.planner._apply(fn)
         return result
 
     @property
@@ -202,7 +228,7 @@ def exp_policy(self):
 
     @exp_policy.setter
     def exp_policy(self, value):
-        from core.decomposition import ExpPolicy, resolve_fixed_iterations
+        from core.runtime.decomposition import ExpPolicy, resolve_fixed_iterations
 
         self._exp_policy = value if isinstance(value, ExpPolicy) else ExpPolicy(value)
         self._exp_fixed_iterations = resolve_fixed_iterations(self._exp_policy, self.dtype, self.n)
@@ -254,10 +280,9 @@ def embed_vector(self, vectors: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Multivector coefficients [..., dim].
         """
-        g1_idx = (1 << torch.arange(self.n, device=vectors.device)).long()
-        mv = torch.zeros(*vectors.shape[:-1], self.dim, device=vectors.device, dtype=vectors.dtype)
-        mv.scatter_(-1, g1_idx.expand_as(vectors), vectors)
-        return mv
+        g1_idx = self._basis_vector_indices(vectors.device)
+        mv = vectors.new_zeros(*vectors.shape[:-1], self.dim)
+        return mv.index_copy(-1, g1_idx, vectors)
 
     def get_grade_norms(self, mv: torch.Tensor) -> torch.Tensor:
         """Calculates norms per grade. Useful for invariant features.
@@ -329,12 +354,16 @@ def _generate_cayley_table(self, device, dtype: torch.dtype = torch.float32):
         else:
             bv_sq_scalar = torch.zeros(0, dtype=cayley_signs.dtype, device=device)
 
-        # Precomputed signs for single-pass wedge and inner product
-        # wedge(A,B) = (AB - BA)/2 uses antisymmetric part of signs
+        # Precomputed signs for single-pass exterior and symmetric products.
+        # wedge(A,B) is the exterior product: the grade-sum part of AB.
+        gi = grade_index.unsqueeze(1)  # [D, 1] - left summation index grade
+        gj_for_result = grade_index[cayley_indices]  # [D, D] - right index j = i^k grade
+        gk = grade_index.unsqueeze(0)  # [1, D] - output index k grade
+        exterior_valid = gk == gi + gj_for_result
+        wedge_gp_signs = gp_signs * exterior_valid.to(dtype=gp_signs.dtype)
+
         # inner(A,B) = (AB + BA)/2 uses symmetric part of signs
-        wedge_cayley_signs = (cayley_signs - cayley_signs.T) / 2.0
         inner_cayley_signs = (cayley_signs + cayley_signs.T) / 2.0
-        wedge_gp_signs = torch.gather(wedge_cayley_signs, 1, cayley_indices)
         inner_gp_signs = torch.gather(inner_cayley_signs, 1, cayley_indices)
 
         # Precomputed signs for commutator [A,B] = AB - BA and
@@ -466,7 +495,7 @@ def _compute_signs(self, indices: torch.Tensor, device, dtype: torch.dtype = tor
 
         return (commutator_sign * metric_sign).to(dtype=dtype)
 
-    def geometric_product(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
+    def geometric_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
         """Computes the Geometric Product.
 
         Uses vectorized gather + broadcast multiply + sum.
@@ -478,6 +507,8 @@ def geometric_product(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: The product AB [..., Dim].
         """
+        if kwargs:
+            return self.projected_product(A, B, op="gp", **kwargs)
         check_multivector(A, self, "geometric_product(A)")
         check_multivector(B, self, "geometric_product(B)")
 
@@ -487,7 +518,7 @@ def geometric_product(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         # result[..., k] = sum_i A[..., i] * B[..., cayley[i,k]] * signs[i,k]
         return torch.matmul(A.unsqueeze(-2), B_gathered * self.gp_signs).squeeze(-2)
 
-    def grade_projection(self, mv: torch.Tensor, grade: int) -> torch.Tensor:
+    def grade_projection(self, mv: torch.Tensor, grade: int, **kwargs) -> torch.Tensor:
         """Isolates a specific grade.
 
         Uses multiplicative masking (mv * float_mask) instead of boolean
@@ -500,12 +531,15 @@ def grade_projection(self, mv: torch.Tensor, grade: int) -> torch.Tensor:
         Returns:
             torch.Tensor: Projected multivector [..., Dim].
         """
+        if kwargs:
+            kwargs.setdefault("output_grades", (int(grade),))
+            return self.planned_unary(mv, op="grade_projection", **kwargs)
         mask = self.grade_masks_float[grade]
         if mask.dtype != mv.dtype:
             mask = mask.to(dtype=mv.dtype)
         return mv * mask
 
-    def reverse(self, mv: torch.Tensor) -> torch.Tensor:
+    def reverse(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
         """Computes the reversion. The Clifford conjugate.
 
         Args:
@@ -514,15 +548,19 @@ def reverse(self, mv: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Reversed multivector [..., Dim].
         """
+        if kwargs:
+            return self.planned_unary(mv, op="reverse", **kwargs)
         rev = self.rev_signs
         if rev.dtype != mv.dtype:
             rev = rev.to(dtype=mv.dtype)
         return mv * rev
 
-    def wedge(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
-        """Computes the wedge (outer) product: A ^ B = (AB - BA)/2.
+    def wedge(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Computes the wedge/exterior product ``A ^ B``.
 
-        Single-pass implementation using precomputed antisymmetric signs.
+        For homogeneous inputs this is the grade-sum part of the
+        geometric product, ``<AB>_{grade(A)+grade(B)}``.  For vectors this
+        coincides with ``(AB - BA) / 2``.
 
         Reference:
             Pence, T., Yamada, D., & Singh, V. (2025). "Composing Linear Layers
@@ -535,6 +573,8 @@ def wedge(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Wedge product A ^ B [..., dim].
         """
+        if kwargs:
+            return self.projected_product(A, B, op="wedge", **kwargs)
         B_gathered = B[..., self.cayley_indices]
         return torch.matmul(A.unsqueeze(-2), B_gathered * self.wedge_gp_signs).squeeze(-2)
 
@@ -559,9 +599,7 @@ def right_contraction(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         bv_idx_exp = self._bv_indices.expand(*A.shape[:-1], -1)
         bv_coeffs = torch.gather(A, -1, bv_idx_exp)  # [..., num_bv]
 
-        # Grade-1 indices: powers of 2 for basis vectors
-        g1_idx = torch.arange(self.n, device=A.device)
-        g1_idx = (1 << g1_idx).long()  # [n]
+        g1_idx = self._basis_vector_indices(A.device)
         g1_idx_exp = g1_idx.expand(*B.shape[:-1], -1)
         v_coeffs = torch.gather(B, -1, g1_idx_exp)  # [..., n]
 
@@ -577,7 +615,19 @@ def right_contraction(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         result.scatter_(-1, g1_idx_exp, result_v)
         return result
 
-    def inner_product(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
+    def _basis_vector_indices(self, device) -> torch.Tensor:
+        indices = self._g1_indices
+        if indices.device != torch.device(device):
+            indices = indices.to(device=device)
+        return indices
+
+    def _bivector_indices_for(self, device) -> torch.Tensor:
+        indices = self._bv_indices
+        if indices.device != torch.device(device):
+            indices = indices.to(device=device)
+        return indices
+
+    def inner_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
         """Computes the inner product: A . B = (AB + BA)/2.
 
         Single-pass implementation using precomputed symmetric signs.
@@ -593,14 +643,15 @@ def inner_product(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Inner product A . B [..., dim].
         """
+        if kwargs:
+            return self.projected_product(A, B, op="inner", **kwargs)
         B_gathered = B[..., self.cayley_indices]
         return torch.matmul(A.unsqueeze(-2), B_gathered * self.inner_gp_signs).squeeze(-2)
 
-    def commutator(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
+    def commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
         """Computes the commutator (Lie bracket): [A, B] = AB - BA.
 
-        Single-pass implementation using precomputed antisymmetric signs
-        (same structure as :meth:`wedge` but without the 1/2 factor).
+        Single-pass implementation using precomputed antisymmetric signs.
 
         Args:
             A (torch.Tensor): Left operand [..., dim].
@@ -609,10 +660,12 @@ def commutator(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Commutator [A, B] [..., dim].
         """
+        if kwargs:
+            return self.projected_product(A, B, op="commutator", **kwargs)
         B_gathered = B[..., self.cayley_indices]
         return torch.matmul(A.unsqueeze(-2), B_gathered * self.comm_gp_signs).squeeze(-2)
 
-    def anti_commutator(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
+    def anti_commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
         """Computes the anti-commutator: {A, B} = AB + BA.
 
         Single-pass implementation using precomputed symmetric signs
@@ -625,6 +678,8 @@ def anti_commutator(self, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Anti-commutator {A, B} [..., dim].
         """
+        if kwargs:
+            return self.projected_product(A, B, op="anti_commutator", **kwargs)
         B_gathered = B[..., self.cayley_indices]
         return torch.matmul(A.unsqueeze(-2), B_gathered * self.anti_comm_gp_signs).squeeze(-2)
 
@@ -799,7 +854,7 @@ def blade_reject(self, mv: torch.Tensor, blade: torch.Tensor) -> torch.Tensor:
         """
         return mv - self.blade_project(mv, blade)
 
-    def grade_involution(self, mv: torch.Tensor) -> torch.Tensor:
+    def grade_involution(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
         """Grade involution (main involution): x_hat = sum (-1)^k <x>_k.
 
         Flips sign of all odd-grade components, preserves even-grade.
@@ -811,12 +866,14 @@ def grade_involution(self, mv: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Involuted multivector [..., dim].
         """
+        if kwargs:
+            return self.planned_unary(mv, op="grade_involution", **kwargs)
         signs = self._involution_signs
         if signs.dtype != mv.dtype:
             signs = signs.to(dtype=mv.dtype)
         return mv * signs
 
-    def clifford_conjugation(self, mv: torch.Tensor) -> torch.Tensor:
+    def clifford_conjugation(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
         """Clifford conjugation: bar{x} = grade_involution(reverse(x)).
 
         Combines reversion and grade involution. For a k-blade:
@@ -830,6 +887,8 @@ def clifford_conjugation(self, mv: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Conjugated multivector [..., dim].
         """
+        if kwargs:
+            return self.planned_unary(mv, op="clifford_conjugation", **kwargs)
         cs = self.conj_signs
         if cs.dtype != mv.dtype:
             cs = cs.to(dtype=mv.dtype)
@@ -1057,15 +1116,23 @@ def _exp_compiled_safe(self, B: torch.Tensor) -> torch.Tensor:
         Returns:
             torch.Tensor: Rotor exp(B) [..., dim].
         """
-        from core.decomposition import compiled_safe_decomposed_exp
+        from core.runtime.decomposition import compiled_safe_decomposed_exp
 
         R_closed = self._exp_bivector_closed(B)
         R_decomposed = compiled_safe_decomposed_exp(self, B, fixed_iterations=self._exp_fixed_iterations)
 
-        BB = self.geometric_product(B, B)
-        # Subtract scalar part, check if residual is negligible
-        scalar_part = self.grade_projection(BB, 0)
-        non_scalar_energy = (BB - scalar_part).norm(dim=-1, keepdim=True)
+        # For bivectors, B*B has only scalar and grade-4 components; the
+        # grade-4 energy is therefore the simplicity residual.
+        grade4 = self.projected_product(
+            B,
+            B,
+            op="gp",
+            left_grades=(2,),
+            right_grades=(2,),
+            output_grades=(4,),
+            compact_output=True,
+        )
+        non_scalar_energy = grade4.norm(dim=-1, keepdim=True)
         is_simple = non_scalar_energy < self.eps * 100
 
         return torch.where(is_simple, R_closed, R_decomposed)
diff --git a/core/runtime/context.py b/core/runtime/context.py
new file mode 100644
index 0000000..44c1630
--- /dev/null
+++ b/core/runtime/context.py
@@ -0,0 +1,155 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Lightweight algebra context for planned high-dimensional execution."""
+
+from __future__ import annotations
+
+from typing import Iterable, Optional
+
+import torch
+
+from core.foundation.basis import normalize_grades
+from core.foundation.device import resolve_device, resolve_dtype
+from core.foundation.layout import AlgebraSpec, GradeLayout
+from core.planning.planner import GradePlanner
+from core.planning.policy import DEFAULT_PLANNING_LIMITS, FULL_LAYOUT_MAX_N, PlanningLimits
+from core.runtime.projected import AlgebraRuntimeMixin
+
+
+class AlgebraContext(AlgebraRuntimeMixin):
+    """Signature and planning host without dense Cayley-table materialization."""
+
+    def __init__(
+        self,
+        p: int,
+        q: int = 0,
+        r: int = 0,
+        *,
+        device="cuda",
+        dtype: torch.dtype = torch.float32,
+        default_grades: Optional[Iterable[int]] = None,
+        allow_full_layout_products: Optional[bool] = None,
+        planning_limits: Optional[PlanningLimits] = None,
+    ):
+        if p < 0 or q < 0 or r < 0:
+            raise ValueError(f"signature counts must be non-negative, got Cl({p},{q},{r})")
+
+        self.p = int(p)
+        self.q = int(q)
+        self.r = int(r)
+        self.n = self.p + self.q + self.r
+        self.dim = 1 << self.n
+        self.num_grades = self.n + 1
+        self.spec = AlgebraSpec(self.p, self.q, self.r)
+        self._device = torch.device(resolve_device(device) if str(device) == "auto" else device)
+        self._dtype = resolve_dtype(dtype)
+        requested_full_layout = False if allow_full_layout_products is None else bool(allow_full_layout_products)
+        self.allow_full_layout_products = requested_full_layout and self.n <= FULL_LAYOUT_MAX_N
+        self.planning_limits = DEFAULT_PLANNING_LIMITS if planning_limits is None else planning_limits
+        self._default_grades = None if default_grades is None else normalize_grades(default_grades, self.n)
+        self._default_layout: Optional[GradeLayout] = None
+        self._g1_indices_cache: dict[str, torch.Tensor] = {}
+        self.planner = GradePlanner(self)
+        self._sync_eps()
+
+    @property
+    def device(self):
+        """Return the context device used for planned executor buffers."""
+        return self._device
+
+    @property
+    def dtype(self) -> torch.dtype:
+        """Return the context floating-point dtype."""
+        return self._dtype
+
+    def bivector_squared_signs(self, *, device=None, dtype: Optional[torch.dtype] = None) -> torch.Tensor:
+        """Return ``(e_ab)^2`` signs in canonical grade-2 layout order."""
+        return self.planner.bivector_squared_signs(
+            device=self.device if device is None else device,
+            dtype=self.dtype if dtype is None else dtype,
+        )
+
+    def _apply(self, fn):
+        """Apply a PyTorch module-style device/dtype transform to cached executors."""
+        probe = fn(torch.empty((), device=self.device, dtype=self.dtype))
+        self._device = probe.device
+        if probe.dtype.is_floating_point:
+            self._dtype = probe.dtype
+        self._sync_eps()
+        self._g1_indices_cache.clear()
+        self.planner._apply(fn)
+        return self
+
+    def to(self, device=None, dtype=None):
+        """Move the context and cached executors."""
+        if device is not None:
+            self._device = torch.device(resolve_device(device) if str(device) == "auto" else device)
+        if dtype is not None:
+            self._dtype = resolve_dtype(dtype)
+        self._sync_eps()
+        self._g1_indices_cache.clear()
+        self.planner.clear_cache()
+        return self
+
+    def geometric_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Plan and execute a geometric product."""
+        return self.projected_product(A, B, op="gp", **kwargs)
+
+    def wedge(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Plan and execute an exterior product."""
+        return self.projected_product(A, B, op="wedge", **kwargs)
+
+    def inner_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Plan and execute the symmetric inner product route."""
+        return self.projected_product(A, B, op="inner", **kwargs)
+
+    def commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Plan and execute a commutator product."""
+        return self.projected_product(A, B, op="commutator", **kwargs)
+
+    def anti_commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Plan and execute an anti-commutator product."""
+        return self.projected_product(A, B, op="anti_commutator", **kwargs)
+
+    def grade_projection(self, mv: torch.Tensor, grade: int, **kwargs) -> torch.Tensor:
+        """Project a dense multivector tensor to one grade."""
+        kwargs.setdefault("output_grades", (int(grade),))
+        return self.planned_unary(mv, op="grade_projection", **kwargs)
+
+    def embed_vector(self, vectors: torch.Tensor) -> torch.Tensor:
+        """Embed grade-1 vector coordinates into dense multivector coefficients."""
+        if vectors.shape[-1] != self.n:
+            raise ValueError(f"vectors last dimension must be {self.n}, got {vectors.shape[-1]}")
+        output = vectors.new_zeros(*vectors.shape[:-1], self.dim)
+        return output.index_copy(-1, self._basis_vector_indices(vectors.device), vectors)
+
+    def _basis_vector_indices(self, device) -> torch.Tensor:
+        resolved = torch.device(device)
+        key = str(resolved)
+        cached = self._g1_indices_cache.get(key)
+        if cached is None:
+            cached = torch.tensor([1 << bit for bit in range(self.n)], dtype=torch.long, device=resolved)
+            self._g1_indices_cache[key] = cached
+        return cached
+
+    def reverse(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Reverse dense or compact multivector coefficients."""
+        return self.planned_unary(mv, op="reverse", **kwargs)
+
+    def grade_involution(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply grade involution to dense or compact multivector coefficients."""
+        return self.planned_unary(mv, op="grade_involution", **kwargs)
+
+    def clifford_conjugation(self, mv: torch.Tensor, **kwargs) -> torch.Tensor:
+        """Apply Clifford conjugation to dense or compact multivector coefficients."""
+        return self.planned_unary(mv, op="clifford_conjugation", **kwargs)
+
+    def _sync_eps(self) -> None:
+        finfo = torch.finfo(self.dtype)
+        self.eps = float(finfo.eps)
+        self.eps_sq = float(finfo.eps**2)
diff --git a/core/decomposition.py b/core/runtime/decomposition.py
similarity index 94%
rename from core/decomposition.py
rename to core/runtime/decomposition.py
index a4880d4..9f57cd1 100644
--- a/core/decomposition.py
+++ b/core/runtime/decomposition.py
@@ -313,7 +313,7 @@ def compiled_safe_decomposed_exp(
     with torch.no_grad():
         decomp = _decompose_compiled_safe(algebra, b.detach(), k=k_actual, fixed_iterations=fixed_iterations)
 
-    bv_mask = algebra.grade_masks[2]
+    bv_indices = _bivector_indices(algebra, b.device)
 
     # Re-project live bivector and compose rotors
     result = identity
@@ -322,8 +322,8 @@ def compiled_safe_decomposed_exp(
         plane_norm = b_i_detached.norm(dim=-1, keepdim=True).clamp(min=algebra.eps_sq)
         plane_dir = b_i_detached / plane_norm
 
-        bv_live = residual[..., bv_mask]
-        plane_bv = plane_dir[..., bv_mask]
+        bv_live = torch.index_select(residual, -1, bv_indices)
+        plane_bv = torch.index_select(plane_dir, -1, bv_indices)
         coeff = (bv_live * plane_bv).sum(dim=-1, keepdim=True)
 
         b_i_live = coeff * plane_dir
@@ -333,3 +333,14 @@ def compiled_safe_decomposed_exp(
         result = algebra.geometric_product(result, R_i)
 
