diff --git a/.github/workflows/continued-frame-cuts-186.yml b/.github/workflows/continued-frame-cuts-186.yml
new file mode 100644
index 0000000000..993b7ab696
--- /dev/null
+++ b/.github/workflows/continued-frame-cuts-186.yml
@@ -0,0 +1,27 @@
+name: Verify continued frame cuts on PR186
+on:
+  push:
+    paths:
+      - 'research/continued-frame-cuts-186/**'
+      - 'research/coordinated-frames-and-entrance-banks/**'
+      - '.github/workflows/continued-frame-cuts-186.yml'
+  pull_request:
+    paths:
+      - 'research/continued-frame-cuts-186/**'
+      - 'research/coordinated-frames-and-entrance-banks/**'
+      - '.github/workflows/continued-frame-cuts-186.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  verify:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 20
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: '3.13'
+      - run: python3 -B research/continued-frame-cuts-186/verify.py
diff --git a/research/balanced-nullity-banks-163/AUDIT-TARGETS.json b/research/balanced-nullity-banks-163/AUDIT-TARGETS.json
new file mode 100644
index 0000000000..31acd84088
--- /dev/null
+++ b/research/balanced-nullity-banks-163/AUDIT-TARGETS.json
@@ -0,0 +1,54 @@
+[
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_apply",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_zero",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_comp_disjoint",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_involutive",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_id",
+  "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_commute",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_eq_sum",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.weighted_apply",
+  "Mathproof.BeyondNLogN.TakeoverBit.weighted_comp",
+  "Mathproof.BeyondNLogN.TakeoverBit.weighted_id",
+  "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_eq",
+  "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_apply",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_idempotent",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_disjoint",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_sum",
+  "Mathproof.BeyondNLogN.TakeoverBit.coordinateBank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_bijective",
+  "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_roles_injective",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_block_capacity",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_block_capacity",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.inherited_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.removed_exterior_count",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_max_rank",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.normalized_volume",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_deficit",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.actual_role_counts",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_weighted_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_weighted_bank_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_eq_address",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_full",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_selected_contents",
+  "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_undeferred_contents",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.inherited_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.removed_exterior_count",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_rank_mass",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_max_rank",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.normalized_volume",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_deficit",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_role_counts",
+  "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_bank_and_inventory_certificate",
+  "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt",
+  "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve",
+  "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.row_reserve_value",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.core_slacks_positive",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_slacks_positive",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.strengthened_public_margin",
+  "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_concrete_guards"
+]
diff --git a/research/balanced-nullity-banks-163/BankAndBalanced.lean b/research/balanced-nullity-banks-163/BankAndBalanced.lean
new file mode 100644
index 0000000000..b681bd7acf
--- /dev/null
+++ b/research/balanced-nullity-banks-163/BankAndBalanced.lean
@@ -0,0 +1,625 @@
+/- Publication snapshot: actual bank endpoints and conditional finite arithmetic only. -/
+import Mathlib.LinearAlgebra.Prod
+import Mathlib.Tactic.Abel
+import Mathlib.Algebra.BigOperators.Group.List.Defs
+import Mathlib.Data.List.Pairwise
+import Mathlib.Data.List.FinRange
+import Mathlib.Data.Fintype.Card
+import Mathlib.Data.Nat.Basic
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.FinCases
+import Mathlib.Tactic.GCongr
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity
+import Mathlib.Data.Real.Basic
+import Mathlib.Tactic.NormNum
+
+
+/-! Exact partial bit address interchanges for split-projector bank packing.
+No supplier, routing, recurrence, precision, or tape theorem is assumed here. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit
+
+variable {R V : Type*} [CommRing R] [AddCommGroup V] [Module R V]
+
+/-- Exchange the P-address component between two arbitrary incoming banks. -/
+def partialSwap (P : V →ₗ[R] V) : (V × V) →ₗ[R] (V × V) where
+  toFun z := (z.1 - P z.1 + P z.2, z.2 - P z.2 + P z.1)
+  map_add' a b := by
+    apply Prod.ext <;> simp only [Prod.fst_add, Prod.snd_add, map_add] <;> abel
+  map_smul' r z := by
+    apply Prod.ext <;> simp only [Prod.smul_fst, Prod.smul_snd, map_smul,
+      RingHom.id_apply, smul_add, smul_sub]
+
+@[simp] theorem partialSwap_apply (P : V →ₗ[R] V) (z : V × V) :
+    partialSwap P z = (z.1 - P z.1 + P z.2, z.2 - P z.2 + P z.1) := rfl
+
+@[simp] theorem partialSwap_zero : partialSwap (0 : V →ₗ[R] V) = LinearMap.id := by
+  apply LinearMap.ext
+  intro z
+  simp
+
+/-- Every disjoint projector pair composes by adding its exchanged components.
