[amaybaum]

WIP / RESEARCH ONLY — SUBSTATE-2, guiding draft PR for issue #1010, under AGENTS.md §A.43. Separate from SUBSTATE-1 #1009 and Yang–Mills action audit #1003.

Drafting snapshot D: main at f6989a27815c4d715223eebf2859422c1231e3f8.
Current source head: 947761dd40c17e4fdedcd8972a0b4e53e8bfba42 (accepted administrative correction integrated by fast-forward).
Scope: 35 changed files, ALL under verification/ (research records, three Lean research modules, bridge root imports and Lean-manuscript census classification). No manuscript, book, physical-premise, frozen receipt, workflow or main changes.

NATIVE-PADIC-1 — source-test finding (writer round)

• Preregistered before the test in verification/programmes/substratum/recursive-substates/native-padic-bridge/PREREGISTRATION.md. Compared the coefficient-1 finite cubic scalar reference wave update formula from papers/oi_lattice_code/trace_out_fractional/wave_eqn.c with a hypothetical compatible family of site alphabets Z/(2^n). The extension is a candidate mathematical lift, not an adopted physical OI substratum. SM §4.1 leaves the physical coefficient open; the congruence argument covers integer or compatible 2-adic coefficients, while the Lean module covers coefficient 1.
• Exact all-state theorem (written proof): for every m<n, coordinatewise reduction mod 2^m commutes with the wave step and its inverse (six-term integer nearest-neighbor sum minus previous slice). Induction gives the same for all times. Thus extra fine digits cannot affect any passive readout factoring through a fixed low-m quotient, regardless of prior. Every full finite low-m two-slice factor is a permutation, so any fixed low-m observable has an exact finite common visible return, even if the whole inverse-limit hidden carrier does not.
• Independent exact controls: native_padic_probe.py and RESULTS.json retain 9,344 modular arithmetic checks on a 27-site L=3 cube, q=2..64 (powers of two), 12 fixed preparations per q, 16 forward/inverse steps for all lower-stage reductions, and high-digit perturbations. This is finite regression evidence, not an exhaustive enumeration over q^54 states.
• Nonvacuous alternative: adding a fixed non-native bit1-to-bit0 output kick at the origin produces a reversible two-slice update that does distinguish initial states agreeing in their low bit. The modified interaction is not the actual OI wave rule, an implemented native-Y gate, quantum interference or Yang–Mills dynamics.
• Lean scope: OIBridge.NativePadicReduction.lean proves the finite-stencil additive-homomorphism projection identities for the wave step and intended inverse-map formula; the inverse property is not proved there. Imported into the root and classified kernel-only in the maintained manuscript census. Build and release-gate checks pass in fresh ordinary PR CI for integrated head 947761dd; no p-adic physical carrier or operational quantum claim is formalized.
• Independent physical payoff gate: an actual native recursive scale-mixing mechanism must be sourced from the existing OI substratum and public observer-access contract, with repeatable full-record interventions, classical memory control and Wilson-versus-squared action controls. Neither deeper digits alone nor passive coarse-graining supplies the missing F2 operational quantum resource or a Yang–Mills action.

CI/governance: fresh ordinary PR run 41284458598 for integrated source head 947761dd passed: 25 jobs succeeded, 1 expected skip; Build and Release gate succeeded. Previous source-head run 41222551849, associated with source head 4e7ddb70, passed: 25 jobs succeeded, 1 expected skip; Mathlib job 124599945935 Build and Release gate both succeeded. This tests a synthetic merge, not an F/E exact-head attestation. No manuscript promotion, manual workflow dispatch or merge.

FIXED-SCALE-1 — actual writer result on this head

One fixed reversible infinite-depth rule (research-only):

T(a,r,c)=(a XOR r XOR c[0], r, shift(c)), where c is a bi-infinite independent fair bit tape, r is a persistent hidden bit, and the four-symbol observer records (a,c[0]). The map is a continuous bijection preserving product-fair measure, with an explicit inverse.

Exact paper-level infinite-model deductions (independent acceptance at c811e876 is author-reported; see provenance qualification below):

• For every positive integer time t, each row of the rooted Γ_t equals (1/4,1/4,1/4,1/4). This family is P-divisible and never returns to identity at a positive common time; P(O_t=O_0)=1/4.
• Nevertheless, I(O_{t-1};O_{t+1}|O_t)=1 bit for all t>=1, since the earlier two records reveal the static r-bit whereas the current record alone does not. The full-record parity identity a_{t-1} XOR a_{t+1}=c_{t-1} XOR c_t follows.
• A fixed time t depends on finitely many tape coordinates, but initial-depth dependence grows without bound as t increases. This uses one fixed rule, unlike the earlier family of depth-m flips.