     return result
+
+
+def _bivector_indices(algebra, device) -> torch.Tensor:
+    if hasattr(algebra, "_bivector_indices_for"):
+        return algebra._bivector_indices_for(device)
+    indices = getattr(algebra, "_bv_indices", None)
+    if indices is not None:
+        if indices.device != torch.device(device):
+            indices = indices.to(device=device)
+        return indices
+    return algebra.grade_indices((2,), device=device)
diff --git a/core/runtime/metric.py b/core/runtime/metric.py
new file mode 100644
index 0000000..8d1c557
--- /dev/null
+++ b/core/runtime/metric.py
@@ -0,0 +1,517 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Metric definitions for Clifford algebras.
+
+Provides distances, norms, and inner products that respect
+the metric signature.
+"""
+
+from typing import Iterable, Optional
+
+import torch
+
+from core.foundation.layout import GradeLayout
+from core.foundation.module import AlgebraLike
+from core.runtime.accessors import compact_values
+from core.runtime.accessors import hermitian_signs as _layout_hermitian_signs
+
+
+def _hermitian_signs(
+    algebra: AlgebraLike,
+    layout: Optional[GradeLayout] = None,
+    *,
+    grades: Optional[Iterable[int]] = None,
+    device=None,
+    dtype: Optional[torch.dtype] = None,
+) -> torch.Tensor:
+    """Return Hermitian sign tensor for a dense or compact layout.
+
+    The Hermitian inner product on Cl(p,q) is:
+        <A, B>_H = sum_I (conj_sign_I * metric_sign_I) * a_I * b_I
+
+    This is precomputed as ``conj_signs * diagonal(cayley_signs)``
+    and registered as a buffer on the algebra in ``_init_derived_tables()``.
+
+    Returns:
+        Sign tensor [Dim] with values +1, -1, or 0 (null blades).
+    """
+    return _layout_hermitian_signs(algebra, layout=layout, grades=grades, device=device, dtype=dtype)
+
+
+def inner_product(algebra: AlgebraLike, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
+    """Compute the scalar product via projection onto grade 0.
+
+    Computes <A B>_0.
+
+    Args:
+        algebra (CliffordAlgebra): The algebra instance.
+        A (torch.Tensor): First multivector [Batch, Dim].
+        B (torch.Tensor): Second multivector [Batch, Dim].
+
+    Returns:
+        torch.Tensor: Scalar part [Batch, 1].
+    """
+    return algebra.projected_geometric_product(A, B, output_grades=(0,), compact_output=True)
+
+
+def induced_norm(algebra: AlgebraLike, A: torch.Tensor) -> torch.Tensor:
+    """Compute the induced norm respecting the metric signature.
+
+    Computes ||A|| = sqrt(|<A ~A>_0|).
+
+    Args:
+        algebra (CliffordAlgebra): The algebra instance.
+        A (torch.Tensor): Multivector [Batch, Dim].
+
+    Returns:
+        torch.Tensor: Norm [Batch, 1].
+    """
+    A_rev = algebra.reverse(A)
+    # Scalar product <A A~>_0
+    sq_norm = inner_product(algebra, A, A_rev)
+
+    # In mixed signatures, sq_norm can be negative.
+    # We return sqrt(|sq_norm|)
+    return torch.sqrt(torch.abs(sq_norm))
+
+
+def geometric_distance(algebra: AlgebraLike, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
+    """Computes geometric distance.
+
+    dist(A, B) = ||A - B||.
+
+    Args:
+        algebra (CliffordAlgebra): The algebra instance.
+        A (torch.Tensor): First multivector.
+        B (torch.Tensor): Second multivector.
+
+    Returns:
+        torch.Tensor: Distance.
+    """
+    diff = A - B
+    return induced_norm(algebra, diff)
+
+
+def grade_purity(algebra: AlgebraLike, A: torch.Tensor, grade: int) -> torch.Tensor:
+    """Checks the purity of the grade by examining coefficient energy.
+
+    Purity = ||<A>_k||^2 / ||A||^2.
+
+    Args:
+        algebra (CliffordAlgebra): The algebra instance.
+        A (torch.Tensor): Multivector [..., Dim].
+        grade (int): Target grade.
+
+    Returns:
+        torch.Tensor: Purity score [0, 1].
+    """
+    # Compute energies (using standard squared norm of coefficients for stability)
+    grade_masks = getattr(algebra, "grade_masks_float", None)
+    if grade_masks is not None and A.shape[-1] == getattr(algebra, "dim"):
+        mask = grade_masks[int(grade)]
+        if mask.device != A.device or mask.dtype != A.dtype:
+            mask = mask.to(device=A.device, dtype=A.dtype)
+        energy_k = (A * A * mask).sum(dim=-1)
+    else:
+        A_k = algebra.grade_projection(A, grade)
+        energy_k = (A_k**2).sum(dim=-1)
+    energy_total = (A**2).sum(dim=-1).clamp(min=algebra.eps)
+
+    return energy_k / energy_total
+
+
+def mean_active_grade(algebra: AlgebraLike, A: torch.Tensor) -> torch.Tensor:
+    """Average grade. Identifies the grade where the majority of the energy resides.
+
+    Mean Grade = Sum(k * ||<A>_k||^2) / ||A||^2.
+
+    Args:
+        algebra (CliffordAlgebra): The algebra instance.
+        A (torch.Tensor): Multivector.
+
+    Returns:
+        torch.Tensor: Average grade index.
+    """
+    grade_energies = _dense_grade_energies(algebra, A)
+    if grade_energies is None:
+        energy_total = (A**2).sum(dim=-1).clamp(min=algebra.eps)
+        weighted_sum = torch.zeros_like(energy_total)
+        for k in range(algebra.n + 1):
+            A_k = algebra.grade_projection(A, k)
+            energy_k = (A_k**2).sum(dim=-1)
+            weighted_sum += k * energy_k
+        return weighted_sum / energy_total
+
+    weights = torch.arange(algebra.n + 1, device=A.device, dtype=grade_energies.dtype)
+    weighted_sum = (grade_energies * weights).sum(dim=-1)
+    energy_total = grade_energies.sum(dim=-1).clamp(min=algebra.eps)
+    return weighted_sum / energy_total
+
+
+# Hermitian Metrics for Mixed-Signature Algebras
+#
+# In Cl(p,q) with q > 0, the standard norm <A~A>_0 can be negative
+# because basis blades involving negative-signature dimensions square
+# to -1. This breaks gradient-based optimization.
+#
+# The Hermitian inner product uses the algebraically proper formula:
+#
+#   <A, B>_H = <bar{A} B>_0 = Sum_I (conj_sign_I * metric_sign_I) * a_I * b_I
+#
+# where conj_sign_I is the Clifford conjugation sign and metric_sign_I
+# is the basis blade self-product sign. We precompute these signs once
+# via _hermitian_signs(). For Euclidean algebras Cl(p,0), all signs are
+# +1 and this reduces to the simple coefficient inner product.
+#
+# Additionally, we provide the Clifford conjugate (bar involution)
+# and the signature-aware trace form for algebraic computations.
+
+
+def clifford_conjugate(algebra: AlgebraLike, mv: torch.Tensor) -> torch.Tensor:
+    """Clifford conjugation (bar involution).
+
+    Combines reversion with grade involution:
+        A_bar_k = (-1)^k * (-1)^{k(k-1)/2} * A_k
+
+    This is the natural *-involution on Cl(p,q). Useful for
+    algebraic computations (e.g., spinor norms, Lipschitz groups).
+
+    Args:
+        algebra: The algebra instance.
+        mv: Multivector [..., Dim].
+
+    Returns:
+        Conjugated multivector [..., Dim].
+    """
+    return algebra.clifford_conjugation(mv)
+
+
+def hermitian_inner_product(
+    algebra: AlgebraLike,
+    A: torch.Tensor,
+    B: torch.Tensor,
+    *,
+    layout: Optional[GradeLayout] = None,
+    left_layout: Optional[GradeLayout] = None,
+    right_layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+    left_grades: Optional[Iterable[int]] = None,
+    right_grades: Optional[Iterable[int]] = None,
+) -> torch.Tensor:
+    """Hermitian inner product on Cl(p,q): <bar{A} B>_0.
+
+    <A, B>_H = Sum_I (conj_sign_I * metric_sign_I) * a_I * b_I
+
+    Uses precomputed sign arrays so that the result equals the scalar
+    part of the geometric product of the Clifford conjugate of A with B.
+    For Euclidean algebras (q=0), all signs are +1 and this reduces to
+    the simple coefficient inner product Sum a_I b_I.
+
+    Args:
+        algebra: The algebra instance.
+        A: First multivector [..., Dim].
+        B: Second multivector [..., Dim].
+
+    Returns:
+        Scalar inner product [..., 1].
+    """
+    A_values, B_values, resolved = _aligned_pair_values(
+        algebra,
+        A,
+        B,
+        layout=layout,
+        left_layout=left_layout,
+        right_layout=right_layout,
+        grades=grades,
+        left_grades=left_grades,
+        right_grades=right_grades,
+    )
+    return _hermitian_inner_values(algebra, A_values, B_values, resolved)
+
+
+def hermitian_norm(
+    algebra: AlgebraLike,
+    A: torch.Tensor,
+    *,
+    layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+) -> torch.Tensor:
+    """Hermitian norm: ||A||_H = sqrt(|<A, A>_H|).
+
+    Always real and non-negative for any signature.
+    Uses abs() since the signed inner product can produce negative
+    self-products in mixed-signature algebras.
+
+    Args:
+        algebra: The algebra instance.
+        A: Multivector [..., Dim].
+
+    Returns:
+        Norm [..., 1]. Always >= 0.
+    """
+    values, resolved = compact_values(algebra, A, layout=layout, grades=grades)
+    sq = _hermitian_inner_values(algebra, values, values, resolved)
+    return torch.sqrt(torch.abs(sq))
+
+
+def hermitian_distance(
+    algebra: AlgebraLike,
+    A: torch.Tensor,
+    B: torch.Tensor,
+    *,
+    layout: Optional[GradeLayout] = None,
+    left_layout: Optional[GradeLayout] = None,
+    right_layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+    left_grades: Optional[Iterable[int]] = None,
+    right_grades: Optional[Iterable[int]] = None,
+) -> torch.Tensor:
+    """Hermitian distance: d_H(A, B) = ||A - B||_H.
+
+    Positive-definite metric distance for any signature.
+    Satisfies: non-negativity, symmetry, triangle inequality, identity.
+
+    Args:
+        algebra: The algebra instance.
+        A: First multivector [..., Dim].
+        B: Second multivector [..., Dim].
+
+    Returns:
+        Distance [..., 1]. Always >= 0.
+    """
+    A_values, B_values, resolved = _aligned_pair_values(
+        algebra,
+        A,
+        B,
+        layout=layout,
+        left_layout=left_layout,
+        right_layout=right_layout,
+        grades=grades,
+        left_grades=left_grades,
+        right_grades=right_grades,
+    )
+    diff = A_values - B_values
+    sq = _hermitian_inner_values(algebra, diff, diff, resolved)
+    return torch.sqrt(torch.abs(sq))
+
+
+def hermitian_angle(
+    algebra: AlgebraLike,
+    A: torch.Tensor,
+    B: torch.Tensor,
+    *,
+    layout: Optional[GradeLayout] = None,
+    left_layout: Optional[GradeLayout] = None,
+    right_layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+    left_grades: Optional[Iterable[int]] = None,
+    right_grades: Optional[Iterable[int]] = None,
+) -> torch.Tensor:
+    """Hermitian angle between multivectors.
+
+    cos(theta) = <A, B>_H / (||A||_H * ||B||_H)
+
+    Args:
+        algebra: The algebra instance.
+        A: First multivector [..., Dim].
+        B: Second multivector [..., Dim].
+
+    Returns:
+        Angle in radians [..., 1].
+    """
+    A_values, B_values, resolved = _aligned_pair_values(
+        algebra,
+        A,
+        B,
+        layout=layout,
+        left_layout=left_layout,
+        right_layout=right_layout,
+        grades=grades,
+        left_grades=left_grades,
+        right_grades=right_grades,
+    )
+    signs = _signs_like(algebra, resolved, A_values, B_values)
+    ip = (signs * A_values * B_values).sum(dim=-1, keepdim=True)
+    sq_a = (signs * A_values * A_values).sum(dim=-1, keepdim=True)
+    sq_b = (signs * B_values * B_values).sum(dim=-1, keepdim=True)
+    # Use sqrt(sq_a * sq_b) instead of sqrt(sq_a)*sqrt(sq_b) to avoid
+    # float32 precision loss from two separate sqrt operations.
+    denom = torch.sqrt(torch.abs(sq_a) * torch.abs(sq_b)).clamp(min=algebra.eps)
+    cos_theta = ip / denom
+    cos_theta = torch.clamp(cos_theta, -1.0, 1.0)
+    return torch.acos(cos_theta)
+
+
+def grade_hermitian_norm(
+    algebra: AlgebraLike,
+    A: torch.Tensor,
+    grade: int,
+    *,
+    layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+) -> torch.Tensor:
+    """Hermitian norm restricted to a single grade.
+
+    ||<A>_k||_H = sqrt(Sum_{I: |I|=k} a_I**2)
+
+    Measures the energy contribution of a specific grade
+    in a signature-independent way.
+
+    Args:
+        algebra: The algebra instance.
+        A: Multivector [..., Dim].
+        grade: Target grade.
+
+    Returns:
+        Grade-specific norm [..., 1].
+    """
+    values, source_layout = compact_values(algebra, A, layout=layout, grades=grades)
+    grade_layout = algebra.layout((int(grade),))
+    grade_values = grade_layout.convert(values, source_layout)
+    sq = _hermitian_inner_values(algebra, grade_values, grade_values, grade_layout)
+    return torch.sqrt(torch.abs(sq))
+
+
+def hermitian_grade_spectrum(
+    algebra: AlgebraLike,
+    A: torch.Tensor,
+    *,
+    layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+) -> torch.Tensor:
+    """Full Hermitian grade spectrum.
+
+    Returns |<A_k, A_k>_H| for each grade k = 0, ..., n.
+    Uses abs() to ensure non-negative values in mixed signatures.
+
+    Args:
+        algebra: The algebra instance.
+        A: Multivector [..., Dim].
+
+    Returns:
+        Grade energies [..., n+1]. Each entry >= 0.
+    """
+    values, source_layout = compact_values(algebra, A, layout=layout, grades=grades)
+    signs = _signs_like(algebra, source_layout, values, values)
+    signed_energy = signs * values * values
+    flat = signed_energy.reshape(-1, source_layout.dim)
+    grade_ids = source_layout.grade_indices_tensor(device=values.device).unsqueeze(0).expand_as(flat)
+    spectrum = signed_energy.new_zeros(flat.shape[0], algebra.n + 1)
+    spectrum.scatter_add_(1, grade_ids, flat)
+    return spectrum.reshape(*values.shape[:-1], algebra.n + 1).abs()
+
+
+def _aligned_pair_values(
+    algebra: AlgebraLike,
+    A,
+    B,
+    *,
+    layout: Optional[GradeLayout] = None,
+    left_layout: Optional[GradeLayout] = None,
+    right_layout: Optional[GradeLayout] = None,
+    grades: Optional[Iterable[int]] = None,
+    left_grades: Optional[Iterable[int]] = None,
+    right_grades: Optional[Iterable[int]] = None,
+) -> tuple[torch.Tensor, torch.Tensor, GradeLayout]:
+    """Compact two values into one static layout without dense materialization."""
+    shared_left_layout = left_layout if left_layout is not None else layout
+    shared_right_layout = right_layout if right_layout is not None else layout
+    shared_left_grades = left_grades if left_grades is not None else grades
+    shared_right_grades = right_grades if right_grades is not None else grades
+
+    A_values, A_layout = compact_values(algebra, A, layout=shared_left_layout, grades=shared_left_grades)
+    B_values, B_layout = compact_values(algebra, B, layout=shared_right_layout, grades=shared_right_grades)
+    resolved = A_layout if A_layout == B_layout else _union_layout(algebra, A_layout, B_layout)
+    if A_layout != resolved:
+        A_values = resolved.convert(A_values, A_layout)
+    if B_layout != resolved:
+        B_values = resolved.convert(B_values, B_layout)
+    return A_values, B_values, resolved
+
+
+def _union_layout(algebra: AlgebraLike, left: GradeLayout, right: GradeLayout) -> GradeLayout:
+    basis = set(left.basis_indices).union(right.basis_indices)
+    grades = sorted({index.bit_count() for index in basis})
+    return algebra.layout(grades)
+
+
+def _hermitian_inner_values(
+    algebra: AlgebraLike,
+    A_values: torch.Tensor,
+    B_values: torch.Tensor,
+    layout: GradeLayout,
+) -> torch.Tensor:
+    signs = _signs_like(algebra, layout, A_values, B_values)
+    return (signs * A_values * B_values).sum(dim=-1, keepdim=True)
+
+
+def _signs_like(
+    algebra: AlgebraLike,
+    layout: GradeLayout,
+    A_values: torch.Tensor,
+    B_values: torch.Tensor,
+) -> torch.Tensor:
+    dtype = torch.promote_types(A_values.dtype, B_values.dtype)
+    return _hermitian_signs(algebra, layout=layout, device=A_values.device, dtype=dtype)
+
+
+def _dense_grade_energies(algebra: AlgebraLike, A: torch.Tensor) -> Optional[torch.Tensor]:
+    grade_index = getattr(algebra, "grade_index", None)
+    if grade_index is None or A.shape[-1] != getattr(algebra, "dim"):
+        return None
+    if grade_index.device != A.device:
+        grade_index = grade_index.to(device=A.device)
+
+    flat = (A * A).reshape(-1, algebra.dim)
+    idx = grade_index.unsqueeze(0).expand_as(flat)
+    energies = flat.new_zeros(flat.shape[0], algebra.n + 1)
+    energies.scatter_add_(1, idx, flat)
+    return energies.reshape(*A.shape[:-1], algebra.n + 1)
+
+
+def signature_trace_form(algebra: AlgebraLike, A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
+    """Signature-aware trace form: <~A B>_0.
+
+    The standard Clifford algebra scalar product. NOT positive-definite
+    in mixed signatures. Use hermitian_inner_product for optimization.
+
+    This form is signature-aware and useful for:
+    - Rotor normalization (R~R = 1)
+    - Versor validation
+    - Spinor norm computation
+
+    Args:
+        algebra: The algebra instance.
+        A: First multivector [..., Dim].
+        B: Second multivector [..., Dim].
+
+    Returns:
+        Scalar trace form [..., 1]. Can be negative in mixed signatures.
+    """
+    A_rev = algebra.reverse(A)
+    prod = algebra.geometric_product(A_rev, B)
+    return prod[..., 0:1]
+
+
+def signature_norm_squared(algebra: AlgebraLike, A: torch.Tensor) -> torch.Tensor:
+    """Signature-aware squared norm: <A~A>_0.
+
+    Can be negative in mixed-signature algebras. Returns the raw value
+    without absolute value, preserving causal structure information.
+
+    For Cl(n,0): always non-negative.
+    For Cl(p,q) with q>0: sign encodes causal character.
+
+    Args:
+        algebra: The algebra instance.
+        A: Multivector [..., Dim].
+
+    Returns:
+        Signed squared norm [..., 1].
+    """
+    return signature_trace_form(algebra, A, A)
diff --git a/core/runtime/multivector.py b/core/runtime/multivector.py
new file mode 100644
index 0000000..9fc2231
--- /dev/null
+++ b/core/runtime/multivector.py
@@ -0,0 +1,563 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Object-oriented multivector wrapper with operator overloading."""
+
+from __future__ import annotations
+
+import torch
+
+from core.foundation.layout import GradeLayout
+from core.foundation.module import AlgebraLike
+from core.runtime.accessors import as_multivector as _as_multivector
+from core.runtime.accessors import materialize_dense
+
+
+class Multivector:
+    """Object-oriented multivector wrapper with operator overloading.
+
+    Wraps a raw coefficient tensor and its parent algebra kernel,
+    exposing every core algebra operation as a method or Python operator.
+
+    Attributes:
+        algebra (AlgebraLike): The backend.
+        tensor (torch.Tensor): Dense coefficients [..., Dim].
+        values (torch.Tensor): Optional compact lane values [..., layout.dim].
+        layout (GradeLayout): Optional compact layout for ``values``.
+    """
+
+    __slots__ = ("_tensor", "algebra", "layout", "values")
+
+    def __init__(
+        self,
+        algebra: AlgebraLike,
+        tensor: torch.Tensor = None,
+        *,
+        values: torch.Tensor = None,
+        layout: GradeLayout = None,
+    ):
+        self.algebra = algebra
+        self.layout = layout
+        if layout is None:
+            if tensor is None:
+                raise ValueError("tensor is required when layout is not provided")
+            self._tensor = tensor
+            self.values = None
+        else:
+            self._check_layout(layout)
+            if values is None:
+                if tensor is None:
+                    raise ValueError("values or tensor is required when layout is provided")
+                values = layout.compact(tensor)
+            if values.shape[-1] != layout.dim:
+                raise ValueError(f"compact values last dimension must be {layout.dim}, got {values.shape[-1]}")
+            self._tensor = None
+            self.values = values
+
+    @classmethod
+    def from_tensor(
+        cls,
+        algebra: AlgebraLike,
+        tensor: torch.Tensor,
+        *,
+        grades=None,
+        layout: GradeLayout = None,
+    ) -> Multivector:
+        """Wrap dense coefficients or compact coefficients with declared layout metadata."""
+        if layout is None and grades is None:
+            return cls(algebra, tensor)
+        return _as_multivector(algebra, tensor, layout=layout, grades=grades)
+
+    @classmethod
+    def from_compact(
+        cls,
+        algebra: AlgebraLike,
+        values: torch.Tensor,
+        *,
+        grades=None,
+        layout: GradeLayout = None,
+    ) -> Multivector:
+        """Create a compact multivector from active lane values."""
+        if layout is None and grades is None:
+            raise ValueError("layout or grades is required for compact multivectors")
+        resolved = algebra.resolve_layout(layout=layout, grades=grades, warn_full=False)
+        return cls(algebra, values=values, layout=resolved)
+
+    @classmethod
+    def from_vectors(cls, algebra: AlgebraLike, vectors: torch.Tensor) -> Multivector:
+        """Promote vectors to compact grade-1 multivectors."""
+        layout = algebra.layout((1,))
+        if vectors.shape[-1] != layout.dim:
+            raise ValueError(f"vectors last dimension must be {layout.dim}, got {vectors.shape[-1]}")
+        return cls(algebra, values=vectors, layout=layout)
+
+    @classmethod
+    def scalar(
+        cls, algebra: AlgebraLike, value: float | torch.Tensor, batch_shape: tuple[int, ...] = ()
+    ) -> Multivector:
+        """Creates a scalar multivector (grade 0 only)."""
+        layout = algebra.layout((0,))
+        values = torch.as_tensor(value, device=algebra.device, dtype=algebra.dtype)
+        if batch_shape:
+            target_shape = torch.Size(batch_shape)
+            if values.ndim == 0:
+                values = values.expand(*batch_shape).clone()
+            elif values.shape == torch.Size((*batch_shape, 1)):
+                return cls(algebra, values=values, layout=layout)
+            elif values.shape != target_shape:
+                values = values.expand(*batch_shape).clone()
+        if values.ndim == 0:
+            values = values.reshape(1)
+        elif values.shape[-1] != 1:
+            values = values.unsqueeze(-1)
+        return cls(algebra, values=values, layout=layout)
+
+    def __repr__(self):
+        storage = "compact" if self.is_compact else "dense"
+        return (
+            f"Multivector(shape={self.shape}, storage={storage}, "
+            f"algebra=Cl({self.algebra.p},{self.algebra.q},{self.algebra.r}))"
+        )
+
+    @property
+    def tensor(self) -> torch.Tensor:
+        """Dense coefficient tensor.
+
+        This property is an explicit dense boundary. Planned paths that operate
+        on compact data should use ``values`` or ``coefficients`` directly.
+        """
+        if self._tensor is not None:
+            return self._tensor
+        # Do not call this inside core operations that can preserve compact
+        # ``values`` and ``layout``; materialization belongs at API boundaries.
+        return materialize_dense(self.algebra, self)
+
+    @tensor.setter
+    def tensor(self, value: torch.Tensor) -> None:
+        self._tensor = value
+        self.values = None
+        self.layout = None
+
+    @property
+    def is_compact(self) -> bool:
+        """Whether this multivector stores compact grade lanes."""
+        return self.layout is not None
+
+    @property
+    def is_dense(self) -> bool:
+        """Whether this multivector stores dense coefficients."""
+        return not self.is_compact
+
+    @property
+    def grades(self) -> tuple[int, ...] | None:
+        """Active grades when compact metadata is available."""
+        return None if self.layout is None else self.layout.grades
+
+    @property
+    def lane_count(self) -> int:
+        """Number of stored coefficient lanes."""
+        return self.coefficients.shape[-1]
+
+    @property
+    def storage(self) -> str:
+        """Storage mode name."""
+        return "compact" if self.is_compact else "dense"
+
+    @property
+    def coefficients(self) -> torch.Tensor:
+        """Return the active storage tensor without dense materialization."""
+        return self.values if self.is_compact else self._tensor
+
+    def dense(self) -> Multivector:
+        """Return a dense-storage multivector."""
+        return Multivector(self.algebra, self.tensor)
+
+    def compact(self, grades) -> Multivector:
+        """Return a compact-storage multivector containing ``grades``."""
+        layout = self.algebra.layout(grades)
+        return self.with_layout(layout)
+
+    def with_layout(self, layout: GradeLayout) -> Multivector:
+        """Return this multivector represented by ``layout``."""
+        self._check_layout(layout)
+        if self.layout == layout:
+            return Multivector(self.algebra, values=self.values, layout=layout)
+        if self.is_compact:
+            return Multivector(self.algebra, values=layout.convert(self.values, self.layout), layout=layout)
+        return Multivector(self.algebra, values=layout.compact(self.tensor), layout=layout)
+
+    def _check_layout(self, layout: GradeLayout) -> None:
+        spec = layout.spec
+        if (spec.p, spec.q, spec.r) != (self.algebra.p, self.algebra.q, self.algebra.r):
+            raise ValueError(
+                f"Layout mismatch: Cl({spec.p},{spec.q},{spec.r}) vs "
+                f"Cl({self.algebra.p},{self.algebra.q},{self.algebra.r})"
+            )
+
+    def _check_algebra(self, other: Multivector) -> None:
+        s, o = self.algebra, other.algebra
+        if (s.p, s.q, s.r) != (o.p, o.q, o.r):
+            raise ValueError(f"Algebra mismatch: Cl({s.p},{s.q},{s.r}) vs Cl({o.p},{o.q},{o.r})")
+
+    def _wrap(self, tensor: torch.Tensor) -> Multivector:
+        return Multivector(self.algebra, tensor)
+
+    def _wrap_compact(self, values: torch.Tensor, layout: GradeLayout) -> Multivector:
+        return Multivector(self.algebra, values=values, layout=layout)
+
+    def _values_for_layout(self, layout: GradeLayout) -> torch.Tensor:
+        self._check_layout(layout)
+        if self.is_compact:
+            return layout.convert(self.values, self.layout)
+        return layout.compact(self.tensor)
+
+    def _combined_layout(self, other: Multivector) -> GradeLayout:
+        left = self.layout if self.is_compact else self.algebra.layout()
+        right = other.layout if other.is_compact else other.algebra.layout()
+        basis = set(left.basis_indices).union(right.basis_indices)
+        grades = sorted({index.bit_count() for index in basis})
+        return self.algebra.layout(grades)
+
+    def __add__(self, other):
+        if isinstance(other, Multivector):
+            self._check_algebra(other)
+            if self.is_compact or other.is_compact:
+                layout = self._combined_layout(other)
+                return self._wrap_compact(self._values_for_layout(layout) + other._values_for_layout(layout), layout)
+            return self._wrap(self.tensor + other.tensor)
+        if isinstance(other, (int, float, torch.Tensor)):
+            if self.is_compact:
+                return self._wrap_compact(self.values + other, self.layout)
+            return self._wrap(self.tensor + other)
+        return NotImplemented
+
+    def __radd__(self, other):
+        return self.__add__(other)
+
+    def __sub__(self, other):
+        if isinstance(other, Multivector):
+            self._check_algebra(other)
+            if self.is_compact or other.is_compact:
+                layout = self._combined_layout(other)
+                return self._wrap_compact(self._values_for_layout(layout) - other._values_for_layout(layout), layout)
+            return self._wrap(self.tensor - other.tensor)
+        if isinstance(other, (int, float, torch.Tensor)):
+            if self.is_compact:
+                return self._wrap_compact(self.values - other, self.layout)
+            return self._wrap(self.tensor - other)
+        return NotImplemented
+
+    def __rsub__(self, other):
+        if isinstance(other, (int, float, torch.Tensor)):
+            if self.is_compact:
+                return self._wrap_compact(other - self.values, self.layout)
+            return self._wrap(other - self.tensor)
+        return NotImplemented
+
+    def __neg__(self):
+        if self.is_compact:
+            return self._wrap_compact(-self.values, self.layout)
+        return self._wrap(-self.tensor)
+
+    def __mul__(self, other):
+        """Geometric product ``A * B``, or scalar scaling."""
+        if isinstance(other, Multivector):
+            self._check_algebra(other)
+            return self.geometric_product(other)
+        if isinstance(other, (int, float)):
+            if self.is_compact:
+                return self._wrap_compact(self.values * other, self.layout)
+            return self._wrap(self.tensor * other)
+        if isinstance(other, torch.Tensor):
+            if self.is_compact:
+                return self._wrap_compact(self.values * other, self.layout)
+            return self._wrap(self.tensor * other)
+        return NotImplemented
+
+    def __rmul__(self, other):
+        if isinstance(other, (int, float, torch.Tensor)):
+            if self.is_compact:
+                return self._wrap_compact(self.values * other, self.layout)
+            return self._wrap(self.tensor * other)
+        return NotImplemented
+
+    def __truediv__(self, other):
+        if isinstance(other, (int, float)):
+            if self.is_compact:
+                return self._wrap_compact(self.values / other, self.layout)
+            return self._wrap(self.tensor / other)
+        if isinstance(other, torch.Tensor):
+            if self.is_compact:
+                return self._wrap_compact(self.values / other, self.layout)
+            return self._wrap(self.tensor / other)
+        return NotImplemented
+
+    def __xor__(self, other):
+        """Wedge/exterior product ``A ^ B``."""
+        if isinstance(other, Multivector):
+            self._check_algebra(other)
+            return self.wedge(other)
+        return NotImplemented
+
+    def __or__(self, other):
+        """Inner product ``A | B``."""
+        if isinstance(other, Multivector):
+            self._check_algebra(other)
+            return self.inner(other)
+        return NotImplemented
+
+    def __invert__(self):
+        """Reversion ``~A``."""
+        return self.reverse()
+
+    def grade(self, k: int) -> Multivector:
+        """Extract the grade-k component."""
+        if self.is_compact:
+            layout = self.algebra.layout((int(k),))
+            if not self.layout.contains_grade(k):
+                values = self.values.new_zeros(*self.values.shape[:-1], layout.dim)
+                return self._wrap_compact(values, layout)
+            values, output_layout = self.algebra.planned_unary(
+                self.values,
+                op="grade_projection",
+                input_layout=self.layout,
+                output_layout=layout,
+                input_compact=True,
+                compact_output=True,
+                return_layout=True,
+            )
+            return self._wrap_compact(values, output_layout)
+        return self._wrap(self.algebra.grade_projection(self.tensor, k))
+
+    def reverse(self) -> Multivector:
+        """Reversion (same as ``~self``)."""
+        if self.is_compact:
+            values, layout = self.algebra.planned_unary(
+                self.values,
+                op="reverse",
+                input_layout=self.layout,
+                input_compact=True,
+                compact_output=True,
+                return_layout=True,
+            )
+            return self._wrap_compact(values, layout)
+        return self._wrap(self.algebra.reverse(self.tensor))
+
+    def grade_involution(self) -> Multivector:
+        """Grade involution (main involution): flips odd-grade signs."""
+        if self.is_compact:
+            values, layout = self.algebra.planned_unary(
+                self.values,
+                op="grade_involution",
+                input_layout=self.layout,
+                input_compact=True,
+                compact_output=True,
+                return_layout=True,
+            )
+            return self._wrap_compact(values, layout)
+        return self._wrap(self.algebra.grade_involution(self.tensor))
+
+    def clifford_conjugation(self) -> Multivector:
+        """Clifford conjugation: grade_involution(reverse(x))."""
+        if self.is_compact:
+            values, layout = self.algebra.planned_unary(
+                self.values,
+                op="clifford_conjugation",
+                input_layout=self.layout,
+                input_compact=True,
+                compact_output=True,
+                return_layout=True,
+            )
+            return self._wrap_compact(values, layout)
+        return self._wrap(self.algebra.clifford_conjugation(self.tensor))
+
+    def dual(self) -> Multivector:
+        """Hodge dual: maps grade-k to grade-(n-k)."""
+        # Dense-only algebra APIs below intentionally cross the dense boundary.
+        # Planned high-dimensional paths should use grade-declared primitives.
+        return self._wrap(self.algebra.dual(self.tensor))
+
+    def inverse(self) -> Multivector:
+        """Blade inverse: B^{-1} = ~B / <B~B>_0."""
+        return self._wrap(self.algebra.blade_inverse(self.tensor))
+
+    def geometric_product(self, other: Multivector) -> Multivector:
+        """Explicit geometric product (same as ``self * other``)."""
+        self._check_algebra(other)
+        if self.is_compact or other.is_compact:
+            return self.projected_product(other, op="gp")
+        return self._wrap(self.algebra.geometric_product(self.tensor, other.tensor))
+
+    def projected_product(
+        self,
+        other: Multivector,
+        *,
+        output_grades=None,
+        op: str = "gp",
+        left_grades=None,
+        right_grades=None,
+    ) -> Multivector:
+        """Grade-projected product using compact layouts when available."""
+        self._check_algebra(other)
+        left_layout = self.layout if self.is_compact else None
+        right_layout = other.layout if other.is_compact else None
+        left_grades = left_grades if left_grades is not None else _layout_grades(left_layout)
+        right_grades = right_grades if right_grades is not None else _layout_grades(right_layout)
+
+        values, layout = self.algebra.projected_product(
+            self.values if self.is_compact else self.tensor,
+            other.values if other.is_compact else other.tensor,
+            left_grades=left_grades,
+            right_grades=right_grades,
+            output_grades=output_grades,
+            left_layout=left_layout,
+            right_layout=right_layout,
+            op=op,
+            left_compact=self.is_compact,
+            right_compact=other.is_compact,
+            return_layout=True,
+        )
+        return self._wrap_compact(values, layout)
+
+    def wedge(self, other: Multivector) -> Multivector:
+        """Wedge/exterior product (same as ``self ^ other``)."""
+        self._check_algebra(other)
+        if self.is_compact or other.is_compact:
+            return self.projected_product(other, op="wedge")
+        return self._wrap(self.algebra.wedge(self.tensor, other.tensor))
+
+    def inner(self, other: Multivector) -> Multivector:
+        """Inner product (same as ``self | other``)."""
+        self._check_algebra(other)
+        if self.is_compact or other.is_compact:
+            return self.projected_product(other, op="inner")
+        return self._wrap(self.algebra.inner_product(self.tensor, other.tensor))
+
+    def left_contraction(self, other: Multivector) -> Multivector:
+        """Left contraction: ``self _| other``."""
+        self._check_algebra(other)
+        return self._wrap(self.algebra.left_contraction(self.tensor, other.tensor))
+
+    def right_contraction(self, other: Multivector) -> Multivector:
+        """Right contraction: ``self |_ other``."""
+        self._check_algebra(other)
+        return self._wrap(self.algebra.right_contraction(self.tensor, other.tensor))
+
+    def commutator(self, other: Multivector) -> Multivector:
+        """Commutator (Lie bracket): ``[self, other] = self*other - other*self``."""
+        self._check_algebra(other)
+        if self.is_compact or other.is_compact:
+            return self.projected_product(other, op="commutator")
+        return self._wrap(self.algebra.commutator(self.tensor, other.tensor))
+
+    def anti_commutator(self, other: Multivector) -> Multivector:
+        """Anti-commutator: ``{self, other} = self*other + other*self``."""
+        self._check_algebra(other)
+        if self.is_compact or other.is_compact:
+            return self.projected_product(other, op="anti_commutator")
+        return self._wrap(self.algebra.anti_commutator(self.tensor, other.tensor))
+
+    def norm(self) -> torch.Tensor:
+        """Induced metric norm (returns scalar tensor)."""
+        from core.runtime.metric import induced_norm
+
+        return induced_norm(self.algebra, self.tensor)
+
+    def norm_sq(self) -> torch.Tensor:
+        """Squared norm: <x * ~x>_0 (returns scalar tensor)."""
+        return self.algebra.norm_sq(self.tensor)
+
+    def get_grade_norms(self) -> torch.Tensor:
+        """Per-grade L2 norms."""
+        if self.is_compact:
+            flat = self.values.pow(2).reshape(-1, self.layout.dim)
+            grade_ids = self.layout.grade_indices_tensor(device=self.values.device).unsqueeze(0).expand_as(flat)
+            result = flat.new_zeros(flat.shape[0], self.algebra.num_grades)
+            result.scatter_add_(1, grade_ids, flat)
+            return result.reshape(*self.values.shape[:-1], self.algebra.num_grades).clamp(min=self.algebra.eps).sqrt()
+        return self.algebra.get_grade_norms(self.tensor)
+
+    def exp(self) -> Multivector:
+        """Exponential map (bivector -> rotor)."""
+        return self._wrap(self.algebra.exp(self.tensor))
+
+    def sandwich(self, x: Multivector) -> Multivector:
+        """Sandwich product: ``self * x * ~self``.
+
+        Falls back to two geometric products when the tensor shapes
+        don't match the optimized [N, D] + [N, C, D] layout.
+        """
+        self._check_algebra(x)
+        R, xt = self.tensor, x.tensor
+        # Optimized path: R is [N, D], x is [N, C, D]
+        if R.dim() == 2 and xt.dim() == 3:
+            return self._wrap(self.algebra.sandwich_product(R, xt))
+        # General fallback: two GPs
+        R_rev = self.algebra.reverse(R)
+        return self._wrap(self.algebra.geometric_product(self.algebra.geometric_product(R, xt), R_rev))
+
+    def reflect(self, n: Multivector) -> Multivector:
+        """Reflect self through hyperplane orthogonal to vector n."""
+        self._check_algebra(n)
+        return self._wrap(self.algebra.reflect(self.tensor, n.tensor))
+
+    def versor_product(self, x: Multivector) -> Multivector:
+        """General versor action: ``hat(self) * x * self^{-1}``."""
+        self._check_algebra(x)
+        return self._wrap(self.algebra.versor_product(self.tensor, x.tensor))
+
+    def blade_project(self, blade: Multivector) -> Multivector:
+        """Project onto blade subspace: ``(self · B) B^{-1}``."""
+        self._check_algebra(blade)
+        return self._wrap(self.algebra.blade_project(self.tensor, blade.tensor))
+
+    def blade_reject(self, blade: Multivector) -> Multivector:
+        """Reject from blade subspace: ``self - proj_B(self)``."""
+        self._check_algebra(blade)
+        return self._wrap(self.algebra.blade_reject(self.tensor, blade.tensor))
+
+    def to(self, *args, **kwargs) -> Multivector:
+        """Move/cast the underlying tensor (same API as ``torch.Tensor.to``)."""
+        if self.is_compact:
+            return Multivector(self.algebra, values=self.values.to(*args, **kwargs), layout=self.layout)
+        return self._wrap(self.tensor.to(*args, **kwargs))
+
+    def detach(self) -> Multivector:
+        """Detach from computation graph."""
+        if self.is_compact:
+            return Multivector(self.algebra, values=self.values.detach(), layout=self.layout)
+        return self._wrap(self.tensor.detach())
+
+    def clone(self) -> Multivector:
+        """Clone the underlying tensor."""
+        if self.is_compact:
+            return Multivector(self.algebra, values=self.values.clone(), layout=self.layout)
+        return self._wrap(self.tensor.clone())
+
+    def requires_grad_(self, requires_grad: bool = True) -> Multivector:
+        """Set requires_grad in-place."""
+        if self.is_compact:
+            self.values.requires_grad_(requires_grad)
+        else:
+            self.tensor.requires_grad_(requires_grad)
+        return self
+
+    @property
+    def shape(self) -> torch.Size:
+        return self.values.shape if self.is_compact else self.tensor.shape
+
+    @property
+    def device(self) -> torch.device:
+        return self.values.device if self.is_compact else self.tensor.device
+
+    @property
+    def dtype(self) -> torch.dtype:
+        return self.values.dtype if self.is_compact else self.tensor.dtype
+
+
+def _layout_grades(layout: GradeLayout) -> tuple[int, ...] | None:
+    return None if layout is None else layout.grades
diff --git a/core/runtime/projected.py b/core/runtime/projected.py
new file mode 100644
index 0000000..da3e863
--- /dev/null
+++ b/core/runtime/projected.py
@@ -0,0 +1,261 @@
+# Versor: Universal Geometric Algebra Neural Network
+# Copyright (C) 2026 Eunkyum Kim <nemonanconcode@gmail.com>
+#
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+#
+
+"""Shared projected-product facade for algebra hosts."""
+
+from __future__ import annotations
+
+from typing import Iterable, Optional
+
+import torch
+
+from core.foundation.layout import GradeLayout
+from core.foundation.validation import check_multivector
+from core.runtime.accessors import default_layout as _default_layout
+from core.runtime.accessors import grade_indices as _grade_indices
+from core.runtime.accessors import hermitian_signs as _hermitian_signs
+from core.runtime.accessors import materialize_dense
+from core.runtime.accessors import resolve_layout as _resolve_layout
+
+
+class AlgebraRuntimeMixin:
+    """Shared runtime protocol for dense kernels and planned contexts."""
+
+    def layout(self, grades: Optional[Iterable[int]] = None) -> GradeLayout:
+        """Return a compact grade layout or the algebra's default layout."""
+        if grades is None:
+            return self.default_layout()
+        return self.planner.layout(grades)
+
+    def default_layout(self) -> GradeLayout:
+        """Return the default layout using the central fallback policy."""
+        return _default_layout(self)
+
+    def product_executor(
+        self,
+        *,
+        left_grades,
+        right_grades,
+        op: str = "gp",
+        output_grades=None,
+        dtype: Optional[torch.dtype] = None,
+        device=None,
+        cache: bool = True,
+    ):
+        """Return a preplanned product executor suitable for ``torch.compile``."""
+        if dtype is None:
+            dtype = getattr(self, "dtype", torch.float32)
+        if device is None:
+            device = getattr(self, "device", None)
+        return self.planner.product_executor(
+            op=op,
+            left_grades=left_grades,
+            right_grades=right_grades,
+            output_grades=output_grades,
+            dtype=dtype,
+            device=device,
+            cache=cache,
+        )
+
+    def resolve_layout(
+        self,
+        *,
+        layout: Optional[GradeLayout] = None,
+        grades: Optional[Iterable[int]] = None,
+        mv=None,
+        allow_full: bool = True,
+        warn_full: bool = True,
+    ) -> GradeLayout:
+        """Resolve static layout metadata for tensors or multivectors."""
+        return _resolve_layout(
+            self,
+            layout=layout,
+            grades=grades,
+            mv=mv,
+            allow_full=allow_full,
+            warn_full=warn_full,
+        )
+
+    def grade_indices(self, grades: Iterable[int], *, device=None) -> torch.Tensor:
+        """Return canonical dense basis indices for ``grades``."""
+        return _grade_indices(self, grades, device=self.device if device is None else device)
+
+    def hermitian_signs(
+        self,
+        layout: Optional[GradeLayout] = None,
+        *,
+        grades: Optional[Iterable[int]] = None,
+        device=None,
+        dtype: Optional[torch.dtype] = None,
+    ) -> torch.Tensor:
+        """Return Hermitian signs for a dense or compact layout."""
+        return _hermitian_signs(self, layout=layout, grades=grades, device=device, dtype=dtype)
+
+    def projected_product(
+        self,
+        A: torch.Tensor,
+        B: torch.Tensor,
+        *,
+        left_grades=None,
+        right_grades=None,
+        output_grades=None,
+        left_layout=None,
+        right_layout=None,
+        output_layout=None,
+        op: str = "gp",
+        left_compact: bool = False,
+        right_compact: bool = False,
+        compact_output: bool = False,
+        return_layout: bool = False,
+        pairwise: bool = False,
+    ):
+        """Compute a declared grade-restricted product through a static executor.
+
+        By default, operands are multiplied elementwise over broadcastable
+        prefix dimensions. Set ``pairwise=True`` when the dimension before each
+        compact lane axis is an independent left/right item axis.
+        """
+        left_layout = self._declared_layout(left_grades, left_layout)
+        right_layout = self._declared_layout(right_grades, right_layout)
+        if not left_compact and left_layout is not None and A.shape[-1] == left_layout.dim:
+            left_compact = left_layout.dim != self.dim
+        if not right_compact and right_layout is not None and B.shape[-1] == right_layout.dim:
+            right_compact = right_layout.dim != self.dim
+        if not left_compact:
+            check_multivector(A, self, "projected_product(A)")
+        if not right_compact:
+            check_multivector(B, self, "projected_product(B)")
+
+        request = self.planner.product_request(
+            A,
+            B,
+            left_grades=left_grades,
+            right_grades=right_grades,
+            output_grades=output_grades,
+            left_layout=left_layout,
+            right_layout=right_layout,
+            output_layout=output_layout,
+            op=op,
+            left_compact=left_compact,
+            right_compact=right_compact,
+        )
+        executor = self.planner.product_executor_for_request(request)
+
+        if pairwise:
+            values = self._execute_pairwise_product(A, B, request, executor)
+        else:
+            values = self._execute_elementwise_product(A, B, request, executor)
+
+        if return_layout:
+            return values, executor.output_layout
+        if compact_output:
+            return values
+        return materialize_dense(self, values, layout=executor.output_layout)
+
+    def projected_geometric_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs):
+        """Projected geometric product convenience wrapper."""
+        return self.projected_product(A, B, op="gp", **kwargs)
+
+    def projected_wedge(self, A: torch.Tensor, B: torch.Tensor, **kwargs):
+        """Projected wedge product convenience wrapper."""
+        return self.projected_product(A, B, op="wedge", **kwargs)
+
+    def projected_inner_product(self, A: torch.Tensor, B: torch.Tensor, **kwargs):
+        """Projected inner product convenience wrapper."""
+        return self.projected_product(A, B, op="inner", **kwargs)
+
+    def projected_commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs):
+        """Projected commutator convenience wrapper."""
+        return self.projected_product(A, B, op="commutator", **kwargs)
+
+    def projected_anti_commutator(self, A: torch.Tensor, B: torch.Tensor, **kwargs):
+        """Projected anti-commutator convenience wrapper."""
+        return self.projected_product(A, B, op="anti_commutator", **kwargs)
+
+    def planned_unary(
+        self,
+        values: torch.Tensor,
+        *,
+        op: str,
+        input_grades=None,
+        output_grades=None,
+        input_layout: Optional[GradeLayout] = None,
+        output_layout: Optional[GradeLayout] = None,
+        input_compact: bool = False,
+        compact_output: bool = False,
+        return_layout: bool = False,
+    ):
+        """Execute a unary operation through the shared static grade planner."""
+        request = self.planner.unary_request(
+            values,
+            op=op,
+            input_grades=input_grades,
+            output_grades=output_grades,
+            input_layout=input_layout,
+            output_layout=output_layout,
+            input_compact=input_compact,
+        )
+        executor = self.planner.unary_executor_for_request(request)
+        output = executor.forward_compact(values) if request.input_compact else executor(values)
+
+        if return_layout:
+            return output, executor.output_layout
+        if compact_output:
+            return output
+        return materialize_dense(self, output, layout=executor.output_layout)
+
+    def _declared_layout(self, grades, layout):
+        if layout is not None:
+            return layout
+        if grades is not None:
+            return self.layout(grades)
+        default_grades = getattr(self, "_default_grades", None)
+        if default_grades is None:
+            return None
+        return self.layout(default_grades)
+
+    def _execute_elementwise_product(self, left, right, request, executor):
+        if request.left_compact or request.right_compact:
+            left_values = left if request.left_compact else executor.left_layout.compact(left)
+            right_values = right if request.right_compact else executor.right_layout.compact(right)
+            self._check_elementwise_prefix(left_values, right_values)
+            return executor.forward_compact(left_values, right_values)
+
+        self._check_elementwise_prefix(left, right)
+        return executor(left, right)
+
+    def _execute_pairwise_product(self, left, right, request, executor):
+        left_values = left if request.left_compact else executor.left_layout.compact(left)
+        right_values = right if request.right_compact else executor.right_layout.compact(right)
+        self._check_pairwise_prefix(left_values, right_values)
+        return executor.forward_pairwise_compact(left_values, right_values)
+
+    @staticmethod
+    def _check_elementwise_prefix(left: torch.Tensor, right: torch.Tensor) -> None:
+        try:
+            torch.broadcast_shapes(left.shape[:-1], right.shape[:-1])
+        except RuntimeError as exc:
+            raise ValueError(
+                "projected_product elementwise prefixes must be broadcastable; "
+                f"got left prefix {tuple(left.shape[:-1])} and right prefix {tuple(right.shape[:-1])}. "
+                "Use pairwise=True when left and right have distinct item axes."
+            ) from exc
+
+    @staticmethod
+    def _check_pairwise_prefix(left: torch.Tensor, right: torch.Tensor) -> None:
+        if left.ndim < 2 or right.ndim < 2:
+            raise ValueError(
+                "pairwise projected_product requires explicit item axes before the compact lane dimension; "
+                f"got left shape {tuple(left.shape)} and right shape {tuple(right.shape)}"
+            )
+        try:
+            torch.broadcast_shapes(left.shape[:-2], right.shape[:-2])
+        except RuntimeError as exc:
+            raise ValueError(
+                "pairwise projected_product batch prefixes must be broadcastable; "
+                f"got left prefix {tuple(left.shape[:-2])} and right prefix {tuple(right.shape[:-2])}"
+            ) from exc
diff --git a/core/visualizer.py b/core/visualizer.py
index 64d021d..10be229 100644
--- a/core/visualizer.py
+++ b/core/visualizer.py
@@ -12,7 +12,7 @@
 from sklearn.decomposition import PCA
 from sklearn.manifold import TSNE
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 
 class GeneralVisualizer:
diff --git a/docs/Architecture.md b/docs/Architecture.md
new file mode 100644
index 0000000..4ce674f
--- /dev/null
+++ b/docs/Architecture.md
@@ -0,0 +1,307 @@
+# Architecture
+
+Versor is organized around one rule: algebraic identity, tensor layout, runtime
+execution, and training behavior should stay separable. The core owns the
+mathematics and the execution contracts; layers, functionals, and optimizers
+expose those contracts in forms that are convenient for actual model code.
+
+## Framework Layers
+
+```mermaid
+flowchart TB
+    User["User code<br/>layers, functionals, optimizers"]
+
+    subgraph Framework["Versor framework"]
+        Layers["layers/<br/>nn.Module composition"]
+        Functional["functional/<br/>stateless algebra helpers, losses, activations"]
+        Optimizers["optimizers/<br/>manifold-aware parameter updates"]
+    end
+
+    subgraph Core["core/"]
+        Foundation["foundation/<br/>basis rules, layouts, validation, manifold tags"]
+        Planning["planning/<br/>grade requests, lane limits, executors"]
+        Runtime["runtime/<br/>dense algebra, planned context, multivectors"]
+        Config["config.py<br/>algebra factory"]
+        Analysis["analysis/<br/>metric/geodesic/symmetry tooling"]
+    end
+
+    User --> Layers
+    User --> Functional
+    User --> Optimizers
+    Layers --> Runtime
+    Layers --> Functional
+    Layers --> Foundation
+    Functional --> Runtime
+    Optimizers --> Foundation
+    Optimizers --> Runtime
+    Config --> Runtime
+    Runtime --> Planning
+    Runtime --> Foundation
+    Planning --> Foundation
+    Analysis --> Runtime
+```
+
+`core/` is the authority for algebra and execution. `layers/` should be thin
+module wrappers around core operations. `functional/` should expose stateless
+helpers and losses without duplicating runtime logic. `optimizers/` should read
+parameter manifold tags and apply updates; it should not decide product plans.
+
+## Algebra Host Selection
+
+```mermaid
+flowchart LR
+    Factory["make_algebra(p, q, r, kernel)"]
+    Dense["CliffordAlgebra<br/>dense tables, full dim = 2^n"]
+    Context["AlgebraContext<br/>planned compact execution"]
+    Spec["AlgebraSpec<br/>signature and dimension metadata"]
+    Planner["GradePlanner<br/>cached layouts and executors"]
+
+    Factory -->|"small n or explicit dense"| Dense
+    Factory -->|"large n or explicit context"| Context
+    Dense --> Spec
+    Context --> Spec
+    Dense --> Planner
+    Context --> Planner
+    Planner --> Spec
+```
+
+`CliffordAlgebra` owns dense Cayley-table buffers and fast full-layout kernels.
+`AlgebraContext` exposes the same high-level product API but routes products
+through static grade planning by default. Both hosts share the runtime facade, so
+declared products use the same layout, cost, and executor path.
+
+## Tensor And Layout Flow
+
+```mermaid
+flowchart LR
+    Grades["Grade set<br/>for example (1, 2)"]
+    Basis["basis_index_tuple_for_grades(n, grades)"]
+    Layout["GradeLayout<br/>grades, dense basis indices, lane count"]
+    Dense["Dense multivector<br/>last dim = 2^n"]
+    Compact["Compact values<br/>last dim = layout.dim"]
+    Materialized["Dense materialization<br/>zeros outside layout"]
+
+    Grades --> Basis --> Layout
+    Dense -->|"layout.compact"| Compact
+    Compact -->|"layout.dense"| Materialized
+    Layout --> Compact
+    Layout --> Materialized
+```
+
+A compact tensor is not just a shorter tensor. It is coefficient values plus a
+`GradeLayout` identity that defines which dense basis blades the lanes
+represent. Raw tensors do not carry that identity, so framework pipeline code
+must declare `*_grades`, pass layouts, or use `Multivector` wrappers when layout
+metadata needs to travel with values.
+
+## Product Execution Flow
+
+```mermaid
+flowchart TB
+    A["A tensor<br/>dense [*, 2^n] or compact [*, left_lanes]"]
+    B["B tensor<br/>dense [*, 2^n] or compact [*, right_lanes]"]
+    Metadata["Declared metadata<br/>left_grades, right_grades, output_grades<br/>left_layout, right_layout, compact flags"]
+
+    Request["ProductRequest<br/>normalized op, layouts, dtype, device"]
+    Cost["PlanningLimits and cost checks<br/>lanes, output width, estimated pairs"]
+    Tree["GradePlanTree<br/>homogeneous route nodes"]
+    Plan["GradeProductPlan<br/>left lanes, right lanes, output lanes, coefficients"]
+    Exec["GradeProductExecutor<br/>forward, forward_compact, forward_pairwise_compact"]
+    Values["Compact output values<br/>[*, output_lanes]"]
+    DenseOut["Dense materialized output<br/>[*, 2^n]"]
+
+    A --> Request
+    B --> Request
+    Metadata --> Request
+    Request --> Cost
+    Cost --> Tree
+    Tree --> Plan
+    Plan --> Exec
+    A --> Exec
+    B --> Exec
+    Exec --> Values
+    Values -->|"compact_output=True"| CompactReturn["Return compact values"]
+    Values -->|"compact_output=False"| DenseOut
+```
+
+Planning uses static grade metadata and tensor shapes. It does not inspect
+runtime tensor values. This keeps compiled paths stable and avoids
+data-dependent symbolic shape extraction.
+
+## Layer Pipeline Contract
+
+```mermaid
+sequenceDiagram
+    participant Model as Model/Layers
+    participant ProductLayer as ProductLayer
+    participant Algebra as Algebra Runtime API
+    participant Planner as GradePlanner
+    participant Executor as GradeProductExecutor
+    participant Optimizer as Optimizer
+
+    Model->>ProductLayer: forward(left, right)
+    ProductLayer->>Algebra: projected_product(...grades, compact flags)
+    Algebra->>Planner: product_request(...)
+    Planner->>Executor: cached or newly built executor
+    Algebra->>Executor: forward / forward_compact / forward_pairwise_compact
+    Executor-->>Model: dense or compact tensor
+    Model->>Optimizer: loss.backward(); step()
+    Optimizer->>Optimizer: update tagged parameters
+```
+
+Optimizers do not run planning. They update parameters after the forward pass.
+Planning happens when the model calls algebra operations through direct runtime
+APIs, functional helpers, `ProductLayer`, or `Multivector` methods.
+
+## Operator Rules
+
+```mermaid
+flowchart TB
+    Route["Route<br/>left grade r, right grade s"]
+    GP["geometric product<br/>grades abs(r-s), abs(r-s)+2, ..."]
+    Wedge["wedge / exterior<br/>grade r+s only"]
+    Symmetric["inner route<br/>symmetric parity grades"]
+    Comm["commutator<br/>odd swap-parity grades"]
+    Anti["anti-commutator<br/>even swap-parity grades, doubled coefficient"]
+    Pair["Basis pair<br/>left index i, right index j"]
+    Output["output index = i XOR j"]
+    Nonzero["operation_may_be_nonzero<br/>wedge requires no shared basis bit<br/>null self-overlap is zero"]
+    Coeff["operation_coefficient<br/>metric sign and op scale"]
+
+    Route --> GP
+    Route --> Wedge
+    Route --> Symmetric
+    Route --> Comm
+    Route --> Anti
+    GP --> Pair
+    Wedge --> Pair
+    Symmetric --> Pair
+    Comm --> Pair
+    Anti --> Pair
+    Pair --> Output
+    Output --> Nonzero
+    Nonzero --> Coeff
+```
+
+The wedge implementation is the exterior product. For homogeneous inputs:
+
+```text
+A_r ^ B_s = <A_r B_s>_{r+s}
+```
+
+For vectors this coincides with `(AB - BA) / 2`, but higher-grade wedge routes
+follow the grade-sum exterior definition.
+
+## Dense Runtime
+
+```mermaid
+flowchart TB
+    Init["CliffordAlgebra init"]
+    Cayley["Cayley indices and signs<br/>basis_product over all pairs"]
+    GP["gp_signs<br/>single-pass geometric product"]
+    Wedge["wedge_gp_signs<br/>grade(output) = grade(left) + grade(right)"]
+    Inner["inner_gp_signs<br/>symmetric sign table"]
+    Comm["comm_gp_signs and anti_comm_gp_signs"]
+    Contractions["left and right contraction helpers"]
+    Product["Runtime product call"]
+    Matmul["gather B by Cayley indices<br/>multiply signs<br/>batched matmul over lanes"]
+
+    Init --> Cayley
+    Cayley --> GP
+    Cayley --> Wedge
+    Cayley --> Inner
+    Cayley --> Comm
+    Cayley --> Contractions
+    GP --> Product
+    Wedge --> Product
+    Inner --> Product
+    Comm --> Product
+    Product --> Matmul
+```
+
+Dense products are table-driven. The input tensors carry coefficients; the
+precomputed tables carry basis multiplication structure.
+
+## Planning Limits
+
+`PlanningLimits` centralizes static guardrails for compact planning:
+
+```python
+from core.planning import PlanningLimits
+from core.runtime.context import AlgebraContext
+
+limits = PlanningLimits(max_lanes=8192, max_pairs=16_000_000)
+algebra = AlgebraContext(32, 0, device="cpu", planning_limits=limits)
+```
+
+`max_lanes` protects compact tensor width. `max_pairs` protects the
+gather/reduce interaction count generated by product plans. Dense algebra hosts
+and planned contexts both accept `planning_limits`, so the same policy object can
+be used across framework construction.
+
+## Analysis Flow
+
+```mermaid
+flowchart LR
+    Data["Input data<br/>vectors or multivectors"]
+    Algebra["Algebra host<br/>dense or context"]
+    Metric["Metric and dimension analysis<br/>norms, distances, effective dimension"]
+    Geodesic["GeodesicFlow<br/>grade-1 wedge to connection bivectors"]
+    Symmetry["SymmetryDetector<br/>group and continuous symmetry"]
+    Comm["CommutatorAnalyzer<br/>bracket spectra and exchange structure"]
+    Report["Analysis result dataclasses"]
+
+    Data --> Algebra
+    Algebra --> Metric
+    Algebra --> Geodesic
+    Algebra --> Symmetry
+    Algebra --> Comm
+    Metric --> Report
+    Geodesic --> Report
+    Symmetry --> Report
+    Comm --> Report
+```
+
+Analysis code should call algebra APIs rather than reconstructing basis rules.
+When active grades are known, analysis should pass `left_grades`,
+`right_grades`, and `output_grades` so the planner can avoid full-layout work.
+
+## End-To-End Product Call
+
+```mermaid
+sequenceDiagram
+    participant Caller as Caller
+    participant Algebra as Algebra Runtime API
+    participant Planner as GradePlanner
+    participant Executor as GradeProductExecutor
+    participant Layout as GradeLayout
+
+    Caller->>Algebra: wedge(A, B, left_grades, right_grades, output_grades)
+    Algebra->>Planner: product_request(...)
+    Planner->>Layout: resolve operand and output layouts
+    Planner->>Planner: build or fetch cached executor
+    Planner->>Executor: return executor
+    Algebra->>Executor: forward, forward_compact, or forward_pairwise_compact
+    Executor-->>Algebra: compact output values
+    Algebra->>Layout: materialize dense unless compact_output=True
+    Algebra-->>Caller: tensor output
+```
+
+The same path applies to geometric product, wedge, inner route, commutator, and
+anti-commutator. The operator name changes route grades and coefficients, not
+the overall tensor plumbing.
+
+## Framework Verification Map
+
+Framework-level tests are grouped by the behavior they prove:
+
+- Core algebra and dense kernel identities: `tests/test_core.py`
+- Static grade planning and compact execution: `tests/test_grade_plan.py`
+- Multivector layout-preserving wrappers: `tests/test_multivector.py`
+- Layer pipeline and optimizer integration: `tests/test_framework_pipeline.py`
+- Functional product helpers: `tests/test_functional_products.py`
+- Optimizer manifold grouping and factories: `tests/test_riemannian_optimizer.py`
+
+Performance checks live in `benchmarks/`. The framework pipeline benchmark
+measures dense-vs-compact products, planned contexts, pairwise compact products,
+and composed layer pipelines.
diff --git a/docs/api/core.md b/docs/api/core.md
index 1970930..3776c78 100644
--- a/docs/api/core.md
+++ b/docs/api/core.md
@@ -3,19 +3,22 @@
 The mathematical kernel of Versor.
 