+The incoming pair is arbitrary, so correlated dirty contents are included. -/
+theorem partialSwap_comp_disjoint (P Q : V →ₗ[R] V)
+    (hPQ : P.comp Q = 0) (hQP : Q.comp P = 0) :
+    (partialSwap P).comp (partialSwap Q) = partialSwap (P + Q) := by
+  have hpq (z : V) : P (Q z) = 0 := by
+    simpa using LinearMap.congr_fun hPQ z
+  have hqp (z : V) : Q (P z) = 0 := by
+    simpa using LinearMap.congr_fun hQP z
+  apply LinearMap.ext
+  intro z
+  rcases z with ⟨x,y⟩
+  apply Prod.ext <;>
+    simp only [LinearMap.comp_apply, partialSwap_apply, map_add, map_sub,
+      LinearMap.add_apply, hpq] <;> abel
+
+/-- A completed split-projector core is its own exact inverse. -/
+theorem partialSwap_involutive (P : V →ₗ[R] V) (hP : P.comp P = P) :
+    Function.Involutive (partialSwap P) := by
+  have hpp (z : V) : P (P z) = P z := by
+    simpa using LinearMap.congr_fun hP z
+  intro z
+  rcases z with ⟨x,y⟩
+  apply Prod.ext <;> simp only [partialSwap_apply, map_add, map_sub, hpp] <;> abel
+
+@[simp] theorem partialSwap_id (z : V × V) :
+    partialSwap (LinearMap.id : V →ₗ[R] V) z = (z.2,z.1) := by
+  simp
+
+/-- Split-projector factors commute at the actual address-map level. -/
+theorem partialSwap_commute (P Q : V →ₗ[R] V)
+    (hPQ : P.comp Q = 0) (hQP : Q.comp P = 0) :
+    (partialSwap P).comp (partialSwap Q) =
+      (partialSwap Q).comp (partialSwap P) := by
+  rw [partialSwap_comp_disjoint P Q hPQ hQP,
+    partialSwap_comp_disjoint Q P hQP hPQ, add_comm]
+
+
+
+
+/-- Execute completed cores in any explicit chronological bank schedule. -/
+def bankSwap : List (V →ₗ[R] V) → ((V × V) →ₗ[R] (V × V))
+  | [] => LinearMap.id
+  | P :: Ps => (partialSwap P).comp (bankSwap Ps)
+
+private theorem comp_sum_zero (P : V →ₗ[R] V) (Ps : List (V →ₗ[R] V))
+    (h : ∀ Q ∈ Ps, P.comp Q = 0) : P.comp Ps.sum = 0 := by
+  induction Ps with
+  | nil => simp
+  | cons Q Ps ih =>
+    have hQ := h Q (by simp)
+    have ht : ∀ A ∈ Ps, P.comp A = 0 := fun A hA => h A (by simp [hA])
+    simp only [List.sum_cons, LinearMap.comp_add, hQ, ih ht, zero_add]
+
+private theorem sum_comp_zero (P : V →ₗ[R] V) (Ps : List (V →ₗ[R] V))
+    (h : ∀ Q ∈ Ps, Q.comp P = 0) : Ps.sum.comp P = 0 := by
+  induction Ps with
+  | nil => simp
+  | cons Q Ps ih =>
+    have hQ := h Q (by simp)
+    have ht : ∀ A ∈ Ps, A.comp P = 0 := fun A hA => h A (by simp [hA])
+    simp only [List.sum_cons, LinearMap.add_comp, hQ, ih ht, zero_add]
+
+/-- A whole shared bank executes exactly the swap on the sum of its disjoint
+address blocks. The statement is on every input, not only clean scratch. -/
+theorem bankSwap_eq_sum (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0)) :
+    bankSwap Ps = partialSwap Ps.sum := by
+  induction Ps with
+  | nil => simp [bankSwap]
+  | cons P Ps ih =>
+    rcases List.pairwise_cons.mp h with ⟨hp, ht⟩
+    rw [bankSwap, ih ht, List.sum_cons]
+    exact partialSwap_comp_disjoint P Ps.sum
+      (comp_sum_zero P Ps (fun Q hQ => (hp Q hQ).1))
+      (sum_comp_zero P Ps (fun Q hQ => (hp Q hQ).2))
+
+/-- A complete split partition needs no exterior auxiliary child. -/
+theorem bankSwap_full (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0))
+    (hsum : Ps.sum = LinearMap.id) (z : V × V) :
+    bankSwap Ps z = (z.2,z.1) := by
+  rw [bankSwap_eq_sum Ps h, hsum]
+  exact partialSwap_id z
+
+/-- One common ancestor-dependent weighted chart, retained for the whole bank. -/
+def weighted (A : (V × V) ≃ₗ[R] (V × V))
+    (T : (V × V) →ₗ[R] (V × V)) : (V × V) →ₗ[R] (V × V) :=
+  A.toLinearMap.comp (T.comp A.symm.toLinearMap)
+