Executed exact finite controls: fixed-scale-skew/fixed_scale_probe.py exhausts cyclic tape lengths 3–9 (32 to 2,048 full states), checks two-way inverses, all 32 supported three-time words at 1/32, uniform rooted maps prior to the wrap, exact 1-bit conditional memory, depth witnesses, finite cyclic return at 2n (not an infinite-period result), and a decoupled/removed-r Markov control with zero conditional memory. All local controls passed in normal and optimized Python; the checked script contains no assert statements.

Strong matching negative control (added after preregistration): a separate reversible four-symbol iid shift has the same complete rooted two-time family Γ_t=J_4/4 for positive t, while all 64 three-time records have probability 1/64 rather than 32 records at 1/32. Hence the three-record total variation is exactly 1/2, with conditional memory 0 vs 1. matched_control.py checks all cyclic iid tapes of lengths 3–7 (up to 16,384 states), normally and optimized, with exact rational results.

Evidence: research files fixed-scale-skew/PREREGISTRATION.md, REPORT.md, fixed_scale_probe.py, RESULTS.json, matched_control.py, MATCHED.json, and replay.py. Source/result Git blob hashes match the locally tested bytes; the joint replay driver checks explicit SHA-256 digests and rejects nonzero exit, stderr, or byte mismatch.

Science boundary: classical memory, not native OI physical dynamics, operational QM, a gauge field, Yang–Mills action or log-periodic prediction. Product-fair tape prior and physical access to a static r-bit via coarse records are model inputs, not derived from OI. The full continuous/infinite statement is a written mathematical proof; no Lean certificate for it or physical-source validation. Independent acceptance is author-reported with the qualification below. The pre-existing conditional finite-depth RecursiveFiniteFactor module builds in ordinary PR CI; this does not formalize the infinite-model proof.

Governance: all 35 changed files are under verification/; no manuscripts, publication claim, merge, F/E designation or manual workflow dispatch. Current-head ordinary CI is green as reported above.

Exact finite results — locally reproducible, not native OI

• A — Z_2 odometer: Levels 1…9 exhaustively checked. The low-bit readout swaps deterministically, Γ₂=I; deeper digits cannot influence that compatible quotient. Classical negative control, no P-indivisibility.
• B1 — a fixed depth-m backflow: m∈{1,2,3,5,7}, all 256-bitstring configurations per choice checked. Static hidden bit controls the visible flip; Γ₁=J/2, Γ₂=I, with exactly I(X₀;X₂|X₁)=1 bit and four length-three words of probability 1/4. Each m defines a different model with finite coupling depth, not one scale-free law. This is a classical memory witness, NOT quantum coherence.
• B2 — scale conveyor: Finite cyclic shifts N=3…9 exhaustively checked, including iid cylinder words below the return horizon; each has period N. In the separate infinite bi-infinite Bernoulli-shift model, all forward observations are iid and P(X_L=X₀)=1/2 for each positive L. Hence no shared exact a.e. visible period despite reversible continuous dynamics. Do NOT pass the finite-period N to the infinite limit.
• C — profinite gauge link tower: Complete four-link inventories Z/2^n, n=1…4, up to 65,536 states, with gauge covariance under all four generator shifts, compatible reduction, reversible Haar-preserving link translation and nontrivial gauge-invariant plaquette holonomy. The lowest-bit visible factor swaps and returns after 2. w and w² remain different gauge-invariant plaquette functions (exact rational q=4 control). This abelian profinite group is NOT connected U(1), SU(2) or SU(3), and it supplies NO Yang–Mills action.

Theorem and Lean scope

Paper proof (not yet Lean): On a compact profinite inverse limit, every continuous finite-valued observation depends on a finite stage (clopen-cylinder compactness proof in ANALYSIS.md). Under continuous dynamics, each fixed-time observed record factors through a finite stage. If an exact common pointwise visible period L is independently supplied, one finite depth governs the whole infinite observed word. This is NOT Poincaré recurrence, not an almost-everywhere automatic conclusion, and not a result for arbitrary measurable/discontinuous observations.

Lean scope: OIBridge.RecursiveFiniteFactor proves only the conditional finite-family depth bound and all-time extension assuming a supplied periodic record family, plus involution of a hidden-controlled visible Boolean flip. The topological factorization premise remains unformalized. The file is imported by OIBridge.lean and entered as kernel-only in the maintained census. Build and release-gate checks pass in fresh ordinary PR CI for integrated head 947761dd. No topological or physical theorem is promoted.

Reproduction / provenance

python3 -I -B verification/programmes/substratum/recursive-substates/replay.py

The local deterministic script passed normally and with python -O, with output bytes exactly matching committed RESULTS.json. File content hashes of both source and expected result were checked against the working files that produced the local result. The driver rejects a nonzero child exit, stderr or byte mismatch. Source SHA-256 6beb25044fc6966a954adc186d1a136f42fedd78c938c2683983316b886af666; results SHA-256 8b949aa7cbd688f7c41fa079f761424afea98700e387db42c2b15320174e2b64.