 ## Algebra
-::: core.algebra.CliffordAlgebra
+::: core.runtime.algebra.CliffordAlgebra
+
+## Algebra Config
+::: core.config
 
 ## Module
-::: core.module.CliffordModule
+::: core.foundation.module.CliffordModule
 
 ## Multivector
-::: core.multivector.Multivector
+::: core.runtime.multivector.Multivector
 
 ## Metric
-::: core.metric
+::: core.runtime.metric
 
 ## Decomposition
-::: core.decomposition
+::: core.runtime.decomposition
 
 ## Analysis
 ::: core.analysis.MetricSearch
diff --git a/docs/design_guide.md b/docs/design_guide.md
index b52c632..6926e0d 100644
--- a/docs/design_guide.md
+++ b/docs/design_guide.md
@@ -75,7 +75,7 @@ The canonical Geometric Blade Network block, annotated:
 
 ```python
 import torch.nn as nn
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers.primitives.linear import CliffordLinear
 from layers.primitives.rotor import RotorLayer
 from layers.primitives.normalization import CliffordLayerNorm
@@ -147,7 +147,7 @@ Versor models are intentionally hybrid. Standard `nn.Linear` and `CliffordLinear
 ```python
 import torch
 import torch.nn as nn
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers.primitives.linear import CliffordLinear
 from layers.primitives.rotor import RotorLayer
 from layers.primitives.normalization import CliffordLayerNorm
@@ -237,7 +237,7 @@ End-to-end: choose algebra, build a 3-layer GBN, train, evaluate. Runs in under
 ```python
 import torch
 import torch.nn as nn
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers.primitives.linear import CliffordLinear
 from layers.primitives.rotor import RotorLayer
 from layers.primitives.normalization import CliffordLayerNorm
diff --git a/docs/index.md b/docs/index.md
index 74b8163..11f47b6 100644
--- a/docs/index.md
+++ b/docs/index.md
@@ -1,6 +1,6 @@
 # Versor: A PyTorch Framework for Geometric Algebra Deep Learning
 
-[![License](https://img.shields.io/badge/License-Apache%202.0-blue.svg)](LICENSE) [![Python](https://img.shields.io/badge/python-3.9+-blue.svg)](https://www.python.org/downloads/) [![PyTorch](https://img.shields.io/badge/PyTorch-2.0+-ee4c2c.svg)](https://pytorch.org/) [![Docs](https://img.shields.io/badge/docs-MkDocs-brightgreen)](https://concode0.github.io/Versor/) [![DOI](https://zenodo.org/badge/1149480519.svg)](https://doi.org/10.5281/zenodo.18939518)
+[![License](https://img.shields.io/badge/License-Apache%202.0-blue.svg)](LICENSE) [![Python](https://img.shields.io/badge/python-3.10+-blue.svg)](https://www.python.org/downloads/) [![PyTorch](https://img.shields.io/badge/PyTorch-2.0+-ee4c2c.svg)](https://pytorch.org/) [![Docs](https://img.shields.io/badge/docs-MkDocs-brightgreen)](https://concode0.github.io/Versor/) [![DOI](https://zenodo.org/badge/1149480519.svg)](https://doi.org/10.5281/zenodo.18939518)
 
 > **"There is a ceiling above standard Deep Learning that no one saw. Versor opens the door above it."**
 
@@ -53,7 +53,7 @@ For code examples of each innovation, see [Innovations](innovations.md).
 
 ## Installation
 
-Versor requires Python 3.9+ and PyTorch.
+Versor requires Python 3.10+ and PyTorch.
 
 ```bash
 # Clone the repository
@@ -80,7 +80,7 @@ uv sync --extra all         # everything
 
 ```python
 import torch
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers.primitives.rotor import RotorLayer
 from layers.linear import CliffordLinear
 from functional.activation import GeometricGELU
diff --git a/docs/tutorial.md b/docs/tutorial.md
index 1e3e05b..1792103 100644
--- a/docs/tutorial.md
+++ b/docs/tutorial.md
@@ -7,7 +7,7 @@ A step-by-step guide to using Versor's geometric layers in your own models.
 Everything starts with a `CliffordAlgebra` instance. The signature $(p, q, r)$ determines the geometry:
 