+@[simp] theorem weighted_apply (A : (V × V) ≃ₗ[R] (V × V))
+    (T : (V × V) →ₗ[R] (V × V)) (z : V × V) :
+    weighted A T z = A (T (A.symm z)) := rfl
+
+/-- Chart conjugations telescope exactly, with no restriction on dirty data. -/
+theorem weighted_comp (A : (V × V) ≃ₗ[R] (V × V))
+    (T U : (V × V) →ₗ[R] (V × V)) :
+    (weighted A T).comp (weighted A U) = weighted A (T.comp U) := by
+  apply LinearMap.ext
+  intro z
+  simp
+
+@[simp] theorem weighted_id (A : (V × V) ≃ₗ[R] (V × V)) :
+    weighted A (LinearMap.id : (V × V) →ₗ[R] (V × V)) = LinearMap.id := by
+  apply LinearMap.ext
+  intro z
+  simp
+
+def weightedBank (A : (V × V) ≃ₗ[R] (V × V)) :
+    List (V →ₗ[R] V) → ((V × V) →ₗ[R] (V × V))
+  | [] => LinearMap.id
+  | P :: Ps => (weighted A (partialSwap P)).comp (weightedBank A Ps)
+
+/-- Every paid weighted core is retained in its chronological product. -/
+theorem weightedBank_eq (A : (V × V) ≃ₗ[R] (V × V))
+    (Ps : List (V →ₗ[R] V)) : weightedBank A Ps = weighted A (bankSwap Ps) := by
+  induction Ps with
+  | nil => simp [weightedBank, bankSwap]
+  | cons P Ps ih =>
+    rw [weightedBank, ih, weighted_comp, bankSwap]
+
+/-- Heterogeneous completed-core packing has the exact full weighted endpoint. -/
+theorem weightedBank_full (A : (V × V) ≃ₗ[R] (V × V))
+    (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0))
+    (hsum : Ps.sum = LinearMap.id) (z : V × V) :
+    weightedBank A Ps z = A ((A.symm z).2,(A.symm z).1) := by
+  rw [weightedBank_eq, weighted_apply, bankSwap_full Ps h hsum]
+
+
+
+section CoordinateBanks
+variable {ι κ : Type*} [DecidableEq ι]
+
+/-- Explicit split address blocks from an actual coordinate-owner ledger. -/
+def coordinateProjector (owner : κ → ι) (s : ι) :
+    (κ → R) →ₗ[R] (κ → R) where
+  toFun v k := if owner k = s then v k else 0
+  map_add' v w := by
+    funext k
+    by_cases h : owner k = s <;> simp [h]
+  map_smul' r v := by
+    funext k
+    by_cases h : owner k = s <;> simp [h]
+
+@[simp] theorem coordinateProjector_apply (owner : κ → ι) (s : ι)
+    (v : κ → R) (k : κ) :
+    coordinateProjector (R := R) owner s v k = if owner k = s then v k else 0 := rfl
+
+theorem coordinateProjector_idempotent (owner : κ → ι) (s : ι) :
+    (coordinateProjector (R := R) owner s).comp (coordinateProjector (R := R) owner s) =
+      coordinateProjector (R := R) owner s := by
+  apply LinearMap.ext
+  intro v
+  funext k
+  by_cases h : owner k = s <;> simp [h]
+
+theorem coordinateProjector_disjoint (owner : κ → ι) (s t : ι) (hst : s ≠ t) :
+    (coordinateProjector (R := R) owner s).comp (coordinateProjector (R := R) owner t) = 0 := by
+  apply LinearMap.ext
+  intro v
+  funext k
+  by_cases h : owner k = s
+  · have ht : owner k ≠ t := by simpa [h] using hst
+    simp [h,hst]
+  · simp [h]
+
+private theorem coordinateProjector_sum_apply (owner : κ → ι)
+    (roles : List ι) (hroles : roles.Nodup) (v : κ → R) (k : κ) :
+    (roles.map (coordinateProjector (R := R) owner)).sum v k =
+      if owner k ∈ roles then v k else 0 := by
+  induction roles with
+  | nil => simp
+  | cons s roles ih =>
+    rcases List.nodup_cons.mp hroles with ⟨hs, ht⟩
+    simp only [List.map_cons, List.sum_cons, LinearMap.add_apply, Pi.add_apply,
+      coordinateProjector_apply, ih ht, List.mem_cons]
+    by_cases he : owner k = s
+    · have hn : owner k ∉ roles := by simpa [he] using hs
+      simp [he,hs]
+    · simp [he]
+
+/-- A disjoint exhaustive coordinate ledger gives an actual identity projector. -/
+theorem coordinateProjector_sum (owner : κ → ι) (roles : List ι)
+    (hroles : roles.Nodup) (hcover : ∀ k, owner k ∈ roles) :