CI: current ordinary draft-PR CI is reported above. No manual dispatch. Local Python evidence does not substitute for Lean, physical or whole-CI evidence.

F2-1 closeout and review provenance

• Frozen preregistration 66eb48ca; A★/stage-1 log b86f0547; A★-1 fdeeb2d5. Thirteen screened rules, no admissible candidate. No general no-go or F2 progress follows.
• First Stage-2 attempt 05ecc873, run 41220197866: permanent INCONCLUSIVE (void). Its files remain preserved.
• Separate bounded repair/run-2 1d345126, evidence 2155273f, wording 4e7ddb70: recorded INCONCLUSIVE (gap). The frozen controls, statements and scientific scope are unchanged.
• Independent-review provenance: ACCEPT at c811e876 and final review of 1d345126 are author-reported in 6543745746 and 6546461983. Reviewer identity/time and a retrievable reviewer-authored verdict remain unresolved in the inspected record. This does not establish that review did not occur.
• Read frozen A★ alongside G★ correction 6546748788: its ones-fixing necessity clause is overstated; the scoped thirteen-rule gap outcome is unchanged. F2/L3 remain open.
• Lane B provenance audit found stale mutable status surfaces, not grounds to reopen accepted science. The four-file status-only correction at 947761dd40c17e4fdedcd8972a0b4e53e8bfba42 is integrated as the current PR head. Fresh CI is green and independent integrated-commit review accepted the administrative delta; the missing historical reviewer source remains unresolved.

Decisions / next hard-to-vary gate

Model B1 establishes selective classical depth feedback; B2 shows deeper levels can successively influence records only at the expense of a shared finite-period bound under its stated dynamics. The next physically meaningful candidate must supply one fixed scale-coupling law, observer readout, invariant prior, measurable apparatus/control contract and a non-adjustable independent prediction. Reject any model that assumes quantum operations or selects Wilson action by arbitrary knobs. Finite observed entropy does not establish physical area-law entropy; a Cantor-like metric alone does not yield log-periodic physics.

Governance: draft/research only; no manuscript, publication claim, F/E designation, workflow dispatch, merge or active-PR hold violation. Independent exact-commit review and the applicable CI evidence remain required before any promotion; no promotion is authorized here.

[amaybaum]

Formal hold retired (owner decision, chat 2026-11-03). Formal holds are retired as a scheduling mechanism (#1074, rule 7). This PR enters the ordinary backlog.

• Preserved: head 947761dd40c17e4fdedcd8972a0b4e53e8bfba42 and the existing work, reviews and evidence on this branch. This comment changes nothing on the branch.
• Obligation row: #1074, row SUBSTATE-2: recursive inverse-limit substructure, depth feedback and gauge/quantum controls #1010 (Recursive sub-states and coherence sourcing, SUBSTATE-2). Priority, prerequisites, readiness and authorization for this PR's work are recorded only in that row.
• No permission: this change authorizes no work, execution, integration, publication or claim promotion, and waives no review or CI requirement.
• Earlier statements of this PR's hold, on this PR and elsewhere, are superseded by this comment and are not edited. Any other requirement they state, such as a merge gate, stands.

---

Generated by Claude Code

[amaybaum]

Closing without merging (owner approval, amaybaum, Claude Code session chat, 2026-11-04: this PR is to be closed "without merging, retaining its branch at 947761dd40c17e4fdedcd8972a0b4e53e8bfba42").

• Branch: research/substate-2-recursive-20261030 is kept at 947761dd, with its full history. The commits that the frozen records pin stay reachable from it as provenance.

• Evidence: the 32 branch-only records and modules are retained on main via evidence(SUBSTATE-2, #1010): land #1011's 32 branch-only records and modules #1096 (merge commit 9d022a65aed3a422a0a370a326255e7ba1083620), byte-identical to 947761dd. F2CoherenceControls was already on main via evidence(PC-1): retain PC-1's records and kernel controls on main (D-72) #1089.

• Open obligations: they are in SUBSTATE-2: recursive inverse-limit substructure, depth feedback and gauge/quantum controls #1010 (handoff and retention record). They are:

  • F2-1 run 2: INCONCLUSIVE (gap);
  • run 1: VOID;
  • the G★ correction;
  • unresolved review provenance and the durable re-review gate;
  • the mathematical limitations;
  • physical sourcing.

  F2/L3 stay with OI ↔ QM equivalence: proof gate (dependency index; carries #885) #914 and physical coupling with P0: physical observable carrier and evolution selection #866.

Closing this PR is an administrative act. It establishes no scientific result and resolves no research obligation.

---

Generated by Claude Code