 ```python
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 # 3D Euclidean (rotations in 3-space)
 algebra = CliffordAlgebra(p=3, q=0, r=0, device='cpu')
@@ -183,7 +183,7 @@ All tasks inherit from `BaseTask` and implement 7 methods:
 
 ```python
 from tasks.base import BaseTask
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from functional.loss import GeometricMSELoss
 
 class MyTask(BaseTask):
diff --git a/examples/conf/task/cgenn.yaml b/examples/conf/task/cgenn.yaml
index 70152b0..71ddea5 100644
--- a/examples/conf/task/cgenn.yaml
+++ b/examples/conf/task/cgenn.yaml
@@ -4,7 +4,6 @@ name: "cgenn"
 algebra:
   p: 3
   q: 0
-  device: "cpu"
 
 dataset:
   name: "point_cloud_invariant"
diff --git a/examples/conf/task/clifford_pde.yaml b/examples/conf/task/clifford_pde.yaml
index 6568289..1ca7d10 100644
--- a/examples/conf/task/clifford_pde.yaml
+++ b/examples/conf/task/clifford_pde.yaml
@@ -4,7 +4,6 @@ name: "clifford_pde"
 algebra:
   p: 2
   q: 0
-  device: "cpu"
 
 dataset:
   name: "taylor_green"
diff --git a/examples/conf/task/gatr.yaml b/examples/conf/task/gatr.yaml
index deb8273..dfbdb37 100644
--- a/examples/conf/task/gatr.yaml
+++ b/examples/conf/task/gatr.yaml
@@ -5,7 +5,6 @@ algebra:
   p: 3
   q: 0
   r: 1
-  device: "cpu"
 
 dataset:
   name: "nbody_spring"
diff --git a/examples/conf/task/hyperbolic.yaml b/examples/conf/task/hyperbolic.yaml
index 8a32a1f..1d5cd45 100644
--- a/examples/conf/task/hyperbolic.yaml
+++ b/examples/conf/task/hyperbolic.yaml
@@ -1,9 +1,10 @@
 # @package _global_
 name: "hyperbolic"
+
 algebra:
   p: 1
   q: 1
-  device: "cpu"
+
 dataset:
   name: "lorentz_boost"
   samples: 10000
diff --git a/examples/conf/task/manifold.yaml b/examples/conf/task/manifold.yaml
index 458c145..ab61ae1 100644
--- a/examples/conf/task/manifold.yaml
+++ b/examples/conf/task/manifold.yaml
@@ -1,10 +1,11 @@
 # @package _global_
 name: "manifold"
+
 algebra:
   p: 3
   q: 0
-  device: "cpu"
+
 dataset:
   name: "figure8"
   samples: 1000
-  noise_std: 0.1
\ No newline at end of file
+  noise_std: 0.1
diff --git a/examples/conf/task/sanity.yaml b/examples/conf/task/sanity.yaml
index 1ad2b3e..137455c 100644
--- a/examples/conf/task/sanity.yaml
+++ b/examples/conf/task/sanity.yaml
@@ -1,9 +1,10 @@
 # @package _global_
 name: "sanity"
+
 algebra:
   p: 3
   q: 0
-  device: "cpu"
+
 dataset:
   name: "random_noise"
   samples: 1000
diff --git a/examples/datasets/amass.py b/examples/datasets/amass.py
index 434b10e..81df07d 100644
--- a/examples/datasets/amass.py
+++ b/examples/datasets/amass.py
@@ -9,7 +9,7 @@
 import torch
 from torch.utils.data import Dataset
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 
 class AMASSDataset(Dataset):
diff --git a/examples/datasets/synthetic.py b/examples/datasets/synthetic.py
index 84587c2..3ce5f14 100644
--- a/examples/datasets/synthetic.py
+++ b/examples/datasets/synthetic.py
@@ -9,7 +9,7 @@
 import torch
 from torch.utils.data import Dataset
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 
 class Figure8Dataset(Dataset):
diff --git a/examples/demo.py b/examples/demo.py
index 4914e1a..eebca02 100644
--- a/examples/demo.py
+++ b/examples/demo.py
@@ -25,7 +25,7 @@
 import torch
 import torch.optim as optim
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra
 from layers import BladeSelector, RotorLayer
 
 # Setup Page
@@ -102,7 +102,7 @@ def plot_3d_manifold(data, title, color_data=None):
 # Main app logic
 
 # 1. Generate Data
-algebra = CliffordAlgebra(3, 0, device="cpu")
+algebra = make_algebra(3, 0, device="cpu")
 t = torch.linspace(0, 2 * np.pi, samples)
 x = torch.sin(t)
 y = torch.sin(t) * torch.cos(t)
diff --git a/examples/tasks/cgenn.py b/examples/tasks/cgenn.py
index 259914c..7be6ef0 100644
--- a/examples/tasks/cgenn.py
+++ b/examples/tasks/cgenn.py
@@ -60,8 +60,8 @@
 import torch.nn as nn
 from torch.utils.data import DataLoader, TensorDataset
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.config import make_algebra_from_config
+from core.foundation.module import CliffordModule
 from functional.activation import GeometricSquare
 from layers import (
     BladeSelector,
@@ -205,9 +205,11 @@ class CGENNTask(BaseTask):
     """
 
     def setup_algebra(self):
-        return CliffordAlgebra(
+        return make_algebra_from_config(
+            self.cfg.algebra,
             p=self.cfg.algebra.p,
             q=self.cfg.algebra.q,
+            r=self.cfg.algebra.get("r", 0),
             device=self.device,
         )
 
diff --git a/examples/tasks/clifford_pde.py b/examples/tasks/clifford_pde.py
index 4950672..9c9379e 100644
--- a/examples/tasks/clifford_pde.py
+++ b/examples/tasks/clifford_pde.py
@@ -66,8 +66,8 @@
 import torch.nn as nn
 from torch.utils.data import DataLoader, TensorDataset
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.config import make_algebra_from_config
+from core.foundation.module import CliffordModule
 from functional.activation import GeometricGELU
 from layers import (
     CliffordLayerNorm,
@@ -276,9 +276,11 @@ class CliffordPDETask(BaseTask):
     def setup_algebra(self):
         # Cl(2,0): dim = 2^2 = 4 components
         # grade-0: 1 scalar, grade-1: 2 vectors, grade-2: 1 bivector
-        return CliffordAlgebra(
+        return make_algebra_from_config(
+            self.cfg.algebra,
             p=self.cfg.algebra.p,
             q=self.cfg.algebra.q,
+            r=self.cfg.algebra.get("r", 0),
             device=self.device,
         )
 
diff --git a/examples/tasks/gatr.py b/examples/tasks/gatr.py
index 45c3bfb..7ab4e74 100644
--- a/examples/tasks/gatr.py
+++ b/examples/tasks/gatr.py
@@ -63,7 +63,7 @@
 import torch.nn as nn
 from torch.utils.data import DataLoader, TensorDataset
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra_from_config
 from layers import CliffordLinear, GeometricTransformerBlock
 from layers.adapters.projective import ProjectiveEmbedding
 from tasks.base import BaseTask
@@ -220,7 +220,8 @@ def setup_algebra(self):
         # Cl(3,0,1): 3 Euclidean + 1 degenerate = Projective Geometric Algebra
         # dim = 2^4 = 16 multivector components
         r = self.cfg.algebra.get("r", 1)
-        return CliffordAlgebra(
+        return make_algebra_from_config(
+            self.cfg.algebra,
             p=self.cfg.algebra.p,
             q=self.cfg.algebra.q,
             r=r,
diff --git a/examples/tasks/hyperbolic.py b/examples/tasks/hyperbolic.py
index b30e6fd..34e4059 100644
--- a/examples/tasks/hyperbolic.py
+++ b/examples/tasks/hyperbolic.py
@@ -9,8 +9,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.config import make_algebra_from_config
+from core.foundation.module import CliffordModule
 from core.visualizer import GeneralVisualizer
 from functional.loss import GeometricMSELoss
 from layers import RotorLayer
@@ -44,7 +44,7 @@ def __init__(self, cfg):
 
     def setup_algebra(self):
         """2D Spacetime Cl(1, 1)."""
-        return CliffordAlgebra(p=1, q=1, device=self.device)
+        return make_algebra_from_config(self.cfg.algebra, p=1, q=1, r=0, device=self.device)
 
     def setup_model(self):
         """The Booster."""
diff --git a/examples/tasks/manifold.py b/examples/tasks/manifold.py
index 29fa06c..e3a4ce1 100644
--- a/examples/tasks/manifold.py
+++ b/examples/tasks/manifold.py
@@ -9,8 +9,8 @@
 import torch.nn as nn
 from torch.utils.data import DataLoader
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.config import make_algebra_from_config
+from core.foundation.module import CliffordModule
 from core.visualizer import GeneralVisualizer
 from examples.datasets.synthetic import Figure8Dataset
 from functional.loss import SubspaceLoss
@@ -47,7 +47,13 @@ def __init__(self, cfg):
 
     def setup_algebra(self):
         """3D Euclidean."""
-        return CliffordAlgebra(p=self.cfg.algebra.p, q=self.cfg.algebra.q, device=self.device)
+        return make_algebra_from_config(
+            self.cfg.algebra,
+            p=self.cfg.algebra.p,
+            q=self.cfg.algebra.q,
+            r=self.cfg.algebra.get("r", 0),
+            device=self.device,
+        )
 
     def setup_model(self):
         """The Unbender."""
diff --git a/examples/tasks/sanity_check.py b/examples/tasks/sanity_check.py
index fe7e1bd..394fc30 100644
--- a/examples/tasks/sanity_check.py
+++ b/examples/tasks/sanity_check.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.config import make_algebra_from_config
+from core.foundation.module import CliffordModule
 from core.visualizer import GeneralVisualizer
 from functional.loss import GeometricMSELoss
 from layers import RotorLayer
@@ -40,7 +40,7 @@ def __init__(self, cfg):
 
     def setup_algebra(self):
         """Standard 3D Euclidean."""
-        return CliffordAlgebra(p=3, q=0, device=self.device)
+        return make_algebra_from_config(self.cfg.algebra, p=3, q=0, r=0, device=self.device)
 
     def setup_model(self):
         """Identity Net."""
diff --git a/experiments/_gdo/benchmarks.py b/experiments/_gdo/benchmarks.py
index 850efa0..76c6ef6 100644
--- a/experiments/_gdo/benchmarks.py
+++ b/experiments/_gdo/benchmarks.py
@@ -31,8 +31,8 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from experiments._lib import setup_algebra
 from functional.activation import GeometricGELU
 from layers import (
     BladeSelector,
@@ -57,7 +57,7 @@ class SmallGBNModel(CliffordModule):
     """
 
     def __init__(self, p: int = 3, q: int = 0, channels: int = 4, device: str = "cpu"):
-        algebra = CliffordAlgebra(p, q, device=device)
+        algebra = setup_algebra(p, q, device=device)
         super().__init__(algebra)
         self.norm = CliffordLayerNorm(self.algebra, channels)
         self.rotor = RotorLayer(self.algebra, channels)
@@ -87,7 +87,7 @@ class MultiRotorRegistrationModel(CliffordModule):
     """
 
     def __init__(self, n_clusters: int = 3, points_per_cluster: int = 20, device: str = "cpu"):
-        algebra = CliffordAlgebra(3, 0, device=device)
+        algebra = setup_algebra(3, 0, device=device)
         super().__init__(algebra)
         dim = self.algebra.dim
 
@@ -158,7 +158,7 @@ def __init__(
         batch_size: int = 4,
         device: str = "cpu",
     ):
-        algebra = CliffordAlgebra(p, q, device=device)
+        algebra = setup_algebra(p, q, device=device)
         super().__init__(algebra)
         self.vocab_size = vocab_size
         self.seq_len = seq_len
diff --git a/experiments/_gdo/controller.py b/experiments/_gdo/controller.py
index 4773066..3c06593 100644
--- a/experiments/_gdo/controller.py
+++ b/experiments/_gdo/controller.py
@@ -9,7 +9,7 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from optimizers.riemannian import MANIFOLD_EUCLIDEAN
 
 from .config import GDOConfig
diff --git a/experiments/_gdo/harness.py b/experiments/_gdo/harness.py
index 17db9e2..affdcc8 100644
--- a/experiments/_gdo/harness.py
+++ b/experiments/_gdo/harness.py
@@ -9,7 +9,7 @@
 import torch.nn as nn
 from torch.optim import Optimizer
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers import MultiRotorLayer, RotorLayer
 from optimizers.riemannian import ExponentialSGD, RiemannianAdam
 
diff --git a/experiments/_gdo/optimizer.py b/experiments/_gdo/optimizer.py
index 4412466..1bceb7f 100644
--- a/experiments/_gdo/optimizer.py
+++ b/experiments/_gdo/optimizer.py
@@ -8,7 +8,7 @@
 import torch.nn as nn
 from torch.optim import Optimizer
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from optimizers.riemannian import (
     MANIFOLD_EUCLIDEAN,
     MANIFOLD_SPHERE,
diff --git a/experiments/_gdo/parameter_groups.py b/experiments/_gdo/parameter_groups.py
index 2686079..12fb99f 100644
--- a/experiments/_gdo/parameter_groups.py
+++ b/experiments/_gdo/parameter_groups.py
@@ -8,7 +8,6 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
 from core.analysis import (
     CommutatorAnalyzer as CoreCommutatorAnalyzer,
 )
@@ -16,6 +15,7 @@
     GeodesicFlow,
     SpectralAnalyzer,
 )
+from core.runtime.algebra import CliffordAlgebra
 from layers import MultiRotorLayer, RotorLayer
 
 from .config import GDOConfig
diff --git a/experiments/_gdo/pre_exploration.py b/experiments/_gdo/pre_exploration.py
index ee07428..fece19b 100644
--- a/experiments/_gdo/pre_exploration.py
+++ b/experiments/_gdo/pre_exploration.py
@@ -24,7 +24,6 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
 from core.analysis import (
     CommutatorAnalyzer as CoreCommutatorAnalyzer,
 )
@@ -43,6 +42,8 @@
     SpectralResult,
     SymmetryResult,
 )
+from core.foundation.module import AlgebraLike
+from experiments._lib import setup_algebra
 
 from .config import GDOConfig
 from .parameter_groups import GeometricParameterController
@@ -124,7 +125,7 @@ class PreExplorationAnalyzer:
 
     def __init__(
         self,
-        algebra: Optional[CliffordAlgebra] = None,
+        algebra: Optional[AlgebraLike] = None,
         n_samples: int = 200,
         sample_radius: float = 0.5,
         device: str = "cpu",
@@ -520,7 +521,7 @@ def analyze(
         if dim_result is not None and dim_result.intrinsic_dim >= 2:
             try:
                 land_dim = min(dim_result.intrinsic_dim, 6)
-                temp_algebra = CliffordAlgebra(land_dim, 0, device=self.device)
+                temp_algebra = setup_algebra(land_dim, 0, device=self.device)
                 reduced = eda.reduce(sampled, land_dim)
                 mv_land = temp_algebra.embed_vector(reduced)
                 k = min(8, mv_land.shape[0] - 1)
diff --git a/experiments/_lib.py b/experiments/_lib.py
index 837d90f..b6470f8 100644
--- a/experiments/_lib.py
+++ b/experiments/_lib.py
@@ -40,8 +40,9 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.metric import hermitian_grade_spectrum
+from core.config import make_algebra
+from core.foundation.module import AlgebraLike
+from core.runtime.metric import hermitian_grade_spectrum
 from functional.activation import GeometricGELU
 from layers import CliffordLayerNorm, CliffordLinear, RotorLayer
 
@@ -67,9 +68,30 @@ def set_seed(seed: int, deterministic: bool = False) -> None:
 # ---------------------------------------------------------------------------
 
 
-def setup_algebra(p: int, q: int = 0, r: int = 0, device: str = "cpu") -> CliffordAlgebra:
-    """One-line ``CliffordAlgebra(p, q, r, device=device).to(device)`` wrapper."""
-    return CliffordAlgebra(p=p, q=q, r=r, device=device).to(device)
+def setup_algebra(
+    p: int,
+    q: int = 0,
+    r: int = 0,
+    device: str = "cpu",
+    *,
+    dtype: torch.dtype | str = torch.float32,
+    kernel: str = "auto",
+    dense_threshold: int = 8,
+    exp_policy: str = "balanced",
+    fixed_iterations: Optional[int] = None,
+) -> AlgebraLike:
+    """Construct the shared experiment algebra through the core factory."""
+    return make_algebra(
+        p=p,
+        q=q,
+        r=r,
+        kernel=kernel,
+        dense_threshold=dense_threshold,
+        device=device,
+        dtype=dtype,
+        exp_policy=exp_policy,
+        fixed_iterations=fixed_iterations,
+    )
 
 
 # ---------------------------------------------------------------------------
@@ -208,12 +230,12 @@ def count_parameters(model: nn.Module) -> int:
 # ---------------------------------------------------------------------------
 
 
-def grade1_indices(algebra: CliffordAlgebra) -> List[int]:
+def grade1_indices(algebra: AlgebraLike) -> List[int]:
     """Multivector indices of grade-1 basis elements ``[e1, e2, ..., e_n]``."""
     return [1 << i for i in range(algebra.n)]
 
 
-def extract_grade1(mv: torch.Tensor, algebra: CliffordAlgebra, n: Optional[int] = None) -> torch.Tensor:
+def extract_grade1(mv: torch.Tensor, algebra: AlgebraLike, n: Optional[int] = None) -> torch.Tensor:
     """Slice grade-1 components from a multivector ``[..., dim] → [..., n]``.
 
     ``n`` defaults to ``algebra.n`` (all grade-1 slots). Inverse of
@@ -228,7 +250,7 @@ def extract_grade1(mv: torch.Tensor, algebra: CliffordAlgebra, n: Optional[int]
 # ---------------------------------------------------------------------------
 
 
-def gbn_residual_block(algebra: CliffordAlgebra, channels: int) -> nn.ModuleDict:
+def gbn_residual_block(algebra: AlgebraLike, channels: int) -> nn.ModuleDict:
     """The four-step block every GBN experiment shares.
 
     Returns ``{'norm', 'rotor', 'act', 'linear'}`` — no skip, no outer module
@@ -262,7 +284,7 @@ def apply_residual_block(block: nn.ModuleDict, h: torch.Tensor) -> torch.Tensor:
 
 
 @torch.no_grad()
-def mean_grade_spectrum(mv_iter: Iterable[torch.Tensor], algebra: CliffordAlgebra) -> np.ndarray:
+def mean_grade_spectrum(mv_iter: Iterable[torch.Tensor], algebra: AlgebraLike) -> np.ndarray:
     """Mean Hermitian grade spectrum across an iterable of multivectors.
 
     Each element may be any shape ending in ``algebra.dim``; it is flattened
diff --git a/experiments/_templates/inc_template.py b/experiments/_templates/inc_template.py
index 10d2876..5ffd1b4 100644
--- a/experiments/_templates/inc_template.py
+++ b/experiments/_templates/inc_template.py
@@ -50,7 +50,7 @@
 # Bootstrap project root so the file runs both via ``-m`` and as a bare script.
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir, os.pardir)))
 
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
 from experiments._lib import (
     count_parameters,
     ensure_output_dir,
diff --git a/experiments/dbg_linear_basis_mixing.py b/experiments/dbg_linear_basis_mixing.py
index 55f1fb5..2074890 100644
--- a/experiments/dbg_linear_basis_mixing.py
+++ b/experiments/dbg_linear_basis_mixing.py
@@ -55,8 +55,8 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from experiments._lib import (
     RawDefaultsHelpFormatter,
     build_visualization_metadata,
diff --git a/experiments/dbg_lorentz.py b/experiments/dbg_lorentz.py
index 7cc554a..b117626 100644
--- a/experiments/dbg_lorentz.py
+++ b/experiments/dbg_lorentz.py
@@ -56,12 +56,12 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.metric import (
+from core.foundation.module import CliffordModule
+from core.runtime.metric import (
     hermitian_grade_spectrum,
     signature_norm_squared,
     signature_trace_form,
 )
-from core.module import CliffordModule
 from experiments._lib import (
     build_visualization_metadata,
     ensure_output_dir,
diff --git a/experiments/dbg_maxwell_equations.py b/experiments/dbg_maxwell_equations.py
index eff8a56..391203e 100644
--- a/experiments/dbg_maxwell_equations.py
+++ b/experiments/dbg_maxwell_equations.py
@@ -57,8 +57,8 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from experiments._lib import (
     build_visualization_metadata,
     ensure_output_dir,
diff --git a/experiments/dbg_navier_stokes.py b/experiments/dbg_navier_stokes.py
index 1aa86b3..42f6eec 100644
--- a/experiments/dbg_navier_stokes.py
+++ b/experiments/dbg_navier_stokes.py
@@ -56,8 +56,8 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.metric import hermitian_grade_spectrum, hermitian_inner_product
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.metric import hermitian_grade_spectrum, hermitian_inner_product
 from experiments._lib import (
     build_visualization_metadata,
     count_parameters,
diff --git a/experiments/dbg_yang_mills.py b/experiments/dbg_yang_mills.py
index cf56a71..4d55eab 100644
--- a/experiments/dbg_yang_mills.py
+++ b/experiments/dbg_yang_mills.py
@@ -54,9 +54,9 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.algebra import CliffordAlgebra
-from core.metric import hermitian_grade_spectrum, hermitian_inner_product
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.metric import hermitian_grade_spectrum, hermitian_inner_product
 from experiments._lib import (
     build_visualization_metadata,
     count_parameters,
diff --git a/experiments/inc_embed_compress.py b/experiments/inc_embed_compress.py
index dce84c8..4474dd4 100644
--- a/experiments/inc_embed_compress.py
+++ b/experiments/inc_embed_compress.py
@@ -61,11 +61,11 @@
 from datasets import load_dataset
 from sentence_transformers import SentenceTransformer
 
-from core.algebra import CliffordAlgebra
 from core.analysis._types import DimensionResult
 from core.analysis.dimension import DimensionLifter, EffectiveDimensionAnalyzer
 from core.analysis.spectral import SpectralAnalyzer
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from experiments._lib import (
     build_visualization_metadata,
     ensure_output_dir,
diff --git a/experiments/inc_lattice_morph.py b/experiments/inc_lattice_morph.py
index 61c94b3..84b6fa1 100644
--- a/experiments/inc_lattice_morph.py
+++ b/experiments/inc_lattice_morph.py
@@ -48,10 +48,9 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.algebra import CliffordAlgebra
-from core.decomposition import ExpPolicy
-from core.metric import induced_norm
-from core.module import CliffordModule
+from core.foundation.module import AlgebraLike, CliffordModule
+from core.runtime.decomposition import ExpPolicy
+from core.runtime.metric import induced_norm
 from experiments._lib import (
     build_visualization_metadata,
     ensure_output_dir,
@@ -59,6 +58,7 @@
     save_experiment_figure,
     section_header,
     set_seed,
+    setup_algebra,
     signature_metadata,
 )
 from optimizers.riemannian import RiemannianAdam
@@ -118,14 +118,11 @@ class MorphMode(str, Enum):
 class StructureTracker:
     """Computes lattice geometric invariants from basis multivectors."""
 
-    def __init__(self, algebra: CliffordAlgebra):
+    def __init__(self, algebra: AlgebraLike):
         self.algebra = algebra
 
     def compute_volume(self, basis_mvs: torch.Tensor) -> torch.Tensor:
-        """det(L) = ||b_1 ^ b_2 ^ ... ^ b_n|| via iterative outer product.
-
-        Uses grade_projection(GP(blade, b_i), k+1) instead of wedge(),
-        because wedge() is (AB-BA)/2 which only works for two vectors.
+        """det(L) = ||b_1 ^ b_2 ^ ... ^ b_n|| via iterative exterior product.
 
         Args:
             basis_mvs: [n, D] grade-1 multivectors.
@@ -135,9 +132,15 @@ def compute_volume(self, basis_mvs: torch.Tensor) -> torch.Tensor:
         blade = basis_mvs[0]
         current_grade = 1
         for i in range(1, basis_mvs.shape[0]):
-            product = self.algebra.geometric_product(blade, basis_mvs[i])
-            current_grade += 1
-            blade = self.algebra.grade_projection(product, current_grade)
+            next_grade = current_grade + 1
+            blade = self.algebra.wedge(
+                blade,
+                basis_mvs[i],
+                left_grades=(current_grade,),
+                right_grades=(1,),
+                output_grades=(next_grade,),
+            )
+            current_grade = next_grade
         return induced_norm(self.algebra, blade).squeeze(-1)
 
     def compute_gram_matrix(self, basis_mvs: torch.Tensor) -> torch.Tensor:
@@ -212,10 +215,10 @@ class MorphStage(CliffordModule):
     When ``compound_blades >= 2`` and ``n >= 4`` the global and twist rotors
     learn a *sum* of independent simple bivectors per slot, producing
     non-simple bivectors that are handled by ``algebra.exp()`` via the active
-    :class:`~core.decomposition.ExpPolicy`.
+    :class:`~core.runtime.decomposition.ExpPolicy`.
     """
 
-    def __init__(self, algebra: CliffordAlgebra, n: int, compound_blades: int = 1):
+    def __init__(self, algebra: AlgebraLike, n: int, compound_blades: int = 1):
         """
         Args:
             algebra: Clifford algebra instance.
@@ -386,7 +389,7 @@ def inverse(self, basis_mvs: torch.Tensor) -> torch.Tensor:
 class MorphPipeline(nn.Module):
     """Sequential composition of MorphStages with intermediate tracking."""
 
-    def __init__(self, algebra: CliffordAlgebra, n: int, num_stages: int = 3, compound_blades: int = 1):
+    def __init__(self, algebra: AlgebraLike, n: int, num_stages: int = 3, compound_blades: int = 1):
         super().__init__()
         self.compound_blades = compound_blades
         self.stages = nn.ModuleList(
@@ -456,8 +459,7 @@ def __init__(
             p, q = n - 1, 1
         else:
             p, q = n, 0
-        self.algebra = CliffordAlgebra(p=p, q=q, device=device, dtype=dtype).to(device)
-        self.algebra.exp_policy = ExpPolicy.PRECISE
+        self.algebra = setup_algebra(p=p, q=q, device=device, dtype=dtype, exp_policy=ExpPolicy.PRECISE)
         self.signature_q = q
         self.tracker = StructureTracker(self.algebra)
         self.pipeline = MorphPipeline(
@@ -711,7 +713,7 @@ def verify_reconstruction(self, tolerance: float = 1e-5) -> dict:
 class MorphVisualizer:
     """Visualization for lattice morphing."""
 
-    def __init__(self, algebra: CliffordAlgebra, n: int, output_dir: str, metadata: str, args: argparse.Namespace):
+    def __init__(self, algebra: AlgebraLike, n: int, output_dir: str, metadata: str, args: argparse.Namespace):
         self.algebra = algebra
         self.n = n
         self.output_dir = ensure_output_dir(output_dir)
diff --git a/experiments/inc_pendulum_dynamics.py b/experiments/inc_pendulum_dynamics.py
index 5770c87..94178d3 100644
--- a/experiments/inc_pendulum_dynamics.py
+++ b/experiments/inc_pendulum_dynamics.py
@@ -52,9 +52,9 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.algebra import CliffordAlgebra
-from core.metric import hermitian_norm
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.metric import hermitian_norm
 from experiments._lib import (
     apply_residual_block,
     build_visualization_metadata,
diff --git a/experiments/inc_sta_trajectory.py b/experiments/inc_sta_trajectory.py
index 4bda41e..9011840 100644
--- a/experiments/inc_sta_trajectory.py
+++ b/experiments/inc_sta_trajectory.py
@@ -56,9 +56,9 @@
 
 sys.path.insert(0, os.path.abspath(os.path.join(os.path.dirname(__file__), os.pardir)))
 
-from core.algebra import CliffordAlgebra
-from core.metric import signature_norm_squared
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.metric import signature_norm_squared
 from experiments._lib import (
     build_visualization_metadata,
     count_parameters,
diff --git a/functional/__init__.py b/functional/__init__.py
index 595a9b1..13c025c 100644
--- a/functional/__init__.py
+++ b/functional/__init__.py
@@ -3,7 +3,7 @@
 Includes activation functions, loss functions, and orthogonality enforcement.
 """
 
-from .activation import GeometricGELU, GradeSwish
+from .activation import GeometricGELU, GeometricSquare, GradeSwish
 from .loss import (
     BivectorRegularization,
     ChamferDistance,
@@ -15,11 +15,43 @@
     SubspaceLoss,
 )
 from .orthogonality import OrthogonalitySettings, StrictOrthogonality
+from .products import (
+    anti_commutator,
+    clifford_conjugation,
+    commutator,
+    dual,
+    embed_vector,
+    geometric_product,
+    grade_involution,
+    grade_projection,
+    inner_product,
+    norm_sq,
+    product,
+    projected_product,
+    reverse,
+    wedge,
+)
 
 __all__ = [
     # activations
     "GeometricGELU",
+    "GeometricSquare",
     "GradeSwish",
+    # products
+    "product",
+    "projected_product",
+    "geometric_product",
+    "wedge",
+    "inner_product",
+    "commutator",
+    "anti_commutator",
+    "grade_projection",
+    "reverse",
+    "grade_involution",
+    "clifford_conjugation",
+    "dual",
+    "norm_sq",
+    "embed_vector",
     # losses
     "GeometricMSELoss",
     "SubspaceLoss",
diff --git a/functional/activation.py b/functional/activation.py
index c66d84b..92510c2 100644
--- a/functional/activation.py
+++ b/functional/activation.py
@@ -14,7 +14,7 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
 
 
 class GeometricGELU(CliffordModule):
diff --git a/functional/loss.py b/functional/loss.py
index a14a9fd..1a8f365 100644
--- a/functional/loss.py
+++ b/functional/loss.py
@@ -9,8 +9,8 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.metric import hermitian_grade_spectrum
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.metric import hermitian_grade_spectrum
 
 
 class GeometricMSELoss(CliffordModule):
diff --git a/functional/orthogonality.py b/functional/orthogonality.py
index 6730f4c..c906015 100644
--- a/functional/orthogonality.py
+++ b/functional/orthogonality.py
@@ -52,7 +52,7 @@
 import torch
 import torch.nn as nn
 
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
 
 
 @dataclass
diff --git a/functional/products.py b/functional/products.py
new file mode 100644
index 0000000..bc3f254
--- /dev/null
+++ b/functional/products.py
@@ -0,0 +1,130 @@
+"""Stateless geometric algebra product helpers.
+
+These wrappers keep model code concise while preserving the algebra host as the
+single execution authority. Dense kernels, compact planned kernels, and pairwise
+planned kernels all flow through the same public calls.
+"""
+
+from __future__ import annotations
+
+from typing import Any
+
+import torch
+
+_PRODUCT_METHODS = {
+    "gp": ("gp", "geometric_product"),
+    "geometric_product": ("gp", "geometric_product"),
+    "wedge": ("wedge", "wedge"),
+    "outer": ("wedge", "wedge"),
+    "inner": ("inner", "inner_product"),
+    "inner_product": ("inner", "inner_product"),
+    "commutator": ("commutator", "commutator"),
+    "anti_commutator": ("anti_commutator", "anti_commutator"),
+    "anticommutator": ("anti_commutator", "anti_commutator"),
+}
+
+
+def canonical_product_op(op: str) -> str:
+    """Return the planner op name for a supported product alias."""
+    return _resolve_product_op(op)[0]
+
+
+def _resolve_product_op(op: str) -> tuple[str, str]:
+    op_key = op.lower()
+    if op_key not in _PRODUCT_METHODS:
+        supported = ", ".join(sorted(_PRODUCT_METHODS))
+        raise ValueError(f"Unsupported product op {op!r}. Supported ops: {supported}")
+    return _PRODUCT_METHODS[op_key]
+
+
+def product(algebra, left: torch.Tensor, right: torch.Tensor, *, op: str = "gp", **kwargs: Any) -> torch.Tensor:
+    """Apply a binary geometric algebra product.
+
+    Args:
+        algebra: Dense ``CliffordAlgebra`` or planned ``AlgebraContext``.
+        left: Left operand.
+        right: Right operand.
+        op: ``"gp"``, ``"wedge"``, ``"inner"``, ``"commutator"``, or
+            ``"anti_commutator"``.
+        **kwargs: Optional grade/layout declarations accepted by
+            ``algebra.projected_product``.
+
+    Returns:
+        Product values in dense or compact form according to ``kwargs``.
+    """
+    planned_op, method_name = _resolve_product_op(op)
+    if kwargs:
+        return algebra.projected_product(left, right, op=planned_op, **kwargs)
+    return getattr(algebra, method_name)(left, right)
+
+
+def projected_product(
+    algebra,
+    left: torch.Tensor,
+    right: torch.Tensor,
+    *,
+    op: str = "gp",
+    **kwargs: Any,
+) -> torch.Tensor:
+    """Apply a declared grade-restricted product through the planner."""
+    return product(algebra, left, right, op=op, **kwargs)
+
+
+def geometric_product(algebra, left: torch.Tensor, right: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply the geometric product."""
+    return product(algebra, left, right, op="gp", **kwargs)
+
+
+def wedge(algebra, left: torch.Tensor, right: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply the exterior product."""
+    return product(algebra, left, right, op="wedge", **kwargs)
