diff --git a/VERIFICATION.md b/VERIFICATION.md
index d70bde0e3..bb20cf0bc 100644
--- a/VERIFICATION.md
+++ b/VERIFICATION.md
@@ -12,7 +12,7 @@ Stage 1 is in place for every crate in `crates/`. [`formal/extraction/README.md`
 - Rust the toolchain cannot translate is rewritten by small patches applied only while extracting. The committed source is never patched, and CI runs each patched crate's tests, including checks against real proofs, on the patched source.
 - CI fails when a committed extraction no longer matches the Rust, when one no longer typechecks, or when the patched source fails its tests. Every run reports what each extraction trusts: opaque items, untranslated bodies, patches, and the shared standard-library models.
 
-Stage 2 has begun with the native verifier, whose proofs live in `crates/verifier/proofs/verifier/lean/Verifier/Verification/` and are built with every extraction. They show that the extracted field arithmetic is CompPoly's `BF64` and its cubic extension `Ext3`, that `eq_kernel` computes ArkLib's `multilinearWeight`, and that the verifier's level-0 WHIR fold of a queried row is ArkLib's `multilinearCombine` of the row's lanes (`level0_fold`), resting on no opaque item's assumption.
+Stage 2 has begun with the native verifier, whose proofs live in `crates/verifier/proofs/verifier/lean/Verifier/Verification/` and are built with every extraction. They show that the extracted field arithmetic is CompPoly's `BF64` and its cubic extension `Ext3`, that `eq_kernel` computes ArkLib's `multilinearWeight`, and that the verifier's WHIR fold of a queried row is ArkLib's `multilinearCombine` of the row's lanes, at level 0 (`level0_fold`) and at every deeper level (`deeper_fold`), resting on no opaque item's assumption.
 
 ## Introduction
 
diff --git a/crates/verifier/proofs/verifier/lean/Verifier.lean b/crates/verifier/proofs/verifier/lean/Verifier.lean
index 48f8b164e..dd547c2a6 100644
--- a/crates/verifier/proofs/verifier/lean/Verifier.lean
+++ b/crates/verifier/proofs/verifier/lean/Verifier.lean
@@ -1,4 +1,4 @@
 import Verifier.Extraction
 import Verifier.Verification.ProofObligations
 import Verifier.Verification.Field
-import Verifier.Verification.Fold
+import Verifier.Verification.Dot
diff --git a/crates/verifier/proofs/verifier/lean/Verifier/Extraction/Funs.lean b/crates/verifier/proofs/verifier/lean/Verifier/Extraction/Funs.lean
index afd9c5cc8..df9f25411 100644
--- a/crates/verifier/proofs/verifier/lean/Verifier/Extraction/Funs.lean
+++ b/crates/verifier/proofs/verifier/lean/Verifier/Extraction/Funs.lean
@@ -6260,23 +6260,20 @@ def multilinear.eq_eval
     Visibility: public -/
 @[rust_loop_body]
 def multilinear.dot_loop.body
-  (iter : core.iter.adapters.zip.Zip (core.slice.iter.Iter field.E)
-  (core.slice.iter.Iter field.E)) (result : field.E) :
-  RustM (ControlFlow ((core.iter.adapters.zip.Zip (core.slice.iter.Iter
-    field.E) (core.slice.iter.Iter field.E)) × field.E) field.E)
+  (left : Slice field.E) (right : Slice field.E)
+  (iter : core.ops.range.Range Std.Usize) (result : field.E) :
+  RustM (ControlFlow ((core.ops.range.Range Std.Usize) × field.E) field.E)
   := do
   let (o, iter1) ←
-    core.iter.adapters.zip.Zip.Insts.CoreIterTraitsIteratorIteratorPair.next
-      (core.slice.iter.Iter.Insts.CoreIterTraitsIteratorIteratorSharedAT
-      field.E)
-      (core.slice.iter.Iter.Insts.CoreIterTraitsIteratorIteratorSharedAT
-      field.E) iter
+    core.ops.range.Range.Insts.CoreIterTraitsIteratorIterator.next
+      core.Usize.Insts.CoreIterRangeStep iter
   match o with
   | core.option.Option.None => ok (done result)
-  | core.option.Option.Some p =>
-    let (x, y) := p
-    let e ← field.E.Insts.CoreOpsArithMulEE.mul x y
-    let result1 ← field.E.Insts.CoreOpsArithAddAssignE.add_assign result e
+  | core.option.Option.Some i =>
+    let e ← Slice.index_usize left i
+    let e1 ← Slice.index_usize right i
+    let e2 ← field.E.Insts.CoreOpsArithMulEE.mul e e1
+    let result1 ← field.E.Insts.CoreOpsArithAddAssignE.add_assign result e2
     ok (cont (iter1, result1))
 
 /-- [verifier::multilinear::dot]: loop 0:
@@ -6284,12 +6281,13 @@ def multilinear.dot_loop.body
     Visibility: public -/
 @[rust_loop]
 def multilinear.dot_loop
-  (iter : core.iter.adapters.zip.Zip (core.slice.iter.Iter field.E)
-  (core.slice.iter.Iter field.E)) (result : field.E) :
+  (iter : core.ops.range.Range Std.Usize) (left : Slice field.E)
+  (right : Slice field.E) (result : field.E) :
   RustM field.E
   := do
   loop
-    (fun (iter1, result1) => multilinear.dot_loop.body iter1 result1)
+    (fun (iter1, result1) => multilinear.dot_loop.body left right iter1
+      result1)
     (iter, result)
 
 /-- [verifier::multilinear::dot]:
@@ -6301,14 +6299,8 @@ def multilinear.dot
   let right_val ← core.slice.Slice.len right
   massert (left_val = right_val)
   let result ← field.ZERO
-  let i ← core.slice.Slice.iter left
-  let iter ←
-    core.iter.traits.iterator.Iterator.zip.default
-      (core.slice.iter.Iter.Insts.CoreIterTraitsIteratorIteratorSharedAT
-      field.E)
-      (core.SharedASlice.Insts.CoreIterTraitsCollectIntoIteratorSharedATIter
-      field.E) i right
-  multilinear.dot_loop iter result
+  multilinear.dot_loop { start := 0#usize, «end» := left_val } left right
+    result
 
 /-- [verifier::multilinear::padding_weight::{impl core::ops::function::Fn<(usize,), verifier::field::E> for verifier::multilinear::padding_weight::{closure}<'_0>}::call]:
     Source: 'crates/verifier/src/multilinear.rs', lines 91:14-91:49 -/
diff --git a/crates/verifier/proofs/verifier/lean/Verifier/Verification/Dot.lean b/crates/verifier/proofs/verifier/lean/Verifier/Verification/Dot.lean
new file mode 100644
index 000000000..36a824c5f
--- /dev/null
+++ b/crates/verifier/proofs/verifier/lean/Verifier/Verification/Dot.lean
@@ -0,0 +1,140 @@
+import Verifier.Verification.Fold
+
+/-! The verifier's fold of a queried row past level 0: `dot(ext_row(row), eq_kernel(r))` is ArkLib's
+`multilinearCombine` at `r` of the row's lanes, lane `i` being the extension element of the row's words
+`3i, 3i + 1, 3i + 2` (`deeper_fold`). -/
+
+open CoreModels Aeneas
+open Aeneas.Std hiding namespace core alloc
+open RustM Std.Do
+open Verifier.Field
+
+namespace Verifier
+
+/-- `dot`'s sum over its first `i` terms. -/
+def dotSum (l r : List verifier.field.E) (i : ℕ) : BF64.Ext3 :=
+  ∑ t ∈ Finset.range i, toExt3 l[t]! * toExt3 r[t]!