+
+
+def inner_product(algebra, left: torch.Tensor, right: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply the inner product."""
+    return product(algebra, left, right, op="inner", **kwargs)
+
+
+def commutator(algebra, left: torch.Tensor, right: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply the commutator product."""
+    return product(algebra, left, right, op="commutator", **kwargs)
+
+
+def anti_commutator(algebra, left: torch.Tensor, right: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply the anti-commutator product."""
+    return product(algebra, left, right, op="anti_commutator", **kwargs)
+
+
+def grade_projection(algebra, values: torch.Tensor, grade: int, **kwargs: Any) -> torch.Tensor:
+    """Project multivectors to one grade."""
+    return algebra.grade_projection(values, grade, **kwargs)
+
+
+def reverse(algebra, values: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply reversion."""
+    return algebra.reverse(values, **kwargs)
+
+
+def grade_involution(algebra, values: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply grade involution."""
+    return algebra.grade_involution(values, **kwargs)
+
+
+def clifford_conjugation(algebra, values: torch.Tensor, **kwargs: Any) -> torch.Tensor:
+    """Apply Clifford conjugation."""
+    return algebra.clifford_conjugation(values, **kwargs)
+
+
+def dual(algebra, values: torch.Tensor) -> torch.Tensor:
+    """Apply the Hodge dual."""
+    return algebra.dual(values)
+
+
+def norm_sq(algebra, values: torch.Tensor) -> torch.Tensor:
+    """Return the algebraic squared norm."""
+    return algebra.norm_sq(values)
+
+
+def embed_vector(algebra, vectors: torch.Tensor) -> torch.Tensor:
+    """Embed coordinate vectors into the grade-1 subspace."""
+    return algebra.embed_vector(vectors)
diff --git a/layers/__init__.py b/layers/__init__.py
index 2e2061c..457847b 100644
--- a/layers/__init__.py
+++ b/layers/__init__.py
@@ -3,7 +3,7 @@
 Organized into Primitives, Canonical Blocks, and Task-Specific Adapters.
 """
 
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
 
 from .adapters.embedding import MultivectorEmbedding, RotaryBivectorPE
 from .adapters.mother import EntropyGatedAttention, MotherEmbedding, PhaseShiftHead
@@ -13,7 +13,16 @@
 from .primitives.linear import CliffordLinear
 from .primitives.multi_rotor import MultiRotorLayer
 from .primitives.normalization import CliffordLayerNorm
+from .primitives.product import (
+    AntiCommutatorLayer,
+    CommutatorLayer,
+    GeometricProductLayer,
+    InnerProductLayer,
+    ProductLayer,
+    WedgeLayer,
+)
 from .primitives.projection import BladeSelector, GeometricNeutralizer
+from .primitives.reflection import ReflectionLayer
 from .primitives.rotor import RotorLayer
 from .primitives.rotor_gadget import RotorGadget
 
@@ -30,8 +39,15 @@
     "CliffordLinear",
     "RotorGadget",
     "CliffordLayerNorm",
+    "ProductLayer",
+    "GeometricProductLayer",
+    "WedgeLayer",
+    "InnerProductLayer",
+    "CommutatorLayer",
+    "AntiCommutatorLayer",
     "BladeSelector",
     "GeometricNeutralizer",
+    "ReflectionLayer",
     "MultivectorEmbedding",
     "RotaryBivectorPE",
     "MotherEmbedding",
diff --git a/layers/adapters/conformal.py b/layers/adapters/conformal.py
index cc48056..dd6f958 100644
--- a/layers/adapters/conformal.py
+++ b/layers/adapters/conformal.py
@@ -7,8 +7,8 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 
 class ConformalEmbedding(CliffordModule):
diff --git a/layers/adapters/embedding.py b/layers/adapters/embedding.py
index beb6300..1b0bfee 100644
--- a/layers/adapters/embedding.py
+++ b/layers/adapters/embedding.py
@@ -8,8 +8,9 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.manifold import MANIFOLD_SPIN, tag_manifold
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 
 class MultivectorEmbedding(CliffordModule):
@@ -122,7 +123,7 @@ def __init__(
 
         if learnable:
             self.bivector_weights = nn.Parameter(init)
-            self.bivector_weights._manifold = "spin"
+            tag_manifold(self.bivector_weights, MANIFOLD_SPIN)
         else:
             self.register_buffer("bivector_weights", init)
 
diff --git a/layers/adapters/gnn.py b/layers/adapters/gnn.py
index 71b6591..de87277 100644
--- a/layers/adapters/gnn.py
+++ b/layers/adapters/gnn.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 from ..primitives.linear import CliffordLinear
 
diff --git a/layers/adapters/mother.py b/layers/adapters/mother.py
index 29079a0..883fb5a 100644
--- a/layers/adapters/mother.py
+++ b/layers/adapters/mother.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 from ..blocks.attention import GeometricProductAttention
 from ..primitives.normalization import CliffordLayerNorm
diff --git a/layers/adapters/projective.py b/layers/adapters/projective.py
index 4058915..200ca9a 100644
--- a/layers/adapters/projective.py
+++ b/layers/adapters/projective.py
@@ -7,8 +7,8 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 
 class ProjectiveEmbedding(CliffordModule):
diff --git a/layers/blocks/attention.py b/layers/blocks/attention.py
index 3f54985..2caebbe 100644
--- a/layers/blocks/attention.py
+++ b/layers/blocks/attention.py
@@ -11,13 +11,15 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 from ..primitives.linear import CliffordLinear
 
 # Memory-bounded block size for chunked attention computation
 _BLOCK_SIZE = 64
+_G2_BLADE_CHUNK_SIZE = 16
+_SCORE_PRECOMPUTE_LIMIT = 8_000_000
 
 
 class GeometricProductAttention(CliffordModule):
@@ -50,6 +52,8 @@ def __init__(
         causal: bool = True,
         bivector_weight: float = 0.5,
         dropout: float = 0.0,
+        score_blade_chunk_size: int = _G2_BLADE_CHUNK_SIZE,
+        score_precompute_limit: int = _SCORE_PRECOMPUTE_LIMIT,
     ):
         """Sets up geometric product attention.
 
@@ -60,6 +64,10 @@ def __init__(
             causal: Apply causal mask for autoregressive generation.
             bivector_weight: lambda_ weight on bivector score component.
             dropout: Dropout rate on attention weights.
+            score_blade_chunk_size: Grade-2 output blades processed per dense
+                chunk when exact dense scoring is used.
+            score_precompute_limit: Maximum temporary ``K_g2`` elements allowed
+                before exact dense scoring switches to chunked grade-2 blades.
         """
         super().__init__(algebra)
         assert channels % num_heads == 0, f"channels ({channels}) must be divisible by num_heads ({num_heads})"
@@ -69,6 +77,8 @@ def __init__(
         self.head_channels = channels // num_heads
         self.causal = causal
         self.bivector_weight = bivector_weight
+        self.score_blade_chunk_size = max(1, int(score_blade_chunk_size))
+        self.score_precompute_limit = max(0, int(score_precompute_limit))
 
         # Q, K, V projections operate on [B*L, channels, dim]
         self.q_proj = CliffordLinear(algebra, channels, channels)
@@ -78,83 +88,37 @@ def __init__(
 
         self.attn_dropout = nn.Dropout(dropout) if dropout > 0.0 else None
 
-        # Precompute bilinear score tables (replaces pairwise geometric product)
+        # Precompute bilinear score routes (replaces pairwise geometric product)
         self._precompute_score_tables()
 
     def _precompute_score_tables(self):
-        """Precomputes lookup tables for efficient attention scoring.
-
-        Replaces the O(L**2) full pairwise geometric product with direct bilinear
-        forms for grade-0 and grade-2 components of Q * reverse(K):
-
-        Grade-0:  <Q * rev(K)>_0 = Sum_a Q[a] * K[a] * metric_rev[a]
-                  -> simple weighted dot product, no pairwise expansion needed.
-
-        Grade-2:  <Q * rev(K)>_r = Sum_a Q[a] * K[a^r] * g2_sign[r, a]
-                  -> precompute K_g2 once, then batched matmul.
-
-        Memory: ~4 MB peak vs ~256 MB for the naive B_gathered approach.
-        """
+        """Precompute exact dense attention score routes."""
         alg = self.algebra
         D = alg.dim
 
+        if not hasattr(alg, "gp_signs") or not hasattr(alg, "rev_signs"):
+            raise ValueError("GeometricProductAttention currently requires dense CliffordAlgebra inputs.")
+
         # Grade-0 metric: metric_rev[a] = gp_signs[a, 0] * rev_signs[a]
         # gp_signs[a, 0] is the sign when A[a] * B[a] contributes to output blade 0
         metric_rev = alg.gp_signs[:, 0].float() * alg.rev_signs.float()
         self.register_buffer("_metric_rev", metric_rev)  # [D]
 
-        # Grade-2 tables: for each grade-2 blade r, for each A-blade a:
-        #   B-blade  = a XOR r
-        #   sign     = rev_sign[a^r] * gp_signs[a, r]
         g2_blades = [i for i in range(D) if bin(i).count("1") == 2]
-        n_g2 = len(g2_blades)
-        self.n_g2 = n_g2
-
-        if n_g2 > 0:
-            a_idx = torch.arange(D, device=alg.device)
-            r_vals = torch.tensor(g2_blades, dtype=torch.long, device=alg.device)  # [n_g2]
-
-            # b_idx[r, a] = a XOR r_vals[r]
-            b_idx = a_idx.unsqueeze(0) ^ r_vals.unsqueeze(1)  # [n_g2, D]
-
-            # rev_sign at the B-blade position
-            rev_b = alg.rev_signs.float()[b_idx]  # [n_g2, D]
-
-            # gp_signs[a, r_val]: sign when A[a] pairs with B[a^r] to give output r
-            # alg.gp_signs[:, r_vals] -> [D, n_g2]; transpose -> [n_g2, D]
-            gp_ar = alg.gp_signs[:, r_vals].float().T  # [n_g2, D]
-
-            g2_sign = rev_b * gp_ar  # [n_g2, D]
-        else:
-            b_idx = torch.zeros(0, D, dtype=torch.long, device=alg.device)
-            g2_sign = torch.zeros(0, D, device=alg.device)
-
-        self.register_buffer("_g2_b_idx", b_idx)  # [n_g2, D] long
-        self.register_buffer("_g2_sign", g2_sign)  # [n_g2, D] float
+        self.n_g2 = len(g2_blades)
+        self.register_buffer("_g2_blades", torch.tensor(g2_blades, dtype=torch.long, device=alg.device))
+        self.register_buffer("_basis_indices", torch.arange(D, dtype=torch.long, device=alg.device))
 
     def _compute_score(
         self,
         q_head: torch.Tensor,
         k_head: torch.Tensor,
-        k_g2: torch.Tensor,
     ) -> torch.Tensor:
-        """Computes GA attention score using precomputed bilinear form tables.
-
-        Avoids the O(B.H.Lq.Lk.Hc.D.BLOCK) memory of the full pairwise
-        geometric product. Instead:
-
-          Grade-0: score_g0 = Q_weighted @ K^T  (weighted dot product, peak ~1 MB)
-          Grade-2: batched matmul via precomputed k_g2            (peak ~4 MB)
+        """Compute GA attention scores for one query block."""
+        return self._compute_score_dense(q_head, k_head)
 
-        Args:
-            q_head: Query block [B, H, Lq, Hc, D]
-            k_head: Keys        [B, H, Lk, Hc, D]
-            k_g2:   Precomputed [B, H, Lk, Hc, n_g2, D]
-                    k_g2[b,h,j,c,r,d] = K[b,h,j,c, d^r] * g2_sign[r, d]
-
-        Returns:
-            scores: [B, H, Lq, Lk]
-        """
+    def _compute_score_dense(self, q_head: torch.Tensor, k_head: torch.Tensor) -> torch.Tensor:
+        """Exact dense score with automatic full/prechunked grade-2 routing."""
         B, H, Lq, Hc, D = q_head.shape
         Lk = k_head.shape[2]
         n_g2 = self.n_g2
@@ -169,21 +133,15 @@ def _compute_score(
 
         # == Grade-2 score ====================================================
         # ||<Q * rev(K)>_2||_F = sqrt(Sum_c Sum_r (Sum_d Q[c,d]*k_g2[j,c,r,d])^2)
-        # Batched matmul merging (B, H, Hc) into one batch dimension:
-        #   q_2d:     [B*H*Hc, Lq, D]
-        #   k_g2_2d:  [B*H*Hc, Lk*n_g2, D]   (Lk and n_g2 merged, n_g2 varies fast)
-        #   comp:     [B*H*Hc, Lq, Lk*n_g2]
-        # Peak ~4 MB vs ~256 MB for the naive B_gathered approach.
         if n_g2 > 0:
             q_2d = q_head.permute(0, 1, 3, 2, 4).reshape(B * H * Hc, Lq, D)
-            # k_g2: [B, H, Lk, Hc, n_g2, D] -> permute to [B, H, Hc, Lk, n_g2, D]
-            k_g2_t = k_g2.permute(0, 1, 3, 2, 4, 5)
-            k_g2_2d = k_g2_t.reshape(B * H * Hc, Lk * n_g2, D)
-            # [B*H*Hc, Lq, D] @ [B*H*Hc, D, Lk*n_g2] -> [B*H*Hc, Lq, Lk*n_g2]
-            comp = torch.bmm(q_2d, k_g2_2d.transpose(-2, -1))
-            # Sum squared components over n_g2, then sum over Hc -> [B, H, Lq, Lk]
-            comp_sq = comp.reshape(B * H * Hc, Lq, Lk, n_g2).pow(2).sum(-1)  # [B*H*Hc, Lq, Lk]
-            score_g2_sq = comp_sq.reshape(B, H, Hc, Lq, Lk).sum(2)  # [B, H, Lq, Lk]
+
+            full_k_g2_elements = B * H * Lk * Hc * n_g2 * D
+            if full_k_g2_elements <= self.score_precompute_limit:
+                score_g2_sq = self._dense_score_g2_precomputed(q_2d, k_head, B, H, Hc, Lq, Lk, D, n_g2)
+            else:
+                k_2d = k_head.permute(0, 1, 3, 2, 4).reshape(B * H * Hc, Lk, D)
+                score_g2_sq = self._dense_score_g2_chunked(q_2d, k_2d, B, H, Hc, Lq, Lk, D, n_g2)
             score_g2 = score_g2_sq.sqrt()
         else:
             score_g2 = torch.zeros_like(score_g0)
@@ -192,6 +150,39 @@ def _compute_score(
         scale = math.sqrt(self.head_channels * self.algebra.dim)
         return (score_g0 + self.bivector_weight * score_g2) / scale
 
+    def _dense_score_g2_precomputed(self, q_2d, k_head, B, H, Hc, Lq, Lk, D, n_g2):
+        """Dense grade-2 score using one full shifted-key materialization."""
+        r_vals = self._g2_blades
+        b_idx = self._basis_indices.unsqueeze(0) ^ r_vals.unsqueeze(1)
+        rev_b = self.algebra.rev_signs[b_idx].to(dtype=k_head.dtype)
+        gp_ar = self.algebra.gp_signs[:, r_vals].T.to(dtype=k_head.dtype)
+        g2_sign = rev_b * gp_ar
+
+        k_g2 = k_head[..., b_idx] * g2_sign
+        k_g2_2d = k_g2.permute(0, 1, 3, 2, 4, 5).reshape(B * H * Hc, Lk * n_g2, D)
+        comp = torch.bmm(q_2d, k_g2_2d.transpose(-2, -1))
+        comp_sq = comp.reshape(B * H * Hc, Lq, Lk, n_g2).pow(2).sum(-1)
+        return comp_sq.reshape(B, H, Hc, Lq, Lk).sum(2)
+
+    def _dense_score_g2_chunked(self, q_2d, k_2d, B, H, Hc, Lq, Lk, D, n_g2):
+        """Dense grade-2 score using bounded output-blade chunks."""
+        score_g2_sq = q_2d.new_zeros(B, H, Lq, Lk)
+        for start in range(0, n_g2, self.score_blade_chunk_size):
+            end = min(start + self.score_blade_chunk_size, n_g2)
+            r_vals = self._g2_blades[start:end]
+            b_idx = self._basis_indices.unsqueeze(0) ^ r_vals.unsqueeze(1)
+            rev_b = self.algebra.rev_signs[b_idx].to(dtype=k_2d.dtype)
+            gp_ar = self.algebra.gp_signs[:, r_vals].T.to(dtype=k_2d.dtype)
+            g2_sign = rev_b * gp_ar
+
+            k_shifted = torch.index_select(k_2d, -1, b_idx.reshape(-1))
+            k_shifted = k_shifted * g2_sign.reshape(-1)
+            k_g2_2d = k_shifted.reshape(B * H * Hc, Lk * (end - start), D)
+            comp = torch.bmm(q_2d, k_g2_2d.transpose(-2, -1))
+            comp_sq = comp.reshape(B * H * Hc, Lq, Lk, end - start).pow(2).sum(-1)
+            score_g2_sq = score_g2_sq + comp_sq.reshape(B, H, Hc, Lq, Lk).sum(2)
+        return score_g2_sq
+
     def forward(self, x: torch.Tensor, key_padding_mask: torch.Tensor = None) -> torch.Tensor:
         """Computes geometric product attention.
 
@@ -226,11 +217,6 @@ def forward(self, x: torch.Tensor, key_padding_mask: torch.Tensor = None) -> tor
         else:
             causal_mask = None
 
-        # Precompute K_g2 once for all query blocks - much cheaper than recomputing
-        # k_g2[b,h,j,c,r,d] = K[b,h,j,c, d^r_val] * g2_sign[r, d]
-        # Shape: [B, H, L, Hc, n_g2, D]  ~= 768 KB for the small MPS config
-        K_g2 = K[..., self._g2_b_idx] * self._g2_sign  # [B, H, L, Hc, n_g2, D]
-
         # Chunked attention over query positions to bound memory
         output_chunks = []
         for q_start in range(0, L, _BLOCK_SIZE):
@@ -239,7 +225,7 @@ def forward(self, x: torch.Tensor, key_padding_mask: torch.Tensor = None) -> tor
             Q_block = Q[:, :, q_start:q_end]  # [B, H, Lq, Hc, D]
 
             # Compute scores: [B, H, Lq, L]
-            scores = self._compute_score(Q_block, K, K_g2)
+            scores = self._compute_score(Q_block, K)
 
             # Apply causal mask
             if causal_mask is not None:
diff --git a/layers/blocks/multi_rotor_ffn.py b/layers/blocks/multi_rotor_ffn.py
index fbc6891..1feb76f 100644
--- a/layers/blocks/multi_rotor_ffn.py
+++ b/layers/blocks/multi_rotor_ffn.py
@@ -7,8 +7,8 @@
 
 import torch
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from functional.activation import GeometricGELU
 
 from ..primitives.linear import CliffordLinear
diff --git a/layers/blocks/transformer.py b/layers/blocks/transformer.py
index e750083..0c5459f 100644
--- a/layers/blocks/transformer.py
+++ b/layers/blocks/transformer.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 from ..adapters.mother import EntropyGatedAttention
 from ..primitives.normalization import CliffordLayerNorm
diff --git a/layers/primitives/__init__.py b/layers/primitives/__init__.py
index e69de29..451f879 100644
--- a/layers/primitives/__init__.py
+++ b/layers/primitives/__init__.py
@@ -0,0 +1,34 @@
+"""Primitive Clifford neural network layers."""
+
+from .linear import CliffordLinear
+from .multi_rotor import MultiRotorLayer
+from .normalization import CliffordLayerNorm
+from .product import (
+    AntiCommutatorLayer,
+    CommutatorLayer,
+    GeometricProductLayer,
+    InnerProductLayer,
+    ProductLayer,
+    WedgeLayer,
+)
+from .projection import BladeSelector, GeometricNeutralizer
+from .reflection import ReflectionLayer
+from .rotor import RotorLayer
+from .rotor_gadget import RotorGadget
+
+__all__ = [
+    "CliffordLinear",
+    "MultiRotorLayer",
+    "CliffordLayerNorm",
+    "ProductLayer",
+    "GeometricProductLayer",
+    "WedgeLayer",
+    "InnerProductLayer",
+    "CommutatorLayer",
+    "AntiCommutatorLayer",
+    "BladeSelector",
+    "GeometricNeutralizer",
+    "ReflectionLayer",
+    "RotorLayer",
+    "RotorGadget",
+]
diff --git a/layers/primitives/linear.py b/layers/primitives/linear.py
index 6cd5003..a678506 100644
--- a/layers/primitives/linear.py
+++ b/layers/primitives/linear.py
@@ -15,9 +15,9 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
-from core.validation import check_channels, check_multivector
+from core.foundation.module import CliffordModule
+from core.foundation.validation import check_channels, check_multivector
+from core.runtime.algebra import CliffordAlgebra
 
 
 class CliffordLinear(CliffordModule):
diff --git a/layers/primitives/multi_rotor.py b/layers/primitives/multi_rotor.py
index d51d385..0198f0b 100644
--- a/layers/primitives/multi_rotor.py
+++ b/layers/primitives/multi_rotor.py
@@ -13,9 +13,10 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
-from core.validation import check_channels, check_multivector
+from core.foundation.manifold import MANIFOLD_SPIN, tag_manifold
+from core.foundation.module import CliffordModule
+from core.foundation.validation import check_channels, check_multivector
+from core.runtime.algebra import CliffordAlgebra
 
 
 class MultiRotorLayer(CliffordModule):
@@ -63,7 +64,7 @@ def __init__(
 
         self.rotor_grade_weights = nn.Parameter(torch.Tensor(num_rotors, self.num_grade_elements))
         if grade == 2:
-            self.rotor_grade_weights._manifold = "spin"
+            tag_manifold(self.rotor_grade_weights, MANIFOLD_SPIN)
 
         # Mixing weights (Euclidean — intentionally untagged)
         self.weights = nn.Parameter(torch.Tensor(channels, num_rotors))
diff --git a/layers/primitives/normalization.py b/layers/primitives/normalization.py
index c10cf2c..149566b 100644
--- a/layers/primitives/normalization.py
+++ b/layers/primitives/normalization.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 
 
 class CliffordLayerNorm(CliffordModule):
diff --git a/layers/primitives/product.py b/layers/primitives/product.py
new file mode 100644
index 0000000..90c9819
--- /dev/null
+++ b/layers/primitives/product.py
@@ -0,0 +1,138 @@
+"""Module wrappers around declared geometric algebra products."""
+
+from __future__ import annotations
+
+from typing import Iterable, Optional
+
+import torch
+
+from core.foundation.basis import normalize_grades
+from core.foundation.module import CliffordModule
+from functional.products import canonical_product_op, product
+
+
+def _normalize_optional_grades(grades: Optional[Iterable[int]], n: int, *, name: str) -> Optional[tuple[int, ...]]:
+    if grades is None:
+        return None
+    if isinstance(grades, int):
+        grades = (grades,)
+    return normalize_grades(grades, n, name=name)
+
+
+class ProductLayer(CliffordModule):
+    """Apply a geometric algebra product inside ``nn.Module`` graphs.
+
+    The layer is intentionally thin: grade declarations and compact/pairwise
+    behavior are forwarded to ``algebra.projected_product`` when supplied,
+    while the dense no-declaration path uses the algebra's direct kernels.
+    """
+
+    def __init__(
+        self,
+        algebra,
+        *,
+        op: str = "gp",
+        left_grades: Optional[Iterable[int]] = None,
+        right_grades: Optional[Iterable[int]] = None,
+        output_grades: Optional[Iterable[int]] = None,
+        left_compact: bool = False,
+        right_compact: bool = False,
+        compact_output: bool = False,
+        pairwise: bool = False,
+    ):
+        """Initialize a product layer.
+
+        Args:
+            algebra: Dense ``CliffordAlgebra`` or planned ``AlgebraContext``.
+            op: Product route: ``"gp"``, ``"wedge"``, ``"inner"``,
+                ``"commutator"``, or ``"anti_commutator"``.
+            left_grades: Declared input grades for the left operand.
+            right_grades: Declared input grades for the right operand.
+            output_grades: Optional output grade projection.
+            left_compact: Whether the left operand is already compact.
+            right_compact: Whether the right operand is already compact.
+            compact_output: Return compact output instead of dense coefficients.
+            pairwise: Treat the penultimate dimension of each compact operand
+                as independent left/right item axes.
+        """
+        super().__init__(algebra)
+        self.op = canonical_product_op(op)
+        self.left_grades = _normalize_optional_grades(left_grades, algebra.n, name="left_grades")
+        self.right_grades = _normalize_optional_grades(right_grades, algebra.n, name="right_grades")
+        self.output_grades = _normalize_optional_grades(output_grades, algebra.n, name="output_grades")
+        self.left_compact = bool(left_compact)
+        self.right_compact = bool(right_compact)
+        self.compact_output = bool(compact_output)
+        self.pairwise = bool(pairwise)
+
+    def forward(self, left: torch.Tensor, right: torch.Tensor) -> torch.Tensor:
+        """Apply the configured product to ``left`` and ``right``."""
+        kwargs = self._product_kwargs()
+        return product(self.algebra, left, right, op=self.op, **kwargs)
+
+    def _product_kwargs(self) -> dict:
+        kwargs = {}
+        if self.left_grades is not None:
+            kwargs["left_grades"] = self.left_grades
+        if self.right_grades is not None:
+            kwargs["right_grades"] = self.right_grades
+        if self.output_grades is not None:
+            kwargs["output_grades"] = self.output_grades
+        if self.left_compact:
+            kwargs["left_compact"] = True
+        if self.right_compact:
+            kwargs["right_compact"] = True
+        if self.compact_output:
+            kwargs["compact_output"] = True
+        if self.pairwise:
+            kwargs["pairwise"] = True
+        return kwargs
+
+    def extra_repr(self) -> str:
+        parts = [f"op={self.op!r}"]
+        if self.left_grades is not None:
+            parts.append(f"left_grades={self.left_grades}")
+        if self.right_grades is not None:
+            parts.append(f"right_grades={self.right_grades}")
+        if self.output_grades is not None:
+            parts.append(f"output_grades={self.output_grades}")
+        if self.compact_output:
+            parts.append("compact_output=True")
+        if self.pairwise:
+            parts.append("pairwise=True")
+        return ", ".join(parts)
+
+
+class GeometricProductLayer(ProductLayer):
+    """Layer form of the geometric product."""
+
+    def __init__(self, algebra, **kwargs):
+        super().__init__(algebra, op="gp", **kwargs)
+
+
+class WedgeLayer(ProductLayer):
+    """Layer form of the exterior product."""
+
+    def __init__(self, algebra, **kwargs):
+        super().__init__(algebra, op="wedge", **kwargs)
+
+
+class InnerProductLayer(ProductLayer):
+    """Layer form of the inner product."""
+
+    def __init__(self, algebra, **kwargs):
+        super().__init__(algebra, op="inner", **kwargs)
+
+
+class CommutatorLayer(ProductLayer):
+    """Layer form of the commutator product."""
+
+    def __init__(self, algebra, **kwargs):
+        super().__init__(algebra, op="commutator", **kwargs)
+
+
+class AntiCommutatorLayer(ProductLayer):
+    """Layer form of the anti-commutator product."""
+
+    def __init__(self, algebra, **kwargs):
+        super().__init__(algebra, op="anti_commutator", **kwargs)
diff --git a/layers/primitives/projection.py b/layers/primitives/projection.py
index 4034292..4df6aa3 100644
--- a/layers/primitives/projection.py
+++ b/layers/primitives/projection.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from utils.compat import safe_linalg_solve
 
 
diff --git a/layers/primitives/reflection.py b/layers/primitives/reflection.py
index b37f252..0d83b47 100644
--- a/layers/primitives/reflection.py
+++ b/layers/primitives/reflection.py
@@ -8,9 +8,10 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
-from core.validation import check_channels, check_multivector
+from core.foundation.manifold import MANIFOLD_SPHERE, tag_manifold
+from core.foundation.module import CliffordModule
+from core.foundation.validation import check_channels, check_multivector
+from core.runtime.algebra import CliffordAlgebra
 
 
 class ReflectionLayer(CliffordModule):
@@ -46,7 +47,7 @@ def __init__(self, algebra: CliffordAlgebra, channels: int):
         self.num_vectors = algebra.n
 
         self.vector_weights = nn.Parameter(torch.Tensor(channels, self.num_vectors))
-        self.vector_weights._manifold = "sphere"
+        tag_manifold(self.vector_weights, MANIFOLD_SPHERE)
 
         # Cache for eval mode
         self._cached_n = None
diff --git a/layers/primitives/rotor.py b/layers/primitives/rotor.py
index 51732ef..2fc064c 100644
--- a/layers/primitives/rotor.py
+++ b/layers/primitives/rotor.py
@@ -8,9 +8,10 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
-from core.validation import check_channels, check_multivector
+from core.foundation.manifold import MANIFOLD_SPIN, tag_manifold
+from core.foundation.module import CliffordModule
+from core.foundation.validation import check_channels, check_multivector
+from core.runtime.algebra import CliffordAlgebra
 
 
 class RotorLayer(CliffordModule):
@@ -23,7 +24,7 @@ class RotorLayer(CliffordModule):
     Preserves origin. For grade=2, also preserves lengths and angles (isometry).
 
     The exp strategy (closed-form vs decomposition) is controlled by
-    ``algebra.exp_policy`` -- see :class:`core.decomposition.ExpPolicy`.
+    ``algebra.exp_policy`` -- see :class:`core.runtime.decomposition.ExpPolicy`.
 
     Attributes:
         channels (int): Number of versors.
@@ -57,7 +58,7 @@ def __init__(
 
         self.grade_weights = nn.Parameter(torch.Tensor(channels, self.num_grade_elements))
         if grade == 2:
-            self.grade_weights._manifold = "spin"
+            tag_manifold(self.grade_weights, MANIFOLD_SPIN)
 
         # Versor cache for eval mode
         self._cached_V_left = None
diff --git a/layers/primitives/rotor_gadget.py b/layers/primitives/rotor_gadget.py
index 80b653e..bd283ba 100644
--- a/layers/primitives/rotor_gadget.py
+++ b/layers/primitives/rotor_gadget.py
@@ -14,9 +14,10 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
-from core.validation import check_channels, check_multivector
+from core.foundation.manifold import MANIFOLD_SPIN, tag_manifold
+from core.foundation.module import CliffordModule
+from core.foundation.validation import check_channels, check_multivector
+from core.runtime.algebra import CliffordAlgebra
 
 
 class RotorGadget(CliffordModule):
@@ -89,10 +90,10 @@ def __init__(
         # Rotor parameters: bivector coefficients for exponential map
         # Left rotors: [num_rotor_pairs, num_bivectors]
         self.bivector_left = nn.Parameter(torch.randn(num_rotor_pairs, self.num_bivectors) * 0.1)
-        self.bivector_left._manifold = "spin"
+        tag_manifold(self.bivector_left, MANIFOLD_SPIN)
         # Right rotors: [num_rotor_pairs, num_bivectors]
         self.bivector_right = nn.Parameter(torch.randn(num_rotor_pairs, self.num_bivectors) * 0.1)
-        self.bivector_right._manifold = "spin"
+        tag_manifold(self.bivector_right, MANIFOLD_SPIN)
 
         # Channel routing: block diagonal partitioning (paper style)
         # Each rotor pair processes a subset of input channels
diff --git a/mkdocs.yml b/mkdocs.yml
index 8727d0f..d126966 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -37,6 +37,7 @@ nav:
   - Home: index.md
   - Guide:
       - Design Guide: design_guide.md
+      - Architecture: Architecture.md
       - Philosophy: philosophy.md
       - Mathematics: mathematical.md
       - Innovations: innovations.md
diff --git a/models/blocks/gbn.py b/models/blocks/gbn.py
index 79db066..2000f22 100644
--- a/models/blocks/gbn.py
+++ b/models/blocks/gbn.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from functional.activation import GeometricGELU
 from layers import BladeSelector, CliffordLinear, RotorLayer
 
diff --git a/models/blocks/multi_rotor.py b/models/blocks/multi_rotor.py
index 60fb452..145afab 100644
--- a/models/blocks/multi_rotor.py
+++ b/models/blocks/multi_rotor.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from functional.activation import GeometricGELU
 from layers import CliffordLinear, MultiRotorLayer
 
diff --git a/models/blocks/time_series.py b/models/blocks/time_series.py
index f301e9d..f674eaf 100644
--- a/models/blocks/time_series.py
+++ b/models/blocks/time_series.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from layers import CliffordLinear, RotorLayer
 
 
diff --git a/models/deap/eeg_net.py b/models/deap/eeg_net.py
index 6d2e2b9..bf4de26 100644
--- a/models/deap/eeg_net.py
+++ b/models/deap/eeg_net.py
@@ -18,11 +18,13 @@
 (immediate) and Grade-4 (long-range) for VADL prediction.
 """
 
+from typing import Optional
+
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.config import make_algebra, make_algebra_from_config
+from core.foundation.module import AlgebraLike, CliffordModule
 from layers import (
     CliffordLayerNorm,
     GeometricNeutralizer,
@@ -39,7 +41,7 @@ class MultiTargetPhaseShiftHead(CliffordModule):
     each target can independently shift its prediction range.
     """
 