+
+/-- `dot`'s loop invariant after `i` terms. -/
+def dotInv (l r : Slice verifier.field.E) (i : Std.Usize) (s : verifier.field.E) : RustM Prop :=
+  ok (toExt3 s = dotSum l.val r.val i.val)
+
+@[spec]
+theorem dot_loop_spec (l r : Slice verifier.field.E) (n : Std.Usize) (hn : n.val = l.length)
+    (hr : l.length = r.length) (s0 : verifier.field.E) (h0 : toExt3 s0 = 0) :
+    ⦃ ⌜ True ⌝ ⦄
+    verifier.multilinear.dot_loop { start := 0#usize, «end» := n } l r s0
+    ⦃ ⇓ s => ⌜ (dotInv l r n s).holds ⌝ ⦄ := by
+  unfold verifier.multilinear.dot_loop
+  apply Hax.loop_range_spec_unsigned (inv := dotInv l r)
+  · simp
+  · simp [dotInv, dotSum, h0]
+  · intro acc i _ hi hinv
+    simp only [dotInv, holds_ok] at hinv
+    hax_mvcgen [verifier.multilinear.dot_loop.body, E_mul_ok, E_add_assign_ok, dotInv]
+    case vc1.hQ =>
+      rename_i hlt s hs x hx y hy
+      obtain ⟨hxi, rfl⟩ := hx
+      obtain ⟨hyi, rfl⟩ := hy
+      refine ⟨hlt, trivial, hs, ?_⟩
+      simp only [dotInv, holds_ok, toExt3_ofExt3, hinv, hs, dotSum, Finset.sum_range_succ]
+      simp [getElem!_pos, hxi, hyi]
+    case vc1.hQ =>
+      have : i.val = n.val := by scalar_tac
+      rw [hinv, this]
+    all_goals scalar_tac
+
+theorem dot_spec (l r : Slice verifier.field.E) (hr : l.length = r.length) :
+    ⦃ ⌜ True ⌝ ⦄
+    verifier.multilinear.dot l r
+    ⦃ ⇓ s => ⌜ toExt3 s = dotSum l.val r.val l.length ⌝ ⦄ := by
+  hax_mvcgen [verifier.multilinear.dot, ZERO_ok, dotInv, toExt3_ofExt3]
+  case vc1.hn => rename_i h _ _ _ _; rw [h, Slice.len_val]
+  case vc4.hQ.success => rename_i h _ _ _ _ _ hs; rwa [h, Slice.len_val] at hs
+  case vc5.h_fail => rename_i h _ h' hne; exact hne (by rw [h, h']; scalar_tac)
+
+/-- Lane `k` of a row past level 0: its words `3k, 3k + 1, 3k + 2` as one extension element. -/
+def wordsE (w : List verifier.field.K) (k : ℕ) : verifier.field.E := ⟨w[3 * k]!, w[3 * k + 1]!, w[3 * k + 2]!⟩
+
+/-- `ext_row`'s loop invariant after `i` lanes. -/
+def extInv (words : Slice verifier.field.K) (i : Std.Usize) (row : alloc.vec.Vec verifier.field.E) : RustM Prop :=
+  ok (row.val = (List.range i.val).map (wordsE words.val))
+
+@[spec]
+theorem ext_row_loop_spec (words : Slice verifier.field.K) (n : Std.Usize) (hn : 3 * n.val ≤ words.length)
+    (row0 : alloc.vec.Vec verifier.field.E) (h0 : row0.val = []) :
+    ⦃ ⌜ True ⌝ ⦄
+    verifier.whir.ext_row_loop { start := 0#usize, «end» := n } words row0
+    ⦃ ⇓ row => ⌜ (extInv words n row).holds ⌝ ⦄ := by
+  unfold verifier.whir.ext_row_loop
+  apply Hax.loop_range_spec_unsigned (inv := extInv words)
+  · simp
+  · simp [extInv, h0]
+  · intro acc i _ hi hinv
+    simp only [extInv, holds_ok] at hinv
+    have hlen : acc.length = i.val := by
+      show acc.val.length = _