-    def __init__(self, algebra: CliffordAlgebra, channels: int, num_targets: int = 4):
+    def __init__(self, algebra: AlgebraLike, channels: int, num_targets: int = 4):
         super().__init__(algebra)
         self.channels = channels
         self.num_targets = num_targets
@@ -66,7 +68,7 @@ class EEGNet(CliffordModule):
     brain region's Grade-0 is cleaned of its own Grade-2 artifacts independently.
     """
 
-    def __init__(self, group_sizes, profiles=None, device=None, config=None):
+    def __init__(self, group_sizes, profiles=None, device=None, config=None, algebra: Optional[AlgebraLike] = None):
         """Initialize EEGNet.
 
         Args:
@@ -77,16 +79,21 @@ def __init__(self, group_sizes, profiles=None, device=None, config=None):
             device: Torch device.
             config: Hydra DictConfig or plain dict with model hyperparameters.
         """
-        p, q = 3, 1
-        if config is not None:
-            if hasattr(config, "algebra"):
-                p = config.algebra.get("p", 3)
-                q = config.algebra.get("q", 1)
-            elif isinstance(config, dict):
-                p = config.get("p", 3)
-                q = config.get("q", 1)
-
-        algebra = CliffordAlgebra(p, q, device=device)
+        if algebra is None:
+            algebra_config = None
+            if config is not None:
+                if hasattr(config, "algebra"):
+                    algebra_config = config.algebra
+                elif isinstance(config, dict):
+                    algebra_config = config.get("algebra", config)
+
+            p = algebra_config.get("p", 3) if algebra_config is not None else 3
+            q = algebra_config.get("q", 1) if algebra_config is not None else 1
+            r = algebra_config.get("r", 0) if algebra_config is not None else 0
+            if algebra_config is not None:
+                algebra = make_algebra_from_config(algebra_config, p=p, q=q, r=r, device=device)
+            else:
+                algebra = make_algebra(p=p, q=q, r=r, device=device or "cpu")
         super().__init__(algebra)
 
         if config is not None and hasattr(config, "model"):
diff --git a/models/lqa/glr_net.py b/models/lqa/glr_net.py
index 74969a3..f91e075 100644
--- a/models/lqa/glr_net.py
+++ b/models/lqa/glr_net.py
@@ -24,8 +24,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from layers.adapters.embedding import RotaryBivectorPE
 from layers.adapters.mother import MotherEmbedding
 from layers.blocks.transformer import GeometricTransformerBlock
diff --git a/models/lqa/heads.py b/models/lqa/heads.py
index 7eb985d..13bdc3f 100644
--- a/models/lqa/heads.py
+++ b/models/lqa/heads.py
@@ -17,8 +17,8 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from layers.primitives.projection import GeometricNeutralizer
 from layers.primitives.rotor import RotorLayer
 
diff --git a/models/md17/forcenet.py b/models/md17/forcenet.py
index 8c630f9..c96bffa 100644
--- a/models/md17/forcenet.py
+++ b/models/md17/forcenet.py
@@ -9,8 +9,8 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from functional.activation import GeometricGELU, GeometricSquare
 from layers import BladeSelector, CliffordLayerNorm, CliffordLinear, MultiRotorLayer
 
diff --git a/models/sr/estimator.py b/models/sr/estimator.py
index 121890f..b802786 100644
--- a/models/sr/estimator.py
+++ b/models/sr/estimator.py
@@ -16,8 +16,8 @@
 import torch
 from sklearn.base import BaseEstimator, RegressorMixin
 
-from core.algebra import CliffordAlgebra
-from core.decomposition import ExpPolicy
+from core.config import make_algebra
+from core.runtime.decomposition import ExpPolicy
 from models.sr.net import SRGBN
 from models.sr.utils import make_lambdify_fn
 from optimizers.riemannian import RiemannianAdam
@@ -93,7 +93,7 @@ def fit(self, X, y):
         y_t = torch.from_numpy(y_norm).unsqueeze(-1)
 
         n_vars = X.shape[1]
-        algebra = CliffordAlgebra(p=self.p, q=self.q, r=self.r, device="cpu", exp_policy=self.exp_policy)
+        algebra = make_algebra(p=self.p, q=self.q, r=self.r, device="cpu", exp_policy=self.exp_policy)
 
         self.model_ = SRGBN(
             algebra=algebra,
diff --git a/models/sr/grouper.py b/models/sr/grouper.py
index fd79ed9..67a02f9 100644
--- a/models/sr/grouper.py
+++ b/models/sr/grouper.py
@@ -20,7 +20,8 @@
 import numpy as np
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra
+from core.foundation.module import AlgebraLike
 from models.sr.utils import safe_svd, standardize, subsample
 
 logger = logging.getLogger(__name__)
@@ -34,7 +35,7 @@ class VariableGroup:
         var_indices: Indices into original X columns.
         var_names: Human-readable variable names.
         signature: (p, q, r) from MetricSearch.
-        algebra: CliffordAlgebra for this group.
+        algebra: Shared dense algebra or planning context for this group.
         svd_Vt: SVD right-singular vectors for this group (or None).
         mother_offset: Bit offset in mother algebra basis.
         internal_edges: VariableEdge list within this group.
@@ -45,7 +46,7 @@ class VariableGroup:
     var_indices: list
     var_names: list
     signature: tuple
-    algebra: CliffordAlgebra
+    algebra: AlgebraLike
     svd_Vt: np.ndarray = None
     mother_offset: int = 0
     internal_edges: list = field(default_factory=list)
@@ -223,7 +224,7 @@ def _build_relationship_graph(self, X, y, var_names):
             n_analysis,
         )
 
-        algebra = CliffordAlgebra(p, q, r, device=self.device)
+        algebra = make_algebra(p, q, r, device=self.device)
 
         # 5. Embed as grade-1 multivectors
         alg_n = algebra.n
@@ -469,13 +470,13 @@ def build_mother_algebra(self, groups):
         Q = sum(g.signature[1] for g in groups)
         R = sum(g.signature[2] for g in groups)
 
-        if P + Q + R > 12:
-            self._reduce_groups(groups, target_n=12)
+        if P + Q + R > 16:
+            self._reduce_groups(groups, target_n=16)
             P = sum(g.signature[0] for g in groups)
             Q = sum(g.signature[1] for g in groups)
             R = sum(g.signature[2] for g in groups)
 
-        mother = CliffordAlgebra(P, Q, R, device=self.device)
+        mother = make_algebra(P, Q, R, device=self.device)
 
         p_offset = 0
         q_offset = P
@@ -570,7 +571,7 @@ def _single_group(self, X, y, var_names):
 
         p, q, r = safe_metric_search(data, self.device, n_vars)
 
-        algebra = CliffordAlgebra(p, q, r, device=self.device)
+        algebra = make_algebra(p, q, r, device=self.device)
         return VariableGroup(
             var_indices=indices,
             var_names=[var_names[i] for i in indices],
@@ -612,7 +613,7 @@ def _build_group(self, X, y, indices, var_names):
             probe_epochs=20,
         )
 
-        algebra = CliffordAlgebra(p, q, r, device=self.device)
+        algebra = make_algebra(p, q, r, device=self.device)
         return VariableGroup(
             var_indices=indices,
             var_names=names_sub,
@@ -692,7 +693,7 @@ def _reduce_groups(self, groups, target_n=12):
             new_r = 0
             reduction = (p + q + r) - (new_p + new_q + new_r)
             g.signature = (new_p, new_q, new_r)
-            g.algebra = CliffordAlgebra(new_p, new_q, new_r, device=self.device)
+            g.algebra = make_algebra(new_p, new_q, new_r, device=self.device)
             total -= reduction
 
             if reduction == 0:
diff --git a/models/sr/implicit.py b/models/sr/implicit.py
index a7272f1..05e1725 100644
--- a/models/sr/implicit.py
+++ b/models/sr/implicit.py
@@ -24,7 +24,7 @@
 import torch
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra
 from models.sr.net import SRGBN
 from models.sr.utils import safe_sympy_solve
 from optimizers.riemannian import RiemannianAdam
@@ -111,7 +111,7 @@ def probe_best_mode(self, algebra, X, y, geometric_report=None):
 
         # Build implicit algebra: add 1 to p for the y variable
         p, q, r = algebra.p, algebra.q, algebra.r
-        impl_algebra = CliffordAlgebra(p + 1, q, r, device=self.device)
+        impl_algebra = make_algebra(p + 1, q, r, device=self.device)
 
         implicit_loss = self._probe_implicit(impl_algebra, Z)
 
diff --git a/models/sr/net.py b/models/sr/net.py
index a9aa722..8901003 100644
--- a/models/sr/net.py
+++ b/models/sr/net.py
@@ -17,8 +17,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.module import CliffordModule
+from core.foundation.module import CliffordModule
+from core.runtime.algebra import CliffordAlgebra
 from functional.activation import GeometricGELU, GeometricSquare
 from layers import BladeSelector, CliffordLayerNorm, CliffordLinear, RotorLayer
 
diff --git a/models/sr/phases/extraction.py b/models/sr/phases/extraction.py
index ab11032..c062003 100644
--- a/models/sr/phases/extraction.py
+++ b/models/sr/phases/extraction.py
@@ -18,8 +18,8 @@
 import torch
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
 from core.analysis import GeodesicFlow, MetricSearch
+from core.config import make_algebra
 from models.sr.net import SRGBN
 from models.sr.translator import RotorTerm, RotorTranslator
 from models.sr.utils import (
@@ -83,7 +83,7 @@ def _process_group_implicit(self, group, group_idx, prep, X_orig, y_orig, X_norm
 
         # Build augmented algebra (k+1 variables)
         p, q, r = group.signature
-        impl_algebra = CliffordAlgebra(p + 1, q, r, device=self.device)
+        impl_algebra = make_algebra(p + 1, q, r, device=self.device)
 
         # Augmented data Z = [X_group_norm, y_norm]
         X_group_norm = standardize(torch.tensor(X_orig[:, var_indices], dtype=torch.float32, device=self.device))
diff --git a/models/sr/phases/prep.py b/models/sr/phases/prep.py
index b5f5649..d6162ed 100644
--- a/models/sr/phases/prep.py
+++ b/models/sr/phases/prep.py
@@ -17,7 +17,7 @@
 import sympy
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra
 from models.sr.translator import RotorTerm
 from models.sr.utils import (
     make_lambdify_fn,
@@ -130,7 +130,7 @@ def _single_group_fallback(self, X_orig, y_orig, var_names, Vt):
             max_p=max(n_vars, 2),
         )
 
-        algebra = CliffordAlgebra(p, q, r, device=self.device)
+        algebra = make_algebra(p, q, r, device=self.device)
         return VariableGroup(
             var_indices=list(range(n_vars)),
             var_names=var_names or [f"x{i + 1}" for i in range(n_vars)],
diff --git a/models/sr/translator.py b/models/sr/translator.py
index 890ced0..8833177 100644
--- a/models/sr/translator.py
+++ b/models/sr/translator.py
@@ -30,7 +30,7 @@
 import sympy
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from models.sr.utils import LAMBDIFY_MODULES, make_lambdify_fn
 
 logger = logging.getLogger(__name__)
diff --git a/optimizers/__init__.py b/optimizers/__init__.py
index 4524aa9..2eea3d6 100644
--- a/optimizers/__init__.py
+++ b/optimizers/__init__.py
@@ -12,6 +12,7 @@
     RiemannianAdam,
     exponential_retraction,
     group_parameters_by_manifold,
+    make_riemannian_optimizer,
     project_to_tangent_space,
     tag_manifold,
 )
@@ -23,6 +24,7 @@
     "exponential_retraction",
     "tag_manifold",
     "group_parameters_by_manifold",
+    "make_riemannian_optimizer",
     "MANIFOLD_SPIN",
     "MANIFOLD_SPHERE",
     "MANIFOLD_EUCLIDEAN",
diff --git a/optimizers/riemannian.py b/optimizers/riemannian.py
index 474db4a..932ed78 100644
--- a/optimizers/riemannian.py
+++ b/optimizers/riemannian.py
@@ -24,31 +24,14 @@
 import torch.nn as nn
 from torch.optim import Optimizer
 
-MANIFOLD_SPIN = "spin"
-MANIFOLD_SPHERE = "sphere"
-MANIFOLD_EUCLIDEAN = "euclidean"
-
-_VALID_MANIFOLDS = {MANIFOLD_SPIN, MANIFOLD_SPHERE, MANIFOLD_EUCLIDEAN}
-
-
-def tag_manifold(param: nn.Parameter, manifold: str) -> nn.Parameter:
-    """Tag a parameter with its Riemannian manifold type.
-
-    Layers should call this (or set ``param._manifold`` directly) in their
-    ``__init__`` so that :meth:`RiemannianAdam.from_model` can auto-group
-    parameters for correct per-manifold retraction.
-
-    Args:
-        param: The parameter to tag.
-        manifold: One of ``'spin'``, ``'sphere'``, ``'euclidean'``.
-
-    Returns:
-        The same parameter (for chaining).
-    """
-    if manifold not in _VALID_MANIFOLDS:
-        raise ValueError(f"Unknown manifold '{manifold}'. Must be one of {_VALID_MANIFOLDS}")
-    param._manifold = manifold
-    return param
+from core.foundation.manifold import (
+    MANIFOLD_EUCLIDEAN,
+    MANIFOLD_ORDER,
+    MANIFOLD_SPHERE,
+    MANIFOLD_SPIN,
+    tag_manifold,
+    validate_manifold,
+)
 
 
 def group_parameters_by_manifold(
@@ -71,11 +54,50 @@ def group_parameters_by_manifold(
         MANIFOLD_EUCLIDEAN: [],
     }
     for p in model.parameters():
-        manifold = getattr(p, "_manifold", MANIFOLD_EUCLIDEAN)
+        manifold = validate_manifold(getattr(p, "_manifold", MANIFOLD_EUCLIDEAN))
         groups[manifold].append(p)
     return groups
 
 
+def _parameter_groups_for_model(model: nn.Module) -> list[dict]:
+    grouped = group_parameters_by_manifold(model)
+    param_groups = []
+    for manifold in MANIFOLD_ORDER:
+        params = grouped[manifold]
+        if params:
+            param_groups.append({"params": params, "manifold": manifold})
+    if not param_groups:
+        raise ValueError("Model has no parameters")
+    return param_groups
+
+
+def make_riemannian_optimizer(
+    model: nn.Module,
+    algebra,
+    *,
+    optimizer: str = "adam",
+    **kwargs,
+) -> Optimizer:
+    """Create a manifold-aware optimizer from a tagged model.
+
+    Args:
+        model: Model whose parameters may be tagged with ``_manifold``.
+        algebra: Clifford algebra instance used by the optimizer.
+        optimizer: ``"adam"``/``"riemannian_adam"`` or
+            ``"sgd"``/``"exponential_sgd"``.
+        **kwargs: Optimizer-specific keyword arguments.
+
+    Returns:
+        ``RiemannianAdam`` or ``ExponentialSGD`` with per-manifold groups.
+    """
+    key = optimizer.lower().replace("-", "_")
+    if key in {"adam", "riemannian_adam"}:
+        return RiemannianAdam.from_model(model, algebra=algebra, **kwargs)
+    if key in {"sgd", "exponential_sgd"}:
+        return ExponentialSGD.from_model(model, algebra=algebra, **kwargs)
+    raise ValueError("optimizer must be one of 'adam', 'riemannian_adam', 'sgd', or 'exponential_sgd'")
+
+
 class ExponentialSGD(Optimizer):
     """SGD with exponential map retraction for rotor parameters.
 
@@ -154,14 +176,7 @@ def from_model(
         Returns:
             ExponentialSGD instance with per-manifold parameter groups.
         """
-        grouped = group_parameters_by_manifold(model)
-        param_groups = []
-        for manifold in (MANIFOLD_SPIN, MANIFOLD_SPHERE, MANIFOLD_EUCLIDEAN):
-            params = grouped[manifold]
-            if params:
-                param_groups.append({"params": params, "manifold": manifold})
-        if not param_groups:
-            raise ValueError("Model has no parameters")
+        param_groups = _parameter_groups_for_model(model)
         return cls(param_groups, lr=lr, momentum=momentum, algebra=algebra, max_bivector_norm=max_bivector_norm)
 
     @torch.no_grad()
@@ -294,14 +309,7 @@ def from_model(
         Returns:
             RiemannianAdam instance with per-manifold parameter groups.
         """
-        grouped = group_parameters_by_manifold(model)
-        param_groups = []
-        for manifold in (MANIFOLD_SPIN, MANIFOLD_SPHERE, MANIFOLD_EUCLIDEAN):
-            params = grouped[manifold]
-            if params:
-                param_groups.append({"params": params, "manifold": manifold})
-        if not param_groups:
-            raise ValueError("Model has no parameters")
+        param_groups = _parameter_groups_for_model(model)
         return cls(param_groups, lr=lr, betas=betas, eps=eps, algebra=algebra, max_bivector_norm=max_bivector_norm)
 
     @torch.no_grad()
diff --git a/pyproject.toml b/pyproject.toml
index 40c6321..97fc84b 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -3,7 +3,7 @@ name = "versor"
 version = "1.0.0"
 description = "A PyTorch framework for Geometric Algebra Deep Learning"
 readme = "README.md"
-requires-python = ">=3.9"
+requires-python = ">=3.10"
 license = { file = "LICENSE" }
 authors = [
     { name = "Eunkyum Kim", email = "nemonanconcode@gmail.com" },
@@ -24,7 +24,6 @@ classifiers = [
     "Topic :: Scientific/Engineering :: Mathematics",
     "License :: OSI Approved :: Apache Software License",
     "Programming Language :: Python :: 3",
-    "Programming Language :: Python :: 3.9",
     "Programming Language :: Python :: 3.10",
     "Programming Language :: Python :: 3.11",
     "Programming Language :: Python :: 3.12",
@@ -34,6 +33,7 @@ dependencies = [
     "numpy>=1.26.0",
     "hydra-core>=1.3.2",
     "tqdm>=4.67.3",
+    "setuptools>=80.10.2",
 ]
 
 [project.urls]
@@ -77,7 +77,7 @@ all = [
 include = ["core*", "layers*", "models*", "datalib*", "optimizers*", "functional*", "tasks*", "examples*", "experiments*"]
 
 [tool.ruff]
-target-version = "py39"
+target-version = "py310"
 line-length = 120
 indent-width = 4
 extend-exclude = [
diff --git a/tasks/base.py b/tasks/base.py
index 8435e41..9d4c591 100644
--- a/tasks/base.py
+++ b/tasks/base.py
@@ -13,7 +13,7 @@
 from omegaconf import DictConfig
 from tqdm import tqdm
 
-from core.device import DeviceConfig, resolve_device
+from core.foundation.device import DeviceConfig, resolve_device
 from log import get_logger
 
 logger = get_logger(__name__)
@@ -50,6 +50,7 @@ def __init__(self, cfg: DictConfig):
             compile_model=cfg.training.get("compile", False),
             compile_backend=cfg.training.get("compile_backend", None),
             amp=cfg.training.get("amp", False),
+            amp_dtype=cfg.training.get("amp_dtype", None),
             cudnn_benchmark=cfg.training.get("cudnn_benchmark", None),
         )
         self.device = self.device_config.device
@@ -82,7 +83,7 @@ def _resolve_device(device: str) -> str:
         Priority: cuda > mps > cpu.
 
         .. deprecated::
-            Use :func:`core.device.resolve_device` instead.
+            Use :func:`core.foundation.device.resolve_device` instead.
         """
         return resolve_device(device)
 
diff --git a/tasks/deap_eeg.py b/tasks/deap_eeg.py
index 2595096..76ff22c 100644
--- a/tasks/deap_eeg.py
+++ b/tasks/deap_eeg.py
@@ -16,7 +16,7 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra_from_config
 from datalib.deap import get_deap_loaders, get_group_sizes
 from log import get_logger
 from models.deap import EEGNet
@@ -45,7 +45,8 @@ def __init__(self, cfg):
         super().__init__(cfg)
 
     def setup_algebra(self):
-        return CliffordAlgebra(
+        return make_algebra_from_config(
+            self.cfg.algebra,
             p=self.cfg.algebra.get("p", 3),
             q=self.cfg.algebra.get("q", 1),
             r=self.cfg.algebra.get("r", 0),
@@ -64,6 +65,7 @@ def setup_model(self):
             profiles=profiles,
             device=self.device,
             config=self.cfg,
+            algebra=self.algebra,
         )
 
     def _compute_profiles(self, group_sizes):
diff --git a/tasks/lqa.py b/tasks/lqa.py
index 2d049e5..17a4e63 100644
--- a/tasks/lqa.py
+++ b/tasks/lqa.py
@@ -21,7 +21,7 @@
 import torch.nn.functional as F
 from omegaconf import DictConfig
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra_from_config
 from datalib.lqa import get_lqa_loaders
 from functional.loss import AsymmetryLoss, InvolutionConsistencyLoss
 from log import get_logger
@@ -57,7 +57,7 @@ def setup_algebra(self):
         p = self.cfg.algebra.get("p", 4)
         q = self.cfg.algebra.get("q", 1)
         r = self.cfg.algebra.get("r", 0)
-        return CliffordAlgebra(p, q, r, device=self.device)
+        return make_algebra_from_config(self.cfg.algebra, p=p, q=q, r=r, device=self.device)
 
     def setup_model(self):
         """GLRNet with probe-specific head."""
diff --git a/tasks/md17.py b/tasks/md17.py
index 4b78566..7184a17 100644
--- a/tasks/md17.py
+++ b/tasks/md17.py
@@ -8,8 +8,8 @@
 import torch
 import torch.nn as nn
 
-from core.algebra import CliffordAlgebra
-from core.metric import hermitian_grade_spectrum, hermitian_norm
+from core.config import make_algebra_from_config
+from core.runtime.metric import hermitian_grade_spectrum, hermitian_norm
 from datalib.md17 import get_md17_loaders
 from functional.loss import ConservativeLoss, HermitianGradeRegularization
 from log import get_logger
@@ -49,8 +49,13 @@ def __init__(self, cfg):
 
     def setup_algebra(self):
         """Use Cl(3,0,1) PGA for SE(3) rigid-body motions."""
-        exp_policy = self.cfg.model.get("exp_policy", "balanced")
-        return CliffordAlgebra(p=3, q=0, r=self.cfg.algebra.get("r", 1), device=self.device, exp_policy=exp_policy)
+        return make_algebra_from_config(
+            self.cfg.algebra,
+            p=3,
+            q=0,
+            r=self.cfg.algebra.get("r", 1),
+            device=self.device,
+        )
 
     def setup_model(self):
         """Build MD17ForceNet model with PGA motors, dynamic rotors, and RBF."""
diff --git a/tasks/symbolic_regression.py b/tasks/symbolic_regression.py
index 6156402..a7c0f9c 100644
--- a/tasks/symbolic_regression.py
+++ b/tasks/symbolic_regression.py
@@ -20,7 +20,8 @@
 import torch.optim as optim
 from omegaconf import DictConfig
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra_from_config
+from core.foundation.module import AlgebraLike
 from datalib.symbolic_regression import _fetch_pmlb_data, get_dataset_ids, get_sr_loaders, get_sr_raw_splits
 from log import get_logger
 from models.sr import SRGBN
@@ -47,7 +48,7 @@ class SRTask(BaseTask):
         model.hidden_channels   : channel count C
         model.num_layers        : residual block count
         model.num_rotors        : K rotors per MultiRotorLayer
-        model.exp_policy        : exp policy ('balanced', 'precise')
+        algebra.exp_policy      : exp policy ('balanced', 'precise')
         iterative.max_stages    : maximum unbending iterations
         iterative.stage_epochs  : epochs per stage
         iterative.r2_target     : R2 threshold to stop
@@ -71,13 +72,11 @@ def __init__(self, cfg: DictConfig):
 
         # Optional automatic signature discovery
         self._searched_signature = None
-        if cfg.algebra.get("auto", False):
+        if cfg.algebra.get("metric_search", False):
             self._searched_signature = self._run_metric_search(cfg)
 
         # Iterative unbending config
         self.iterative_cfg = dict(cfg.get("iterative", {}))
-        # Remove 'enabled' key if present (legacy compat)
-        self.iterative_cfg.pop("enabled", None)
         # Merge new pipeline config sections
         self.iterative_cfg.update(
             {
@@ -97,13 +96,13 @@ def __init__(self, cfg: DictConfig):
 
         super().__init__(cfg)
 
-        # Override scheduler patience (BaseTask defaults to 3)
-        sched_patience = cfg.training.get("scheduler_patience", 10)
+        # Override scheduler after SR-specific setup.
+        sched_cfg = cfg.training.get("scheduler", {})
         self.scheduler = optim.lr_scheduler.ReduceLROnPlateau(
             self.optimizer,
             mode="min",
-            factor=0.5,
-            patience=sched_patience,
+            factor=sched_cfg.get("factor", 0.5),
+            patience=sched_cfg.get("patience", 10),
         )
 
     def _probe_n_vars(self, cfg):
@@ -155,7 +154,7 @@ def _run_metric_search(self, cfg):
         logger.info(f"MetricSearch: Cl({p},{q},{r}) for {self.dataset_name}")
         return (p, q, r)
 
-    def setup_algebra(self) -> CliffordAlgebra:
+    def setup_algebra(self) -> AlgebraLike:
         """Use searched signature or configured Cl(p,q,r)."""
         if self._searched_signature is not None:
             p, q, r = self._searched_signature
@@ -163,8 +162,13 @@ def setup_algebra(self) -> CliffordAlgebra:
             p = self.cfg.algebra.p
             q = self.cfg.algebra.get("q", 0)
             r = self.cfg.algebra.get("r", 0)
-        exp_policy = self.cfg.model.get("exp_policy", "balanced")
-        return CliffordAlgebra(p=p, q=q, r=r, device=self.device, exp_policy=exp_policy)
+        return make_algebra_from_config(
+            self.cfg.algebra,
+            p=p,
+            q=q,
+            r=r,
+            device=self.device,
+        )
 
     def setup_model(self) -> SRGBN:
         """Build SRGBN with config parameters, optionally auto-sizing."""
diff --git a/tests/conftest.py b/tests/conftest.py
index 581da86..4642036 100644
--- a/tests/conftest.py
+++ b/tests/conftest.py
@@ -1,6 +1,6 @@
 import pytest
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 DEVICE = "cpu"
 
@@ -35,7 +35,6 @@ def algebra_minkowski():
 def algebra_conformal():
     return CliffordAlgebra(p=4, q=1, device=DEVICE)
 
-
 # -- Module-scoped (used by test_geodesic.py - exact name match) ----------
 @pytest.fixture(scope="module")
 def alg2():
diff --git a/tests/test_analysis.py b/tests/test_analysis.py
index 9330b04..6e29e63 100644
--- a/tests/test_analysis.py
+++ b/tests/test_analysis.py
@@ -16,7 +16,6 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
 from core.analysis import (
     AnalysisConfig,
     AnalysisReport,
@@ -33,6 +32,7 @@
     SymmetryResult,
     compute_uncertainty_and_alignment,
 )
+from core.runtime.algebra import CliffordAlgebra
 
 DEVICE = "cpu"
 
diff --git a/tests/test_attention.py b/tests/test_attention.py
new file mode 100644
index 0000000..d3138ef
--- /dev/null
+++ b/tests/test_attention.py
@@ -0,0 +1,74 @@
+import math
+
+import pytest
+import torch
+
+from core.runtime.algebra import CliffordAlgebra
+from layers.blocks.attention import GeometricProductAttention
+
+pytestmark = pytest.mark.unit
+
+DEVICE = "cpu"
+
+
+def _reference_attention_score(algebra, q_head, k_head, bivector_weight):
+    product = algebra.geometric_product(q_head.unsqueeze(3), algebra.reverse(k_head).unsqueeze(2))
+    score_g0 = product[..., 0].sum(-1)
+
+    g2_idx = algebra.grade_masks[2].nonzero(as_tuple=False).squeeze(-1)
+    if g2_idx.numel() > 0:
+        g2 = torch.index_select(product, -1, g2_idx)
+        score_g2 = g2.pow(2).sum(dim=(-1, -2)).sqrt()
+    else:
+        score_g2 = torch.zeros_like(score_g0)
+
+    scale = math.sqrt(q_head.shape[3] * algebra.dim)
+    return (score_g0 + bivector_weight * score_g2) / scale
+
+
+def test_attention_dense_chunked_score_matches_direct_product():
+    algebra = CliffordAlgebra(3, 0, 0, device=DEVICE, dtype=torch.float64)
+    attn = GeometricProductAttention(
+        algebra,
+        channels=4,
+        num_heads=2,
+        causal=False,
+        bivector_weight=0.25,
+        score_blade_chunk_size=1,
+        score_precompute_limit=0,
+    )
+    q_head = torch.randn(2, 2, 3, 2, algebra.dim, dtype=torch.float64)
+    k_head = torch.randn(2, 2, 4, 2, algebra.dim, dtype=torch.float64)
+
+    actual = attn._compute_score(q_head, k_head)
+    expected = _reference_attention_score(algebra, q_head, k_head, attn.bivector_weight)
+
+    assert not hasattr(attn, "_g2_b_idx")
+    assert torch.allclose(actual, expected, atol=1e-12, rtol=1e-12)
+
+
+def test_attention_forward_shape_after_score_refactor():
+    algebra = CliffordAlgebra(3, 0, 0, device=DEVICE, dtype=torch.float32)
+    attn = GeometricProductAttention(algebra, channels=4, num_heads=2, causal=False)
+    x = torch.randn(2, 5, 4, algebra.dim)
+
+    y = attn(x)
+
+    assert y.shape == x.shape
+
+
+@pytest.mark.skipif(not hasattr(torch, "compile"), reason="torch.compile not available")
+def test_attention_dense_score_compiles_fullgraph():
+    algebra = CliffordAlgebra(4, 0, 0, device=DEVICE, dtype=torch.float32)
+    attn = GeometricProductAttention(algebra, channels=4, num_heads=2, causal=False)
+    q_head = torch.randn(1, 2, 3, 2, algebra.dim)
+    k_head = torch.randn(1, 2, 4, 2, algebra.dim)
+
+    def score(q, k):
+        return attn._compute_score(q, k)
+
+    expected = score(q_head, k_head)
+    compiled = torch.compile(score, backend="aot_eager", fullgraph=True)
+    actual = compiled(q_head, k_head)
+
+    assert torch.allclose(actual, expected, atol=1e-6, rtol=1e-6)
diff --git a/tests/test_core.py b/tests/test_core.py
index 2a51445..9296222 100644
--- a/tests/test_core.py
+++ b/tests/test_core.py
@@ -10,7 +10,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 pytestmark = pytest.mark.unit
 
diff --git a/tests/test_decomposition.py b/tests/test_decomposition.py
index c423e48..1e2225c 100644
--- a/tests/test_decomposition.py
+++ b/tests/test_decomposition.py
@@ -15,10 +15,10 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 pytestmark = pytest.mark.unit
-from core.decomposition import (
+from core.runtime.decomposition import (
     ExpPolicy,
     _power_iteration_compiled_safe,
     compiled_safe_decomposed_exp,
diff --git a/tests/test_degenerate_algebra.py b/tests/test_degenerate_algebra.py
index 608473f..8e8d7d7 100644
--- a/tests/test_degenerate_algebra.py
+++ b/tests/test_degenerate_algebra.py
@@ -17,7 +17,7 @@
 
 pytestmark = pytest.mark.unit
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 DEVICE = "cpu"
 
@@ -234,7 +234,7 @@ def test_non_simple_bivector_n4_decomposed_inference(self):
         B[0, bv_indices[0].item()] = 0.3  # e12
         B[0, bv_indices[5].item()] = 0.4  # e34
 
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         alg.exp_policy = ExpPolicy.PRECISE
         with torch.no_grad():
diff --git a/tests/test_exp_signatures.py b/tests/test_exp_signatures.py
index 73a59e4..bbf45aa 100644
--- a/tests/test_exp_signatures.py
+++ b/tests/test_exp_signatures.py
@@ -11,7 +11,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 pytestmark = pytest.mark.unit
 
@@ -443,7 +443,7 @@ class TestExpDecomposedGradient:
 
     def test_decomposed_gradient_cl40(self):
         """EXACT policy should produce finite gradients in Cl(4,0)."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         alg = CliffordAlgebra(4, 0, device=DEVICE)
         alg.exp_policy = ExpPolicy.PRECISE
@@ -461,7 +461,7 @@ def test_decomposed_gradient_cl40(self):
 
     def test_decomposed_gradient_cl15(self):
         """EXACT policy should produce finite gradients in Cl(1,5)."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         alg = CliffordAlgebra(1, 5, device=DEVICE)
         alg.exp_policy = ExpPolicy.PRECISE
@@ -481,7 +481,7 @@ def test_decomposed_gradient_cl15(self):
 
     def test_decomposed_matches_inference(self):
         """EXACT exp with grad should approximate inference result."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         alg = CliffordAlgebra(4, 0, device=DEVICE)
         alg.exp_policy = ExpPolicy.PRECISE
@@ -596,7 +596,7 @@ class TestDecompositionConvergence:
 
     def test_simple_bivector_converges_fast(self):
         """A simple bivector should decompose into 1 component."""
-        from core.decomposition import differentiable_invariant_decomposition
+        from core.runtime.decomposition import differentiable_invariant_decomposition
 
         alg = CliffordAlgebra(4, 0, device=DEVICE)
         bv_mask = alg.grade_masks[2]
@@ -615,7 +615,7 @@ def test_simple_bivector_converges_fast(self):
 
     def test_non_simple_needs_two_components(self):
         """e12 + e34 in Cl(4,0) should decompose into 2 components."""
-        from core.decomposition import differentiable_invariant_decomposition
+        from core.runtime.decomposition import differentiable_invariant_decomposition
 
         alg = CliffordAlgebra(4, 0, device=DEVICE)
         bv_mask = alg.grade_masks[2]
@@ -633,7 +633,7 @@ def test_non_simple_needs_two_components(self):
 
     def test_residual_check_limits_components(self):
         """With tight threshold, simple bivector should yield exactly 1 component."""
-        from core.decomposition import differentiable_invariant_decomposition
+        from core.runtime.decomposition import differentiable_invariant_decomposition
 
         alg = CliffordAlgebra(4, 0, device=DEVICE)
         bv_mask = alg.grade_masks[2]
diff --git a/tests/test_extensions.py b/tests/test_extensions.py
index e3a6fd5..9803ef2 100644
--- a/tests/test_extensions.py
+++ b/tests/test_extensions.py
@@ -8,7 +8,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers import CliffordGraphConv
 from layers.adapters.conformal import ConformalEmbedding
 from layers.adapters.projective import ProjectiveEmbedding
diff --git a/tests/test_framework_pipeline.py b/tests/test_framework_pipeline.py
new file mode 100644
index 0000000..a566a98
--- /dev/null
+++ b/tests/test_framework_pipeline.py
@@ -0,0 +1,140 @@
+import pytest
+import torch
+import torch.nn as nn
+
+from core.planning import PlanningLimits
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.context import AlgebraContext
+from layers import ProductLayer, WedgeLayer
+from optimizers import make_riemannian_optimizer
+
+pytestmark = pytest.mark.unit
+
+
+def test_product_layer_dense_matches_algebra(algebra_3d):
+    left = torch.randn(4, 5, algebra_3d.dim)
+    right = torch.randn(4, 5, algebra_3d.dim)
+    layer = ProductLayer(algebra_3d)
+
+    actual = layer(left, right)
+    expected = algebra_3d.geometric_product(left, right)
+
+    assert torch.allclose(actual, expected)
+
+
+def test_wedge_layer_declared_grades_match_planned_algebra(algebra_3d):
+    left = algebra_3d.embed_vector(torch.randn(4, 5, algebra_3d.n))
+    right = algebra_3d.embed_vector(torch.randn(4, 5, algebra_3d.n))
+    layer = WedgeLayer(
+        algebra_3d,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(2,),
+    )
+
+    actual = layer(left, right)
+    expected = algebra_3d.wedge(left, right, left_grades=(1,), right_grades=(1,), output_grades=(2,))
+
+    assert torch.allclose(actual, expected)
+
+
+def test_product_layer_pairwise_compact_widths_match_dense_reference():
+    context = AlgebraContext(p=5, q=0, device="cpu")
+    dense = CliffordAlgebra(p=5, q=0, device="cpu")
+    left_layout = context.layout((2,))
+    right_layout = context.layout((1,))
+    output_layout = context.layout((3,))
+
+    left = torch.randn(2, 3, left_layout.dim)
+    right = torch.randn(2, 4, right_layout.dim)
+    layer = WedgeLayer(
+        context,
+        left_grades=(2,),
+        right_grades=(1,),
+        output_grades=(3,),
+        left_compact=True,
+        right_compact=True,
+        compact_output=True,
+        pairwise=True,
+    )
+
+    actual = layer(left, right)
+    expected_dense = dense.wedge(
+        left_layout.dense(left).unsqueeze(2),
+        right_layout.dense(right).unsqueeze(1),
+        left_grades=(2,),
+        right_grades=(1,),
+        output_grades=(3,),
+    )
+    expected = output_layout.compact(expected_dense)
+
+    assert actual.shape == (2, 3, 4, output_layout.dim)
+    assert torch.allclose(actual, expected)
+
+
+def test_compact_layer_pipeline_trains_with_riemannian_optimizer_factory():
+    context = AlgebraContext(p=6, q=0, device="cpu")
+    vector_layout = context.layout((1,))
+    trivector_layout = context.layout((3,))
+
+    class CompactPipeline(nn.Module):
+        def __init__(self):
+            super().__init__()
+            self.wedge_vectors = WedgeLayer(
+                context,
+                left_grades=(1,),
+                right_grades=(1,),
+                output_grades=(2,),
+                compact_output=True,
+            )
+            self.wedge_trivector = WedgeLayer(
+                context,
+                left_grades=(2,),
+                right_grades=(1,),
+                output_grades=(3,),
+                left_compact=True,
+                right_compact=True,
+                compact_output=True,
+            )
+            self.scale = nn.Parameter(torch.ones(()))
+
+        def forward(self, left_vector, right_vector, third_vector):
+            bivector = self.wedge_vectors(left_vector, right_vector)
+            return self.scale * self.wedge_trivector(bivector, third_vector)
+
+    dense_left = context.embed_vector(torch.randn(8, context.n))
+    dense_right = context.embed_vector(torch.randn(8, context.n))
+    compact_third = vector_layout.compact(context.embed_vector(torch.randn(8, context.n)))
+    model = CompactPipeline()
+    optimizer = make_riemannian_optimizer(model, context, optimizer="adam", lr=0.01)
+
+    output = model(dense_left, dense_right, compact_third)
+    assert output.shape == (8, trivector_layout.dim)
+    assert len(context.planner._product_executors) == 2
+
+    loss = output.square().mean()
+    loss.backward()
+    optimizer.step()
+
+    assert model.scale.grad is not None
+    assert torch.isfinite(model.scale).all()
+
+
+def test_product_layer_uses_context_planning_limits():
+    limits = PlanningLimits(warn_lanes=32, max_lanes=512, warn_pairs=32, max_pairs=64)
+    context = AlgebraContext(p=16, q=0, device="cpu", planning_limits=limits)
+    vector_layout = context.layout((1,))
+    left = torch.zeros(1, vector_layout.dim)
+    right = torch.zeros(1, vector_layout.dim)
+    layer = ProductLayer(
+        context,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        left_compact=True,
+        right_compact=True,
+        compact_output=True,
+    )
+
+    with pytest.raises(ValueError, match="basis interactions"):
+        layer(left, right)
diff --git a/tests/test_functional_products.py b/tests/test_functional_products.py
new file mode 100644
index 0000000..e2e5dbb
--- /dev/null
+++ b/tests/test_functional_products.py
@@ -0,0 +1,60 @@
+import pytest
+import torch
+
+from functional import (
+    anti_commutator,
+    commutator,
+    geometric_product,
+    grade_projection,
+    inner_product,
+    product,
+    reverse,
+    wedge,
+)
+
+pytestmark = pytest.mark.unit
+
+
+def test_functional_products_match_algebra_dense(algebra_3d):
+    left = torch.randn(3, algebra_3d.dim)
+    right = torch.randn(3, algebra_3d.dim)
+
+    assert torch.allclose(geometric_product(algebra_3d, left, right), algebra_3d.geometric_product(left, right))
+    assert torch.allclose(wedge(algebra_3d, left, right), algebra_3d.wedge(left, right))
+    assert torch.allclose(inner_product(algebra_3d, left, right), algebra_3d.inner_product(left, right))
+    assert torch.allclose(commutator(algebra_3d, left, right), algebra_3d.commutator(left, right))
+    assert torch.allclose(anti_commutator(algebra_3d, left, right), algebra_3d.anti_commutator(left, right))
+
+
+def test_functional_projected_product_compact_output(algebra_3d):
+    left = algebra_3d.embed_vector(torch.randn(4, algebra_3d.n))
+    right = algebra_3d.embed_vector(torch.randn(4, algebra_3d.n))
+    layout = algebra_3d.layout((2,))
+
+    actual = wedge(
+        algebra_3d,
+        left,
+        right,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(2,),
+        compact_output=True,
+    )
+    expected = layout.compact(algebra_3d.wedge(left, right))
+
+    assert actual.shape == (4, layout.dim)
+    assert torch.allclose(actual, expected)
+
+
+def test_functional_unary_helpers(algebra_3d):
+    values = torch.randn(2, algebra_3d.dim)
+
+    assert torch.allclose(reverse(algebra_3d, values), algebra_3d.reverse(values))
+    assert torch.allclose(grade_projection(algebra_3d, values, 1), algebra_3d.grade_projection(values, 1))
+
+
+def test_functional_product_rejects_unknown_op(algebra_3d):
+    values = torch.randn(2, algebra_3d.dim)
+
+    with pytest.raises(ValueError, match="Unsupported product op"):
+        product(algebra_3d, values, values, op="unknown")
diff --git a/tests/test_geodesic.py b/tests/test_geodesic.py
index d76df9b..b726e31 100644
--- a/tests/test_geodesic.py
+++ b/tests/test_geodesic.py
@@ -12,8 +12,8 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
 from core.analysis import DimensionLifter, GeodesicFlow, MetricSearch
+from core.runtime.algebra import CliffordAlgebra
 
 pytestmark = pytest.mark.unit
 
diff --git a/tests/test_grade_plan.py b/tests/test_grade_plan.py
new file mode 100644
index 0000000..cd4d801
--- /dev/null
+++ b/tests/test_grade_plan.py
@@ -0,0 +1,1058 @@
+import pytest
+import torch
+
+from core.config import make_algebra
+from core.foundation.basis import (
+    basis_count_for_grades,
+    basis_index_tuple_for_grades,
+    basis_indices_for_grades,
+    expand_output_grades,
+    geometric_product_output_grades,
+    operation_coefficient,
+    product_output_grades,
+)
+from core.foundation.layout import AlgebraSpec
+from core.planning.flow import GradeFlow
+from core.planning.layouts import build_product_request
+from core.planning.planner import GradePlanner
+from core.planning.policy import PlanningLimits
+from core.planning.product import (
+    GradeProductExecutor,
+    build_grade_product_plan,
+)
+from core.planning.tree import build_grade_plan_tree
+from core.planning.unary import build_unary_request
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.context import AlgebraContext
+from core.runtime.multivector import Multivector
+
+pytestmark = pytest.mark.unit
+
+DEVICE = "cpu"
+
+
+def _project_to_grades(algebra, mv: torch.Tensor, grades: tuple[int, ...]) -> torch.Tensor:
+    result = torch.zeros_like(mv)
+    for grade in grades:
+        result = result + algebra.grade_projection(mv, grade)
+    return result
+
+
+def _grade_only_input(algebra, batch: int, grades: tuple[int, ...], seed: int) -> torch.Tensor:
+    generator = torch.Generator(device=DEVICE).manual_seed(seed)
+    mv = torch.zeros(batch, algebra.dim, dtype=torch.float64)
+    indices = basis_indices_for_grades(algebra.n, grades, device=DEVICE)
+    mv[:, indices] = torch.randn(batch, indices.numel(), dtype=torch.float64, generator=generator) * 0.1
+    return mv
+
+
+def test_grade_expansion_for_common_high_dim_paths():
+    assert geometric_product_output_grades(1, 1, 16) == (0, 2)
+    assert geometric_product_output_grades(2, 1, 16) == (1, 3)
+    assert product_output_grades(2, 1, 16, op="wedge") == (3,)
+    assert product_output_grades(2, 1, 16, op="commutator") == (1,)
+    assert product_output_grades(2, 1, 16, op="anti_commutator") == (3,)
+    assert expand_output_grades((0, 2), (1,), 16, op="gp") == (1, 3)
+    assert expand_output_grades((1,), (1,), 16, op="wedge") == (2,)
+    assert expand_output_grades((1,), (1,), 16, op="gp", project_grades=(0,)) == (0,)
+
+
+def test_operation_coefficients_keep_wedge_as_exterior_product():
+    # e12 and e3 commute, so the antisymmetric formula would vanish.  The
+    # exterior product is instead the grade-sum part of the geometric product.
+    assert operation_coefficient(3, 4, 3, 0, 0, "wedge") == 1.0
+    assert operation_coefficient(3, 4, 3, 0, 0, "commutator") == 0.0
+    assert operation_coefficient(3, 4, 3, 0, 0, "anti_commutator") == 2.0
+
+
+def test_basis_indices_for_grades_are_combinatorial_and_high_dimensional():
+    assert basis_index_tuple_for_grades(4, (1, 2)) == tuple(
+        index for index in range(1 << 4) if index.bit_count() in {1, 2}
+    )
+    high = basis_index_tuple_for_grades(32, (1, 2))
+    assert len(high) == basis_count_for_grades(32, (1, 2))
+    assert high[0] == 1
+    assert high[-1] == (1 << 31) | (1 << 30)
+
+
+def test_basis_tensorization_reports_int64_bitmask_boundary():
+    with pytest.raises(ValueError, match="torch.long basis bitmasks"):
+        basis_indices_for_grades(64, (1,))
+
+
+def test_grade_plan_tree_groups_routes_without_runtime_partition_backend():
+    spec = AlgebraSpec(10, 4, 2)
+    tree = build_grade_plan_tree(
+        spec,
+        left_grades=(1, 2),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        op="gp",
+        chunk_pair_limit=128,
+    )
+
+    assert tree.output_grades == (0, 2)
+    assert [(path.left_grade, path.right_grade, path.output_grades) for path in tree.paths] == [
+        (1, 1, (0, 2)),
+    ]
+    assert tree.path_count == 1
+    assert tree.estimated_pairs == 16 * 16
+    assert tree.estimated_chunks == 2
+    assert tree.path_for_grades(1, 1) is tree.paths[0]
+    assert tree.path_for_grades(2, 1) is None
+
+
+def test_product_request_infers_declared_layouts_and_output_grades():
+    spec = AlgebraSpec(10, 4, 2)
+    left = torch.zeros(2, spec.dim)
+    right = torch.zeros(2, spec.dim)
+
+    request = build_product_request(
+        spec,
+        left,
+        right,
+        left_grades=(1,),
+        right_grades=(1,),
+        op="gp",
+        full_layout_allowed=False,
+    )
+
+    assert request.left_grades == (1,)
+    assert request.right_grades == (1,)
+    assert request.output_grades == (0, 2)
+    assert not request.left_compact
+    assert not request.right_compact
+
+
+def test_product_request_detects_compact_tensors_from_layout_shape():
+    spec = AlgebraSpec(6, 0, 0)
+    layout = spec.layout((1,))
+    left = torch.zeros(2, layout.dim)
+    right = torch.zeros(2, layout.dim)
+
+    request = build_product_request(
+        spec,
+        left,
+        right,
+        left_layout=layout,
+        right_layout=layout,
+        output_grades=(0, 2),
+        op="gp",
+    )
+
+    assert request.left_compact
+    assert request.right_compact
+
+
+def test_unary_request_infers_projection_layout_without_full_layout():
+    spec = AlgebraSpec(10, 4, 2)
+    values = torch.zeros(2, spec.dim)
+
+    request = build_unary_request(
+        spec,
+        values,
+        op="grade_projection",
+        output_grades=(1,),
+        full_layout_allowed=False,
+    )
+
+    assert request.input_grades == (1,)
+    assert request.output_grades == (1,)
+    assert not request.input_compact
+
+
+def test_grade_flow_propagates_embedding_unary_product_and_merge():
+    spec = AlgebraSpec(8, 0, 0)
+    vector = GradeFlow.vector(spec)
+    scalar = GradeFlow.scalar(spec)
+
+    product = vector.product(vector)
+    projected = product.project((2,))
+    merged = scalar.merge(projected)
+
+    assert vector.grades == (1,)
+    assert vector.unary("reverse").grades == (1,)
+    assert product.grades == (0, 2)
+    assert projected.grades == (2,)
+    assert merged.grades == (0, 2)
+
+
+def test_grade_layout_compact_dense_round_trip():
+    spec = AlgebraSpec(4, 1, 1)
+    layout = spec.layout((0, 2))
+    dense = torch.randn(2, spec.dim, dtype=torch.float64, generator=torch.Generator().manual_seed(97))
+
+    values = layout.compact(dense)
+    materialized = layout.dense(values)
+
+    assert values.shape[-1] == layout.dim
+    assert torch.allclose(materialized[..., layout.indices_tensor(device=dense.device)], values)
+    outside = torch.ones(spec.dim, dtype=torch.bool)
+    outside[layout.indices_tensor()] = False
+    assert materialized[..., outside].abs().sum().item() == 0.0
+
+
+@pytest.mark.parametrize("op", ["gp", "wedge", "inner", "commutator", "anti_commutator"])
+def test_static_grade_product_matches_dense_kernel_for_selected_grade_paths(op):
+    algebra = CliffordAlgebra(4, 1, 1, device=DEVICE, dtype=torch.float64)
+    left_grades = (1,)
+    right_grades = (1, 2)
+    output_grades = expand_output_grades(left_grades, right_grades, algebra.n, op=op)
+    plan = build_grade_product_plan(
+        algebra.p,
+        algebra.q,
+        algebra.r,
+        left_grades=left_grades,
+        right_grades=right_grades,
+        output_grades=output_grades,
+        op=op,
+        device=DEVICE,
+        dtype=torch.float64,
+    )
+    product = GradeProductExecutor(plan)
+    A = _grade_only_input(algebra, 3, left_grades, seed=101)
+    B = _grade_only_input(algebra, 3, right_grades, seed=103)
+
+    expected = _project_to_grades(algebra, getattr(algebra, _dense_method_name(op))(A, B), output_grades)
+    actual = product.forward_dense(A, B)
+
+    assert product.pair_count < algebra.dim * algebra.dim
+    assert actual.shape == expected.shape
+    assert torch.allclose(actual, expected, atol=1e-12, rtol=1e-12)
+
+
+def test_wedge_dense_and_planned_paths_are_exterior_product_for_higher_grades():
+    algebra = CliffordAlgebra(3, 0, 0, device=DEVICE, dtype=torch.float64)
+    spec = AlgebraSpec.from_algebra(algebra)
+    layout_2 = spec.layout((2,))
+    layout_1 = spec.layout((1,))
+    layout_3 = spec.layout((3,))
+
+    e12 = torch.zeros(1, algebra.dim, dtype=torch.float64)
+    e12[0, 3] = 1.0
+    e3 = torch.zeros(1, algebra.dim, dtype=torch.float64)
+    e3[0, 4] = 1.0
+    expected = torch.zeros_like(e12)
+    expected[0, 7] = 1.0
+
+    dense = algebra.wedge(e12, e3)
+    compact = algebra.wedge(
+        layout_2.compact(e12),
+        layout_1.compact(e3),
+        left_layout=layout_2,
+        right_layout=layout_1,
+        output_grades=(3,),
+        left_compact=True,
+        right_compact=True,
+        compact_output=True,
+    )
+
+    assert torch.allclose(dense, expected, atol=1e-12, rtol=1e-12)
+    assert torch.allclose(compact, layout_3.compact(expected), atol=1e-12, rtol=1e-12)
+
+
+def test_wedge_chains_as_iterative_exterior_product():
+    algebra = CliffordAlgebra(4, 0, 0, device=DEVICE, dtype=torch.float64)
+    e1 = torch.zeros(1, algebra.dim, dtype=torch.float64)
+    e2 = torch.zeros(1, algebra.dim, dtype=torch.float64)
+    e3 = torch.zeros(1, algebra.dim, dtype=torch.float64)
+    e1[0, 1] = 1.0
+    e2[0, 2] = 1.0
+    e3[0, 4] = 1.0
+    expected = torch.zeros_like(e1)
+    expected[0, 7] = 1.0
+
+    e12 = algebra.wedge(e1, e2, left_grades=(1,), right_grades=(1,), output_grades=(2,))
+    actual = algebra.wedge(e12, e3, left_grades=(2,), right_grades=(1,), output_grades=(3,))
+
+    assert torch.allclose(actual, expected, atol=1e-12, rtol=1e-12)
+
+
+def test_wedge_plan_prunes_grade_route_pairs_before_coefficients():
+    algebra = CliffordAlgebra(6, 0, 0, device=DEVICE, dtype=torch.float64)
+    broad = build_grade_product_plan(
+        algebra.p,
+        algebra.q,
+        algebra.r,
+        left_grades=(2,),
+        right_grades=(1,),
+        output_grades=(1, 3),
+        op="wedge",
+        device=DEVICE,
+        dtype=torch.float64,
+    )
+    exterior_only = build_grade_product_plan(
+        algebra.p,
+        algebra.q,
+        algebra.r,
+        left_grades=(2,),
+        right_grades=(1,),
+        output_grades=(3,),
+        op="wedge",
+        device=DEVICE,
+        dtype=torch.float64,
+    )
+
+    assert broad.pair_count == exterior_only.pair_count
+    assert all(int(index).bit_count() == 3 for index in broad.output_indices.tolist())
+
+
+def test_product_plan_owns_compact_position_buffers():
+    algebra = CliffordAlgebra(4, 1, 0, device=DEVICE, dtype=torch.float64)
+    plan = build_grade_product_plan(
+        algebra.p,
+        algebra.q,
+        algebra.r,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        op="gp",
+        device=DEVICE,
+        dtype=torch.float64,
+    )
+    product = GradeProductExecutor(plan)
+    A = _grade_only_input(algebra, 2, (1,), seed=109)
+    B = _grade_only_input(algebra, 2, (1,), seed=111)
+
+    left_positions = {index: position for position, index in enumerate(plan.left_layout.basis_indices)}
+    right_positions = {index: position for position, index in enumerate(plan.right_layout.basis_indices)}
+    expected_left = torch.tensor([left_positions[int(index)] for index in plan.left_indices], dtype=torch.long)
+    expected_right = torch.tensor([right_positions[int(index)] for index in plan.right_indices], dtype=torch.long)
+
+    assert torch.equal(plan.left_compact_positions.cpu(), expected_left)
+    assert torch.equal(plan.right_compact_positions.cpu(), expected_right)
+    assert torch.allclose(
+        product.forward_compact(plan.left_layout.compact(A), plan.right_layout.compact(B)),
+        product(A, B),
+        atol=1e-12,
+        rtol=1e-12,
+    )
+
+
+def test_product_executor_compact_forward_supports_different_layout_widths():
+    algebra = CliffordAlgebra(4, 1, 0, device=DEVICE, dtype=torch.float64)
+    plan = build_grade_product_plan(
+        algebra.p,
+        algebra.q,
+        algebra.r,
+        left_grades=(1,),
+        right_grades=(1, 2),
+        output_grades=(0, 1, 2, 3),
+        op="gp",
+        device=DEVICE,
+        dtype=torch.float64,
+    )
+    product = GradeProductExecutor(plan)
+    A = _grade_only_input(algebra, 2, (1,), seed=115)
+    B = _grade_only_input(algebra, 2, (1, 2), seed=117)
+
+    compact = product.forward_compact(plan.left_layout.compact(A), plan.right_layout.compact(B))
+    dense = product(A, B)
+
+    assert plan.left_layout.dim != plan.right_layout.dim
+    assert torch.allclose(compact, dense, atol=1e-12, rtol=1e-12)
+
+
+def test_algebra_projected_product_matches_dense_kernel_and_compact_output():
+    algebra = CliffordAlgebra(4, 1, 1, device=DEVICE, dtype=torch.float64)
+    A = _grade_only_input(algebra, 2, (1,), seed=113)
+    B = _grade_only_input(algebra, 2, (1,), seed=127)
+
+    dense_expected = _project_to_grades(algebra, algebra.geometric_product(A, B), (0, 2))
+    dense_actual = algebra.projected_geometric_product(
+        A,
+        B,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+    )
+    compact_actual = algebra.projected_geometric_product(
+        A,
+        B,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        compact_output=True,
+    )
+
+    assert torch.allclose(dense_actual, dense_expected, atol=1e-12, rtol=1e-12)
+    assert compact_actual.shape[-1] == AlgebraSpec.from_algebra(algebra).layout((0, 2)).dim
+
+
+def test_grade_planner_reuses_projected_product_executor():
+    algebra = CliffordAlgebra(4, 1, 1, device=DEVICE, dtype=torch.float64)
+    planner = GradePlanner(algebra)
+
+    first = planner.product_executor(
+        op="gp",
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        dtype=torch.float64,
+        device=DEVICE,
+    )
+    second = planner.product_executor(
+        op="gp",
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        dtype=torch.float64,
+        device=DEVICE,
+    )
+
+    assert first is second
+
+
+def test_algebra_product_executor_returns_preplanned_runtime_handle():
+    algebra = AlgebraContext(6, 0, device=DEVICE, dtype=torch.float32)
+
+    first = algebra.product_executor(
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+    )
+    second = algebra.product_executor(
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+    )
+
+    assert first is second
+    assert first.left_grades == (1,)
+    assert first.right_grades == (1,)
+    assert first.output_grades == (0, 2)
+    assert first.coefficients.dtype == algebra.dtype
+
+
+def test_grade_planner_rekeys_cached_executor_after_dtype_move():
+    algebra = CliffordAlgebra(4, 1, 1, device=DEVICE, dtype=torch.float64)
+    executor = algebra.planner.product_executor(
+        op="gp",
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        dtype=algebra.dtype,
+        device=DEVICE,
+    )
+
+    algebra.to(dtype=torch.float32)
+    moved = algebra.planner.product_executor(
+        op="gp",
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        dtype=algebra.dtype,
+        device=DEVICE,
+    )
+
+    assert moved is executor
+    assert moved.coefficients.dtype == torch.float32
+
+
+def test_multivector_compact_projected_product_keeps_dense_tensor_compatibility():
+    algebra = CliffordAlgebra(4, 1, 1, device=DEVICE, dtype=torch.float64)
+    A = Multivector(algebra, _grade_only_input(algebra, 2, (1,), seed=131)).compact((1,))
+    B = Multivector(algebra, _grade_only_input(algebra, 2, (1,), seed=137)).compact((1,))
+
+    result = A.projected_product(B, output_grades=(0, 2))
+    expected = _project_to_grades(algebra, algebra.geometric_product(A.tensor, B.tensor), (0, 2))
+
+    assert result.is_compact
+    assert result.values.shape[-1] == result.layout.dim
+    assert result.tensor.shape[-1] == algebra.dim
+    assert torch.allclose(result.tensor, expected, atol=1e-12, rtol=1e-12)
+
+
+def test_multivector_compact_projected_product_supports_mixed_dense_operand():
+    algebra = CliffordAlgebra(4, 1, 1, device=DEVICE, dtype=torch.float64)
+    A = Multivector(algebra, _grade_only_input(algebra, 2, (1,), seed=139)).compact((1,))
+    B = Multivector(algebra, _grade_only_input(algebra, 2, (1,), seed=149))
+
+    result = A.projected_product(B, output_grades=(0, 2), right_grades=(1,))
+    expected = _project_to_grades(algebra, algebra.geometric_product(A.tensor, B.tensor), (0, 2))
+
+    assert result.is_compact
+    assert torch.allclose(result.tensor, expected, atol=1e-12, rtol=1e-12)
+
+
+def test_make_algebra_returns_context_above_dense_threshold():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+
+    assert isinstance(algebra, AlgebraContext)
+    assert algebra.n == 16
+    assert not algebra.allow_full_layout_products
+
+
+def test_dense_policy_uses_context_by_default_and_explicit_dense_up_to_twelve():
+    auto_dense = make_algebra(8, 0, 0, device=DEVICE, dtype=torch.float32)
+    auto_context = make_algebra(9, 0, 0, device=DEVICE, dtype=torch.float32)
+    explicit_dense = make_algebra(9, 0, 0, kernel="dense", device=DEVICE, dtype=torch.float32)
+
+    assert isinstance(auto_dense, CliffordAlgebra)
+    assert isinstance(auto_context, AlgebraContext)
+    assert isinstance(explicit_dense, CliffordAlgebra)
+    with pytest.raises(AssertionError):
+        CliffordAlgebra(9, 0, 0, device=DEVICE, dtype=torch.float32)
+    with pytest.raises(AssertionError):
+        make_algebra(13, 0, 0, kernel="dense", device=DEVICE, dtype=torch.float32)
+
+
+def test_dense_kernel_accepts_shared_planned_operation_kwargs():
+    algebra = CliffordAlgebra(4, 1, 1, device=DEVICE, dtype=torch.float64)
+    A = _grade_only_input(algebra, 2, (1,), seed=191)
+    B = _grade_only_input(algebra, 2, (1,), seed=193)
+
+    actual = algebra.geometric_product(
+        A,
+        B,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        compact_output=True,
+    )
+    expected = algebra.projected_geometric_product(
+        A,
+        B,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        compact_output=True,
+    )
+
+    assert torch.allclose(actual, expected, atol=1e-12, rtol=1e-12)
+
+
+def test_dense_and_context_share_layout_indices_and_bivector_metric_signs():
+    dense = CliffordAlgebra(3, 1, 0, device=DEVICE, dtype=torch.float64)
+    context = make_algebra(3, 1, 0, kernel="context", device=DEVICE, dtype=torch.float64)
+
+    assert torch.equal(dense.grade_indices((2,)), context.grade_indices((2,)))
+    assert torch.allclose(dense.bivector_squared_signs(), context.bivector_squared_signs())
+
+
+def test_context_projected_product_handles_high_dim_vector_product():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    A = torch.zeros(1, algebra.dim)
+    B = torch.zeros(1, algebra.dim)
+    A[0, 1] = 1.0
+    B[0, 1] = 1.0
+    B[0, 2] = 1.0
+
+    values, layout = algebra.projected_geometric_product(
+        A,
+        B,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        compact_output=True,
+        return_layout=True,
+    )
+
+    scalar_pos = layout.basis_indices.index(0)
+    bivector_pos = layout.basis_indices.index(3)
+    assert values.shape[-1] == layout.dim
+    assert torch.allclose(values[0, scalar_pos], torch.tensor(1.0))
+    assert torch.allclose(values[0, bivector_pos], torch.tensor(1.0))
+
+
+def test_context_planned_unary_projection_and_reverse_avoid_full_layout():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    mv = torch.zeros(1, algebra.dim)
+    mv[0, 1] = 2.0
+    mv[0, 3] = 5.0
+
+    projected, projected_layout = algebra.grade_projection(mv, 1, compact_output=True, return_layout=True)
+    reversed_bivector = algebra.reverse(
+        mv,
+        input_grades=(2,),
+        compact_output=True,
+    )
+    vector_pos = projected_layout.basis_indices.index(1)
+    bivector_layout = algebra.layout((2,))
+    bivector_pos = bivector_layout.basis_indices.index(3)
+
+    assert torch.allclose(projected[0, vector_pos], torch.tensor(2.0))
+    assert torch.allclose(reversed_bivector[0, bivector_pos], torch.tensor(-5.0))
+
+
+def test_context_planned_unary_compact_reverse():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    layout = algebra.layout((2,))
+    values = torch.arange(layout.dim, dtype=torch.float32).unsqueeze(0)
+
+    actual, output_layout = algebra.reverse(
+        values,
+        input_layout=layout,
+        input_compact=True,
+        compact_output=True,
+        return_layout=True,
+    )
+
+    assert output_layout == layout
+    assert torch.allclose(actual, -values)
+
+
+def test_dense_kernel_planned_unary_handles_compact_layouts():
+    algebra = CliffordAlgebra(6, 0, 0, device=DEVICE, dtype=torch.float32)
+    layout = algebra.layout((2,))
+    values = torch.arange(layout.dim, dtype=torch.float32).unsqueeze(0)
+
+    actual, output_layout = algebra.reverse(
+        values,
+        input_layout=layout,
+        input_compact=True,
+        compact_output=True,
+        return_layout=True,
+    )
+
+    assert output_layout == layout
+    assert torch.allclose(actual, -values)
+
+
+def test_multivector_compact_geometric_product_stays_compact_in_high_dimensions():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    vector_layout = algebra.layout((1,))
+    left = torch.zeros(1, vector_layout.dim)
+    right = torch.zeros(1, vector_layout.dim)
+    left[0, 0] = 1.0
+    right[0, 0] = 1.0
+
+    result = Multivector(algebra, values=left, layout=vector_layout) * Multivector(
+        algebra,
+        values=right,
+        layout=vector_layout,
+    )
+
+    assert result.is_compact
+    assert result.layout.grades == (0, 2)
+    scalar_pos = result.layout.basis_indices.index(0)
+    assert torch.allclose(result.values[0, scalar_pos], torch.tensor(1.0))
+
+
+def test_multivector_compact_binary_wrappers_do_not_unwrap_dense_tensors():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    bivector_layout = algebra.layout((2,))
+    vector_layout = algebra.layout((1,))
+    bivector = Multivector(algebra, values=torch.randn(2, bivector_layout.dim), layout=bivector_layout)
+    vector = Multivector(algebra, values=torch.randn(2, vector_layout.dim), layout=vector_layout)
+
+    results = [
+        (bivector.geometric_product(vector), (1, 3)),
+        (bivector.wedge(vector), (3,)),
+        (bivector.inner(vector), (3,)),
+        (bivector.commutator(vector), (1,)),
+        (bivector.anti_commutator(vector), (3,)),
+    ]
+