+      rw [hinv, List.length_map, List.length_range]
+    hax_mvcgen [verifier.whir.ext_row_loop.body, verifier.field.E.new, extInv]
+    case vc1.h => scalar_tac
+    case vc2 =>
+      rename_i hlt s hs a ha x hx b hb y hy c hc z hz r hr
+      obtain ⟨wa, rfl⟩ := hx
+      obtain ⟨wb, rfl⟩ := hy
+      obtain ⟨wc, rfl⟩ := hz
+      refine ⟨hlt, trivial, hs, ?_⟩
+      simp only [extInv, holds_ok, hr, hinv, hs, List.range_succ, List.map_append, List.map_singleton]
+      congr 2
+      simp only [wordsE]
+      have ha' : a.val = 3 * i.val := by scalar_tac
+      have hb' : b.val = a.val + 1 := by scalar_tac
+      have hc' : c.val = a.val + 2 := by scalar_tac
+      rw [← ha', ← hb', ← hc', getElem!_pos words.val _ wa, getElem!_pos words.val _ wb, getElem!_pos words.val _ wc]
+    case vc1.hQ =>
+      have : i.val = n.val := by scalar_tac
+      rw [hinv, this]
+    all_goals scalar_tac
+
+theorem ext_row_spec (words : Slice verifier.field.K) :
+    ⦃ ⌜ True ⌝ ⦄
+    verifier.whir.ext_row words
+    ⦃ ⇓ row => ⌜ row.val = (List.range (words.length / 3)).map (wordsE words.val) ⌝ ⦄ := by
+  hax_mvcgen [verifier.whir.ext_row, extInv]
+  case vc1.hn => rename_i m h n hn; rw [hn, h]; scalar_tac
+  case vc3.h_ok.success => rename_i m h n hn _ hr; rwa [hn, h, Slice.len_val] at hr
+
+/-- A row's lanes past level 0 as ArkLib's interleaved words at one column: lane `i` is the extension
+element of the row's words `3i, 3i + 1, 3i + 2`. -/
+def extLanes (words : List verifier.field.K) (ϑ : ℕ) : Fin (2 ^ ϑ) → Unit → BF64.Ext3 :=
+  fun i _ => toExt3 (wordsE words i)
+
+theorem deeper_fold (point : Slice verifier.field.E) (hn : point.length ≤ 8)
+    (lw : alloc.vec.Vec verifier.field.E) (hlw : verifier.multilinear.eq_kernel point = ok lw)
+    (words : Slice verifier.field.K) (hwords : words.length = 3 * 2 ^ point.length)
+    (row : alloc.vec.Vec verifier.field.E) (hrow : verifier.whir.ext_row words = ok row)
+    (l w : Slice verifier.field.E) (hl : l.val = row.val) (hw : w.val = lw.val) :
+    ⦃ ⌜ True ⌝ ⦄
+    verifier.multilinear.dot l w
+    ⦃ ⇓ s => ⌜ toExt3 s =
+      multilinearCombine (extLanes words.val point.length) (fun t : Fin point.length => toExt3 point.val[t]!) () ⌝ ⦄ := by
+  obtain ⟨hlen, hval⟩ := Hax.triple_noThrow_elim (eq_kernel_spec point hn) hlw
+  have hr := Hax.triple_noThrow_elim (ext_row_spec words) hrow
+  have h3 : words.length / 3 = 2 ^ point.length := by rw [hwords]; omega
+  rw [h3] at hr
+  have hll : l.length = w.length := by
+    show l.val.length = w.val.length
+    rw [hl, hw, hr, List.length_map, List.length_range]
+    exact hlen.symm
+  have hd := dot_spec l w hll
+  obtain ⟨s, hs⟩ := Hax.triple_noThrow_exists_ok hd
+  have hp := Hax.triple_noThrow_elim hd hs
+  rw [hs]