+    for result, expected_grades in results:
+        assert result.is_compact
+        assert result.layout.grades == expected_grades
+
+
+def test_multivector_compact_addition_merges_layouts_without_dense_materialization():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    vector_layout = algebra.layout((1,))
+    bivector_layout = algebra.layout((2,))
+    vector = Multivector(algebra, values=torch.ones(1, vector_layout.dim), layout=vector_layout)
+    bivector = Multivector(algebra, values=2.0 * torch.ones(1, bivector_layout.dim), layout=bivector_layout)
+
+    result = vector + bivector
+
+    assert result.is_compact
+    assert result.layout.grades == (1, 2)
+    vector_values = vector.with_layout(result.layout).values
+    bivector_values = bivector.with_layout(result.layout).values
+    assert torch.allclose(result.values, vector_values + bivector_values)
+
+
+def test_context_default_grades_drive_compact_product_without_callsite_metadata():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32, default_grades=(1,))
+    vector_layout = algebra.layout()
+    left = torch.zeros(1, vector_layout.dim)
+    right = torch.zeros(1, vector_layout.dim)
+    left[0, 0] = 1.0
+    right[0, 0] = 1.0
+
+    values, output_layout = algebra.geometric_product(
+        left,
+        right,
+        compact_output=True,
+        return_layout=True,
+    )
+
+    assert vector_layout.grades == (1,)
+    assert output_layout.grades == (0, 2)
+    assert torch.allclose(values[0, output_layout.basis_indices.index(0)], torch.tensor(1.0))
+
+
+def test_context_declared_grades_infer_compact_operand_shapes():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    vector_layout = algebra.layout((1,))
+    left = torch.zeros(1, vector_layout.dim)
+    right = torch.zeros(1, vector_layout.dim)
+    left[0, 0] = 1.0
+    right[0, 0] = 1.0
+
+    values, output_layout = algebra.projected_geometric_product(
+        left,
+        right,
+        left_grades=(1,),
+        right_grades=(1,),
+        compact_output=True,
+        return_layout=True,
+    )
+
+    assert output_layout.grades == (0, 2)
+    assert torch.allclose(values[0, output_layout.basis_indices.index(0)], torch.tensor(1.0))
+
+
+def test_context_projected_product_pairwise_mixed_compact_widths():
+    algebra = make_algebra(16, 0, 0, kernel="context", device=DEVICE, dtype=torch.float32)
+    left_layout = algebra.layout((2,))
+    right_layout = algebra.layout((1,))
+    left = torch.randn(3, left_layout.dim)
+    right = torch.randn(4, right_layout.dim)
+
+    values, output_layout = algebra.projected_wedge(
+        left,
+        right,
+        left_layout=left_layout,
+        right_layout=right_layout,
+        output_grades=(3,),
+        pairwise=True,
+        compact_output=True,
+        return_layout=True,
+    )
+    executor = algebra.product_executor(
+        op="wedge",
+        left_grades=(2,),
+        right_grades=(1,),
+        output_grades=(3,),
+        dtype=torch.float32,
+        device=DEVICE,
+    )
+
+    assert output_layout.grades == (3,)
+    assert values.shape == (3, 4, output_layout.dim)
+    assert torch.allclose(values, executor.forward_pairwise_compact(left, right), atol=1e-6, rtol=1e-6)
+
+
+def test_context_projected_product_suggests_pairwise_for_mismatched_item_axes():
+    algebra = make_algebra(16, 0, 0, kernel="context", device=DEVICE, dtype=torch.float32)
+    left_layout = algebra.layout((2,))
+    right_layout = algebra.layout((1,))
+    left = torch.randn(3, left_layout.dim)
+    right = torch.randn(4, right_layout.dim)
+
+    with pytest.raises(ValueError, match="Use pairwise=True"):
+        algebra.projected_wedge(
+            left,
+            right,
+            left_layout=left_layout,
+            right_layout=right_layout,
+            output_grades=(3,),
+            compact_output=True,
+        )
+
+
+def test_context_pairwise_projected_product_requires_item_axes():
+    algebra = make_algebra(16, 0, 0, kernel="context", device=DEVICE, dtype=torch.float32)
+    left_layout = algebra.layout((2,))
+    right_layout = algebra.layout((1,))
+
+    with pytest.raises(ValueError, match="explicit item axes"):
+        algebra.projected_wedge(
+            torch.randn(left_layout.dim),
+            torch.randn(right_layout.dim),
+            left_layout=left_layout,
+            right_layout=right_layout,
+            output_grades=(3,),
+            pairwise=True,
+            compact_output=True,
+        )
+
+
+def test_context_declared_product_requires_compact_output_without_dense_materialization():
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32)
+    vector_layout = algebra.layout((1,))
+    left = torch.zeros(1, vector_layout.dim)
+    right = torch.zeros(1, vector_layout.dim)
+
+    with pytest.raises(ValueError, match="Dense materialization is disabled"):
+        algebra.projected_geometric_product(
+            left,
+            right,
+            left_layout=vector_layout,
+            right_layout=vector_layout,
+        )
+
+
+def test_high_dim_context_requires_declared_layout_for_products():
+    algebra = make_algebra(13, 0, 0, device=DEVICE, dtype=torch.float32)
+    A = torch.zeros(1, algebra.dim)
+    B = torch.zeros(1, algebra.dim)
+
+    with pytest.raises(ValueError, match="Declare active grades"):
+        algebra.geometric_product(A, B)
+
+    with pytest.raises(ValueError, match="Declare active grades"):
+        algebra.reverse(A)
+
+
+def test_context_requires_declared_grades_by_default_even_low_dimensional():
+    context = make_algebra(4, 0, 0, kernel="context", device=DEVICE, dtype=torch.float64)
+
+    with pytest.raises(ValueError, match="Declare active grades"):
+        context.layout()
+
+
+def test_context_warns_for_explicit_implicit_full_layout_fallback_between_eight_and_twelve():
+    context = make_algebra(
+        9,
+        0,
+        0,
+        kernel="context",
+        device=DEVICE,
+        dtype=torch.float32,
+        allow_full_layout_products=True,
+    )
+
+    with pytest.warns(RuntimeWarning, match="implicit full Cl\\(9,0,0\\) layout"):
+        layout = context.layout()
+
+    assert layout.grades == tuple(range(context.n + 1))
+    assert context.allow_full_layout_products
+
+
+def test_low_dim_context_can_use_full_layout_fallback():
+    context = make_algebra(
+        4,
+        0,
+        0,
+        kernel="context",
+        device=DEVICE,
+        dtype=torch.float64,
+        allow_full_layout_products=True,
+    )
+    dense = CliffordAlgebra(4, 0, 0, device=DEVICE, dtype=torch.float64)
+    A = _grade_only_input(dense, 2, (1,), seed=163)
+    B = _grade_only_input(dense, 2, (1,), seed=167)
+
+    actual = context.geometric_product(A, B)
+    expected = dense.geometric_product(A, B)
+
+    assert context.allow_full_layout_products
+    assert torch.allclose(actual, expected, atol=1e-12, rtol=1e-12)
+
+
+def test_context_static_product_cost_limits_raise_before_executor_build():
+    limits = PlanningLimits(warn_lanes=512, max_lanes=512, warn_pairs=512, max_pairs=64)
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32, planning_limits=limits)
+    layout = algebra.layout((1,))
+    left = torch.zeros(1, layout.dim)
+    right = torch.zeros(1, layout.dim)
+
+    with pytest.raises(ValueError, match="basis interactions"):
+        algebra.projected_geometric_product(
+            left,
+            right,
+            left_layout=layout,
+            right_layout=layout,
+            compact_output=True,
+        )
+
+
+def test_direct_product_executor_obeys_static_pair_limits():
+    limits = PlanningLimits(warn_lanes=512, max_lanes=512, warn_pairs=512, max_pairs=64)
+    algebra = make_algebra(16, 0, 0, device=DEVICE, dtype=torch.float32, planning_limits=limits)
+
+    with pytest.raises(ValueError, match="basis interactions"):
+        algebra.planner.product_executor(
+            op="gp",
+            left_grades=(1,),
+            right_grades=(1,),
+            output_grades=(0, 2),
+            dtype=torch.float32,
+            device=DEVICE,
+        )
+
+
+def test_context_static_layout_cost_limit_raises_before_basis_materialization():
+    limits = PlanningLimits(warn_lanes=32, max_lanes=64, warn_pairs=512, max_pairs=1024)
+    algebra = make_algebra(32, 0, 0, device=DEVICE, dtype=torch.float32, planning_limits=limits)
+
+    with pytest.raises(ValueError, match="active lanes"):
+        algebra.layout((1, 2))
+
+
+def test_high_dimensional_vector_product_plan_avoids_full_basis_enumeration():
+    algebra = make_algebra(
+        32,
+        0,
+        0,
+        device=DEVICE,
+        dtype=torch.float32,
+        planning_limits=PlanningLimits(max_lanes=4096, max_pairs=100_000),
+    )
+    vector_layout = algebra.layout((1,))
+    executor = algebra.planner.product_executor(
+        op="gp",
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        dtype=torch.float32,
+        device=DEVICE,
+    )
+
+    assert vector_layout.dim == 32
+    assert executor.output_dim == 1 + 32 * 31 // 2
+    assert executor.pair_count == 32 * 32
+
+
+def test_high_dimensional_product_plan_reports_int64_bitmask_boundary():
+    algebra = make_algebra(
+        64,
+        0,
+        0,
+        device=DEVICE,
+        dtype=torch.float32,
+        planning_limits=PlanningLimits(max_lanes=4096, max_pairs=100_000),
+    )
+
+    with pytest.raises(ValueError, match="Current Torch-backed executors support bitmask tensorization up to n=63"):
+        algebra.planner.product_executor(
+            op="gp",
+            left_grades=(1,),
+            right_grades=(1,),
+            output_grades=(0, 2),
+            dtype=torch.float32,
+            device=DEVICE,
+        )
+
+
+def test_context_static_product_cost_warns_near_configured_limits():
+    limits = PlanningLimits(warn_lanes=512, max_lanes=512, warn_pairs=128, max_pairs=512)
+    algebra = make_algebra(10, 4, 2, device=DEVICE, dtype=torch.float32, planning_limits=limits)
+    layout = algebra.layout((1,))
+    left = torch.zeros(1, layout.dim)
+    right = torch.zeros(1, layout.dim)
+
+    with pytest.warns(RuntimeWarning, match="basis interactions"):
+        values = algebra.projected_geometric_product(
+            left,
+            right,
+            left_layout=layout,
+            right_layout=layout,
+            compact_output=True,
+        )
+
+    assert values.shape[-1] == algebra.layout((0, 2)).dim
+
+
+@pytest.mark.skipif(not hasattr(torch, "compile"), reason="torch.compile not available")
+def test_static_grade_product_compiles_fullgraph_with_aot_eager():
+    algebra = CliffordAlgebra(5, 1, 0, device=DEVICE, dtype=torch.float32)
+    plan = build_grade_product_plan(
+        algebra.p,
+        algebra.q,
+        algebra.r,
+        left_grades=(1,),
+        right_grades=(1,),
+        output_grades=(0, 2),
+        op="gp",
+        device=DEVICE,
+        dtype=torch.float32,
+    )
+    product = GradeProductExecutor(plan)
+    A = _grade_only_input(algebra, 2, (1,), seed=107).to(dtype=torch.float32)
+    B = _grade_only_input(algebra, 2, (1,), seed=109).to(dtype=torch.float32)
+
+    compiled = torch.compile(product, backend="aot_eager", fullgraph=True)
+
+    expected = product(A, B)
+    actual = compiled(A, B)
+
+    assert actual.shape[-1] == product.output_dim
+    assert torch.allclose(actual, expected, atol=1e-6, rtol=1e-6)
+
+
+@pytest.mark.skipif(not hasattr(torch, "compile"), reason="torch.compile not available")
+def test_planned_unary_compiles_fullgraph_with_aot_eager():
+    algebra = make_algebra(6, 0, 0, kernel="context", device=DEVICE, dtype=torch.float32)
+    executor = algebra.planner.unary_executor(
+        op="reverse",
+        input_grades=(2,),
+        dtype=torch.float32,
+        device=DEVICE,
+    )
+    values = _grade_only_input(CliffordAlgebra(6, 0, 0, device=DEVICE), 2, (2,), seed=173).to(dtype=torch.float32)
+
+    compiled = torch.compile(executor, backend="aot_eager", fullgraph=True)
+
+    expected = executor(values)
+    actual = compiled(values)
+
+    assert torch.allclose(actual, expected, atol=1e-6, rtol=1e-6)
+
+
+@pytest.mark.skipif(not hasattr(torch, "compile"), reason="torch.compile not available")
+def test_algebra_projected_product_compiles_fullgraph_after_cache_warm():
+    algebra = CliffordAlgebra(5, 1, 0, device=DEVICE, dtype=torch.float32)
+    A = _grade_only_input(algebra, 2, (1,), seed=151).to(dtype=torch.float32)
+    B = _grade_only_input(algebra, 2, (1,), seed=157).to(dtype=torch.float32)
+
+    def product(x, y):
+        return algebra.projected_geometric_product(
+            x,
+            y,
+            left_grades=(1,),
+            right_grades=(1,),
+            output_grades=(0, 2),
+            compact_output=True,
+        )
+
+    expected = product(A, B)
+    compiled = torch.compile(product, backend="aot_eager", fullgraph=True)
+    actual = compiled(A, B)
+
+    assert torch.allclose(actual, expected, atol=1e-6, rtol=1e-6)
+
+
+@pytest.mark.skipif(not hasattr(torch, "compile"), reason="torch.compile not available")
+def test_context_projected_product_compiles_fullgraph_from_cold_planner_cache():
+    if hasattr(torch, "_dynamo"):
+        torch._dynamo.reset()
+    algebra = AlgebraContext(6, 0, device=DEVICE, dtype=torch.float32)
+    generator = torch.Generator(device=DEVICE).manual_seed(181)
+    left = torch.randn(2, algebra.layout((1,)).dim, dtype=torch.float32, generator=generator)
+    right = torch.randn(2, algebra.layout((1,)).dim, dtype=torch.float32, generator=generator)
+
+    def product(x, y):
+        return algebra.geometric_product(
+            x,
+            y,
+            left_grades=(1,),
+            right_grades=(1,),
+            output_grades=(0, 2),
+            left_compact=True,
+            right_compact=True,
+            compact_output=True,
+        )
+
+    assert not algebra.planner._product_executors
+    compiled = torch.compile(product, backend="aot_eager", fullgraph=True)
+    actual = compiled(left, right)
+    expected = product(left, right)
+
+    assert len(algebra.planner._product_executors) == 1
+    assert torch.allclose(actual, expected, atol=1e-6, rtol=1e-6)
+
+
+def _dense_method_name(op: str) -> str:
+    if op == "gp":
+        return "geometric_product"
+    if op == "inner":
+        return "inner_product"
+    if op == "anti_commutator":
+        return "anti_commutator"
+    return op
diff --git a/tests/test_hermitian_metrics.py b/tests/test_hermitian_metrics.py
index bc01180..a85d58c 100644
--- a/tests/test_hermitian_metrics.py
+++ b/tests/test_hermitian_metrics.py
@@ -3,10 +3,12 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.config import make_algebra
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.multivector import Multivector
 
 pytestmark = pytest.mark.unit
-from core.metric import (
+from core.runtime.metric import (
     _hermitian_signs,
     clifford_conjugate,
     geometric_distance,
@@ -143,6 +145,19 @@ def test_has_negative_signs(self, algebra_minkowski):
         has_negative = (signs < 0).any()
         assert has_negative, "Cl(2,1) should have negative signs"
 
+    def test_compact_context_matches_dense_active_lanes(self):
+        dense = CliffordAlgebra(3, 1, 0, device="cpu", dtype=torch.float64)
+        context = make_algebra(3, 1, 0, kernel="context", device="cpu", dtype=torch.float64)
+        layout = context.layout((1, 2))
+        generator = torch.Generator(device="cpu").manual_seed(719)
+        A = torch.randn(5, layout.dim, dtype=torch.float64, generator=generator)
+        B = torch.randn(5, layout.dim, dtype=torch.float64, generator=generator)
+
+        actual = hermitian_inner_product(context, A, B, layout=layout)
+        expected = hermitian_inner_product(dense, layout.dense(A), layout.dense(B))
+
+        assert torch.allclose(actual, expected, atol=1e-12, rtol=1e-12)
+
 
 class TestHermitianNorm:
     def test_non_negative(self, algebra_minkowski):
@@ -279,6 +294,28 @@ def test_conformal_spectrum(self, algebra_conformal):
         assert spec.shape == (algebra_conformal.n + 1,)
         assert (spec >= -1e-6).all()
 
+    def test_compact_spectrum_fills_inactive_grades(self):
+        context = make_algebra(5, 0, 0, kernel="context", device="cpu", dtype=torch.float32)
+        layout = context.layout((1,))
+        values = torch.ones(2, layout.dim)
+
+        spec = hermitian_grade_spectrum(context, values, layout=layout)
+
+        assert spec.shape == (2, context.n + 1)
+        assert torch.allclose(spec[:, 0], torch.zeros(2))
+        assert torch.allclose(spec[:, 1], torch.full((2,), float(context.n)))
+        assert torch.allclose(spec[:, 2:], torch.zeros(2, context.n - 1))
+
+    def test_compact_multivector_norm_uses_layout_without_dense_materialization(self):
+        context = make_algebra(9, 0, 0, kernel="context", device="cpu", dtype=torch.float32)
+        mv = Multivector.from_vectors(context, torch.ones(3, context.n))
+
+        norm = hermitian_norm(context, mv)
+
+        assert mv.is_compact
+        assert norm.shape == (3, 1)
+        assert torch.allclose(norm, torch.full((3, 1), context.n**0.5))
+
 
 class TestSignatureTraceForm:
     def test_matches_standard_for_euclidean(self, algebra_3d):
diff --git a/tests/test_iterative_unbender.py b/tests/test_iterative_unbender.py
index da87f70..4071ce4 100644
--- a/tests/test_iterative_unbender.py
+++ b/tests/test_iterative_unbender.py
@@ -11,7 +11,7 @@
 
 pytestmark = pytest.mark.slow
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from models.sr.unbender import (
     IterativeUnbender,
     OrthogonalEliminationResult,
diff --git a/tests/test_layer_optimization.py b/tests/test_layer_optimization.py
index bbc573b..ab19ad5 100644
--- a/tests/test_layer_optimization.py
+++ b/tests/test_layer_optimization.py
@@ -8,7 +8,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers import RotorLayer
 
 pytestmark = pytest.mark.unit
diff --git a/tests/test_layers.py b/tests/test_layers.py
index ecdf0e8..e701727 100644
--- a/tests/test_layers.py
+++ b/tests/test_layers.py
@@ -8,10 +8,8 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
-from core.decomposition import ExpPolicy
-from layers import CliffordLinear, MultiRotorLayer, RotorLayer
-from layers.primitives.reflection import ReflectionLayer
+from core.runtime.algebra import CliffordAlgebra
+from layers import CliffordLinear, MultiRotorLayer, ReflectionLayer, RotorLayer
 
 pytestmark = pytest.mark.unit
 
@@ -68,7 +66,7 @@ def test_multi_rotor_invariants(self, algebra_3d):
 
     def test_rotor_layer_exact_policy(self, algebra_3d):
         """Test RotorLayer with EXACT exp policy."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         algebra_3d.exp_policy = ExpPolicy.PRECISE
         x = torch.randn(4, 5, 8)
@@ -86,7 +84,7 @@ def test_rotor_layer_exact_policy(self, algebra_3d):
 
     def test_rotor_layer_policy_vs_standard(self, algebra_3d):
         """Compare RotorLayer with FAST vs EXACT policy."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         layer_a = RotorLayer(algebra_3d, 3)
         layer_b = RotorLayer(algebra_3d, 3)
@@ -107,7 +105,7 @@ def test_rotor_layer_policy_vs_standard(self, algebra_3d):
 
     def test_rotor_layer_backward_exact(self, algebra_3d):
         """Test gradient flow through RotorLayer with EXACT policy."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         algebra_3d.exp_policy = ExpPolicy.PRECISE
 
@@ -129,7 +127,7 @@ def test_rotor_layer_backward_exact(self, algebra_3d):
 
     def test_multi_rotor_layer_exact_policy(self, algebra_3d):
         """Test MultiRotorLayer with EXACT policy."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         algebra_3d.exp_policy = ExpPolicy.PRECISE
         x = torch.randn(4, 5, 8)
@@ -141,7 +139,7 @@ def test_multi_rotor_layer_exact_policy(self, algebra_3d):
 
     def test_multi_rotor_layer_backward_exact(self, algebra_3d):
         """Test gradient flow through MultiRotorLayer with EXACT policy."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         algebra_3d.exp_policy = ExpPolicy.PRECISE
 
@@ -164,7 +162,7 @@ def test_multi_rotor_layer_backward_exact(self, algebra_3d):
 
     def test_rotor_layer_rotor_property(self, algebra_3d):
         """Verify that exp-produced rotors satisfy R * ~R = 1."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         algebra_3d.exp_policy = ExpPolicy.PRECISE
 
diff --git a/tests/test_md17_complete.py b/tests/test_md17_complete.py
index 9093f8a..3a2dfc9 100644
--- a/tests/test_md17_complete.py
+++ b/tests/test_md17_complete.py
@@ -3,11 +3,11 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 pytestmark = pytest.mark.unit
-from core.decomposition import ExpPolicy
-from core.metric import hermitian_grade_spectrum, hermitian_norm
+from core.runtime.decomposition import ExpPolicy
+from core.runtime.metric import hermitian_grade_spectrum, hermitian_norm
 from functional.loss import ConservativeLoss, HermitianGradeRegularization
 from models.md17 import DynamicRotorGenerator, GaussianRBF, MD17ForceNet, MD17InteractionBlock
 
@@ -116,7 +116,7 @@ def test_forward_default(self, algebra):
         assert force.shape == (8, 3)
 
     def test_forward_with_exact_policy(self, algebra):
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         algebra.exp_policy = ExpPolicy.PRECISE
         model = MD17ForceNet(
diff --git a/tests/test_metric_search.py b/tests/test_metric_search.py
index 7bc53c8..0f445e9 100644
--- a/tests/test_metric_search.py
+++ b/tests/test_metric_search.py
@@ -10,7 +10,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 pytestmark = pytest.mark.slow
 from core.analysis import GeodesicFlow, MetricSearch
diff --git a/tests/test_multivector.py b/tests/test_multivector.py
index 3dd8a67..013630c 100644
--- a/tests/test_multivector.py
+++ b/tests/test_multivector.py
@@ -3,8 +3,8 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
-from core.multivector import Multivector
+from core.runtime.algebra import CliffordAlgebra
+from core.runtime.multivector import Multivector
 
 
 @pytest.fixture
@@ -28,7 +28,10 @@ def rand_mv(alg, rng, batch=4):
 def test_from_vectors(alg):
     v = torch.randn(4, 3)
     mv = Multivector.from_vectors(alg, v)
-    assert mv.shape[-1] == 2**alg.n
+    assert mv.is_compact
+    assert mv.grades == (1,)
+    assert mv.shape[-1] == alg.n
+    assert mv.tensor.shape[-1] == 2**alg.n
 
 
 def test_scalar(alg):
diff --git a/tests/test_properties.py b/tests/test_properties.py
index 4486a42..3a6df8e 100644
--- a/tests/test_properties.py
+++ b/tests/test_properties.py
@@ -8,7 +8,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers import CliffordLayerNorm
 
 pytestmark = pytest.mark.unit
diff --git a/tests/test_riemannian_optimizer.py b/tests/test_riemannian_optimizer.py
index 2fdae02..6fe3903 100644
--- a/tests/test_riemannian_optimizer.py
+++ b/tests/test_riemannian_optimizer.py
@@ -14,7 +14,7 @@
 import torch.nn as nn
 import torch.nn.functional as F
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers import MultiRotorLayer, RotorGadget, RotorLayer
 from optimizers.riemannian import (
     MANIFOLD_EUCLIDEAN,
@@ -23,6 +23,7 @@
     ExponentialSGD,
     RiemannianAdam,
     group_parameters_by_manifold,
+    make_riemannian_optimizer,
     tag_manifold,
 )
 
@@ -556,18 +557,18 @@ def test_manifold_tagging(algebra_3d):
     from layers.primitives.reflection import ReflectionLayer
 
     rotor = RotorLayer(algebra_3d, channels=4)
-    assert getattr(rotor.bivector_weights, "_manifold", None) == "spin"
+    assert getattr(rotor.bivector_weights, "_manifold", None) == MANIFOLD_SPIN
 
     reflection = ReflectionLayer(algebra_3d, channels=4)
-    assert getattr(reflection.vector_weights, "_manifold", None) == "sphere"
+    assert getattr(reflection.vector_weights, "_manifold", None) == MANIFOLD_SPHERE
 
     multi = MultiRotorLayer(algebra_3d, channels=4, num_rotors=2)
-    assert getattr(multi.rotor_bivectors, "_manifold", None) == "spin"
+    assert getattr(multi.rotor_bivectors, "_manifold", None) == MANIFOLD_SPIN
     assert not hasattr(multi.weights, "_manifold")  # Euclidean, untagged
 
     gadget = RotorGadget(algebra_3d, in_channels=4, out_channels=8)
-    assert getattr(gadget.bivector_left, "_manifold", None) == "spin"
-    assert getattr(gadget.bivector_right, "_manifold", None) == "spin"
+    assert getattr(gadget.bivector_left, "_manifold", None) == MANIFOLD_SPIN
+    assert getattr(gadget.bivector_right, "_manifold", None) == MANIFOLD_SPIN
 
 
 def test_tag_manifold_helper():
@@ -584,6 +585,17 @@ def test_tag_manifold_helper():
         tag_manifold(p, "invalid")
 
 
+def test_group_parameters_rejects_unknown_manifold():
+    class BadModel(nn.Module):
+        def __init__(self):
+            super().__init__()
+            self.weight = nn.Parameter(torch.randn(2, 3))
+            self.weight._manifold = "bad"
+
+    with pytest.raises(ValueError, match="Unknown manifold"):
+        group_parameters_by_manifold(BadModel())
+
+
 def test_from_model_groups(algebra_3d):
     """Verify from_model creates separate groups per manifold."""
     from layers.primitives.reflection import ReflectionLayer
@@ -612,6 +624,19 @@ def forward(self, x):
     assert "euclidean" in manifolds
 
 
+def test_make_riemannian_optimizer_factory(algebra_3d):
+    layer = RotorLayer(algebra_3d, channels=4)
+
+    adam = make_riemannian_optimizer(layer, algebra_3d, optimizer="adam", lr=0.001)
+    sgd = make_riemannian_optimizer(layer, algebra_3d, optimizer="exponential_sgd", lr=0.01)
+
+    assert isinstance(adam, RiemannianAdam)
+    assert isinstance(sgd, ExponentialSGD)
+
+    with pytest.raises(ValueError, match="optimizer must be"):
+        make_riemannian_optimizer(layer, algebra_3d, optimizer="rmsprop")
+
+
 def test_sphere_retraction(algebra_3d):
     """Verify sphere-tagged params are projected to unit sphere after step."""
     from layers.primitives.reflection import ReflectionLayer
diff --git a/tests/test_rotor_gadget.py b/tests/test_rotor_gadget.py
index b380028..d2a9ab9 100644
--- a/tests/test_rotor_gadget.py
+++ b/tests/test_rotor_gadget.py
@@ -12,7 +12,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from layers import CliffordLinear, RotorGadget
 
 pytestmark = pytest.mark.unit
@@ -286,7 +286,7 @@ class TestExpPolicy:
 
     def test_with_exact_policy(self, algebra_3d):
         """Test layer with EXACT exp policy."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         algebra_3d.exp_policy = ExpPolicy.PRECISE
 
@@ -305,7 +305,7 @@ def test_with_exact_policy(self, algebra_3d):
 
     def test_policy_fast_vs_exact(self, algebra_3d):
         """Compare FAST and EXACT policies (n=3: should match)."""
-        from core.decomposition import ExpPolicy
+        from core.runtime.decomposition import ExpPolicy
 
         torch.manual_seed(42)
         layer_a = RotorGadget(
diff --git a/tests/test_rotor_translate.py b/tests/test_rotor_translate.py
index b64b4e8..780518a 100644
--- a/tests/test_rotor_translate.py
+++ b/tests/test_rotor_translate.py
@@ -7,7 +7,7 @@
 import pytest
 import torch
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 
 pytestmark = pytest.mark.unit
 from models.sr.net import SRGBN
diff --git a/tests/test_symbolic_regression.py b/tests/test_symbolic_regression.py
index 37bddfa..64afd1f 100644
--- a/tests/test_symbolic_regression.py
+++ b/tests/test_symbolic_regression.py
@@ -8,7 +8,7 @@
 import torch
 from omegaconf import OmegaConf
 
-from core.algebra import CliffordAlgebra
+from core.runtime.algebra import CliffordAlgebra
 from datalib.symbolic_regression import (
     BLACKBOX_DATASETS,
     FIRST_PRINCIPLES_DATASETS,
@@ -41,11 +41,22 @@ def small_algebra():
 _TEST_CACHE = "./data/pmlb_cache"
 
 
-def _make_cfg(dataset_name=_TEST_DATASET, hidden_channels=4, num_layers=1, n_samples=200, auto=False):
+def _make_cfg(dataset_name=_TEST_DATASET, hidden_channels=4, num_layers=1, n_samples=200, metric_search=False):
     return OmegaConf.create(
         {
             "name": "sr",
-            "algebra": {"p": 4, "q": 0, "r": 0, "device": "cpu", "auto": auto},
+            "algebra": {
+                "p": 4,
+                "q": 0,
+                "r": 0,
+                "device": "cpu",
+                "dtype": "float32",
+                "exp_policy": "balanced",
+                "kernel": "auto",
+                "metric_search": metric_search,
+                "dense_threshold": 8,
+                "default_grades": None,
+            },
             "dataset": {
                 "dataset_name": dataset_name,
                 "category": "blackbox",
@@ -56,7 +67,6 @@ def _make_cfg(dataset_name=_TEST_DATASET, hidden_channels=4, num_layers=1, n_sam
             "model": {
                 "hidden_channels": hidden_channels,
                 "num_layers": num_layers,
-                "exp_policy": "balanced",
             },
             "training": {
                 "epochs": 1,
@@ -66,6 +76,7 @@ def _make_cfg(dataset_name=_TEST_DATASET, hidden_channels=4, num_layers=1, n_sam
                 "max_bivector_norm": 10.0,
                 "sparsity_weight": 0.01,
                 "seed": 0,
+                "scheduler": {"factor": 0.5, "patience": 10},
             },
             "checkpoint": None,
         }
diff --git a/uv.lock b/uv.lock
index 1cd172e..07c1e35 100644
--- a/uv.lock
+++ b/uv.lock
@@ -1,22 +1,10 @@
 version = 1
 revision = 2
-requires-python = ">=3.9"
+requires-python = ">=3.10"
 resolution-markers = [
-    "python_full_version >= '3.14' and platform_machine == 'ARM64' and sys_platform == 'win32'",
-    "python_full_version >= '3.14' and platform_machine != 'ARM64' and sys_platform == 'win32'",
-    "python_full_version >= '3.14' and sys_platform == 'emscripten'",
-    "python_full_version >= '3.14' and sys_platform != 'emscripten' and sys_platform != 'win32'",
-    "python_full_version >= '3.12' and python_full_version < '3.14' and platform_machine == 'ARM64' and sys_platform == 'win32'",
-    "python_full_version >= '3.12' and python_full_version < '3.14' and platform_machine != 'ARM64' and sys_platform == 'win32'",
-    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform == 'emscripten'",
-    "python_full_version >= '3.12' and python_full_version < '3.14' and sys_platform != 'emscripten' and sys_platform != 'win32'",
-    "python_full_version == '3.11.*' and platform_machine == 'ARM64' and sys_platform == 'win32'",
-    "python_full_version == '3.11.*' and platform_machine != 'ARM64' and sys_platform == 'win32'",
-    "python_full_version == '3.11.*' and sys_platform == 'emscripten'",
-    "python_full_version == '3.11.*' and sys_platform != 'emscripten' and sys_platform != 'win32'",
-    "python_full_version == '3.10.*' and platform_machine == 'ARM64' and sys_platform == 'win32'",
-    "(python_full_version == '3.10.*' and platform_machine != 'ARM64') or (python_full_version == '3.10.*' and sys_platform != 'win32')",
-    "python_full_version < '3.10'",
+    "python_full_version >= '3.12'",
+    "python_full_version == '3.11.*'",
+    "python_full_version < '3.11'",
 ]
 
 [[package]]
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