+  refine Hax.triple_of_ok rfl ?_
+  have hl2 : l.length = 2 ^ point.length := by
+    show l.val.length = _
+    rw [hl, hr, List.length_map, List.length_range]
+  rw [show toExt3 s = _ from hp, dotSum, hl2, Finset.sum_range, multilinearCombine]
+  refine Finset.sum_congr rfl fun i _ => ?_
+  rw [hw, hval i i.isLt, eqw_eq_multilinearWeight _ _ _ i.isLt, hl, hr, extLanes]
+  rw [smul_eq_mul, mul_comm]
+  simp [getElem!_pos]
diff --git a/crates/verifier/proofs/verifier/lean/Verifier/Verification/Std.lean b/crates/verifier/proofs/verifier/lean/Verifier/Verification/Std.lean
index 088823ee0..30db2c68f 100644
--- a/crates/verifier/proofs/verifier/lean/Verifier/Verification/Std.lean
+++ b/crates/verifier/proofs/verifier/lean/Verifier/Verification/Std.lean
@@ -70,6 +70,21 @@ theorem from_elem_triple (x : verifier.field.E) (n : Std.Usize) :
     rw [hm, RustM.match.ok]
     simp only [dite_eq_left hb]
     exact Hax.triple_of_ok (Aeneas.Std.bind_ok _ _) (by rw [← hn]; exact Data.ListN.ListN.from_to_inverse)
+@[spec]
+theorem vec_new_triple (T : Type) :
+    ⦃ ⌜ True ⌝ ⦄ alloc.vec.Vec.new T ⦃ ⇓ v => ⌜ v.val = [] ⌝ ⦄ := by
+  simp only [alloc.vec.Vec.new, rust_primitives.sequence.seq_empty]
+  exact Hax.triple_of_ok (Aeneas.Std.bind_ok _ _) rfl
+
+@[spec]
+theorem vec_push_triple {T : Type} (v : alloc.vec.Vec T) (x : T) (h : v.length < Usize.max) :
+    ⦃ ⌜ True ⌝ ⦄ alloc.vec.Vec.push v x ⦃ ⇓ r => ⌜ r.val = v.val ++ [x] ⌝ ⦄ := by
+  have h' : (v.val ++ [x]).length ≤ Usize.max := by
+    have : v.val.length < Usize.max := h
+    simp; omega
+  simp only [alloc.vec.Vec.push, rust_primitives.sequence.seq_push, dite_eq_left h']
+  exact Hax.triple_of_ok (Aeneas.Std.bind_ok _ _) Data.ListN.ListN.from_to_inverse
+
 theorem getElem!_set {α : Type} [Inhabited α] (l : List α) (a j : ℕ) (x : α) :
     (l.set a x)[j]! = if a = j ∧ j < l.length then x else l[j]! := by
   by_cases h : a = j
diff --git a/crates/verifier/proofs/verifier/patches/pre/08-dot-by-index.patch b/crates/verifier/proofs/verifier/patches/pre/08-dot-by-index.patch
new file mode 100644
index 000000000..ababd005b
--- /dev/null
+++ b/crates/verifier/proofs/verifier/patches/pre/08-dot-by-index.patch
@@ -0,0 +1,15 @@
+diff --git a/crates/verifier/src/multilinear.rs b/crates/verifier/src/multilinear.rs
+index e55ba4a8..ba2ffc4e 100644
+--- a/crates/verifier/src/multilinear.rs
++++ b/crates/verifier/src/multilinear.rs
+@@ -79,8 +79,8 @@ pub fn eq_eval(left: &[E], right: &[E]) -> E {
+ pub fn dot(left: &[E], right: &[E]) -> E {
+     assert_eq!(left.len(), right.len());
+     let mut result = ZERO;
+-    for (&x, &y) in left.iter().zip(right) {
+-        result += x * y;
++    for i in 0..left.len() {
++        result += left[i] * right[i];
+     }
+     result
+ }