+    (roles.map (coordinateProjector (R := R) owner)).sum = LinearMap.id := by
+  apply LinearMap.ext
+  intro v
+  funext k
+  rw [coordinateProjector_sum_apply owner roles hroles v k]
+  simp [hcover k]
+
+/-- The concrete coordinate-owner construction supplies every projector premise
+of the heterogeneous bank endpoint, including arbitrary incoming correlations. -/
+theorem coordinateBank_full (owner : κ → ι) (roles : List ι)
+    (hroles : roles.Nodup) (hcover : ∀ k, owner k ∈ roles)
+    (z : (κ → R) × (κ → R)) :
+    bankSwap (roles.map (coordinateProjector (R := R) owner)) z = (z.2,z.1) := by
+  apply bankSwap_full
+  · rw [List.pairwise_map]
+    exact (List.nodup_iff_pairwise_ne.mp hroles).imp (fun hst =>
+      ⟨coordinateProjector_disjoint owner _ _ hst,
+        coordinateProjector_disjoint owner _ _ (Ne.symm hst)⟩)
+  · exact coordinateProjector_sum owner roles hroles hcover
+
+end CoordinateBanks
+
+section Routing
+variable {G S : Type*} [Group G]
+
+/-- Actual bank address for invocation g and fixed role embedding H_s. -/
+def bankAddress (H : S → G) (g : G) (s : S) : G := g * (H s)⁻¹
+
+/-- Every role visits every physical bank once; no favorable subset is selected. -/
+theorem bankAddress_bijective (H : S → G) (s : S) :
+    Function.Bijective (fun g => bankAddress H g s) := by
+  constructor
+  · intro g g' h
+    exact mul_right_cancel h
+  · intro b
+    refine ⟨b * H s, ?_⟩
+    simp [bankAddress, mul_assoc]
+
+/-- Distinct role embeddings rule out within-invocation physical aliasing. -/
+theorem bankAddress_roles_injective (H : S → G) (hH : Function.Injective H)
+    (g : G) : Function.Injective (bankAddress H g) := by
+  intro s t h
+  apply hH
+  have hi : (H s)⁻¹ = (H t)⁻¹ := mul_left_cancel h
+  exact inv_injective hi
+
+end Routing
+end Mathproof.BeyondNLogN.TakeoverBit
+
+
+
+
+
+/-! Concrete PR155 p12 bit orbit-bank construction and complete paid ledger.
+The inherited all-input local word, stage cover and routing contracts remain
+explicit upstream dependencies; this file proves the new address endpoints,
+coordinate capacities and exact inventory, not an all-size running-time claim. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit.P12
+open Mathproof.BeyondNLogN.TakeoverBit
+
+variable {R : Type*} [CommRing R]
+
+/-- The 18 actual disjoint four-coordinate blocks for retained-gauge cores. -/
+def selectedOwner (j : Fin 72) : Fin 18 := ⟨j.val / 4, by omega⟩
+
+/-- The three actual disjoint 24-coordinate blocks for undeferred cores. -/
+def undeferredOwner (j : Fin 72) : Fin 3 := ⟨j.val / 24, by omega⟩
+
+theorem selected_block_capacity (s : Fin 18) :
+    (Finset.univ.filter (fun j : Fin 72 => selectedOwner j = s)).card = 4 := by
+  fin_cases s <;> decide
+
+theorem undeferred_block_capacity (s : Fin 3) :
+    (Finset.univ.filter (fun j : Fin 72 => undeferredOwner j = s)).card = 24 := by
+  fin_cases s <;> decide
+
+/-- Eighteen completed selected cores cover the entire address module. -/
+theorem selected_bank_full (z : (Fin 72 → R) × (Fin 72 → R)) :
+    bankSwap ((List.finRange 18).map (coordinateProjector (R := R) selectedOwner)) z =
+      (z.2,z.1) := by
+  exact coordinateBank_full selectedOwner (List.finRange 18)
+    (List.nodup_finRange 18) (fun k => by simp) z
+
+/-- Three completed undeferred cores also cover the entire address module. -/
+theorem undeferred_bank_full (z : (Fin 72 → R) × (Fin 72 → R)) :
+    bankSwap ((List.finRange 3).map (coordinateProjector (R := R) undeferredOwner)) z =
+      (z.2,z.1) := by
+  exact coordinateBank_full undeferredOwner (List.finRange 3)
+    (List.nodup_finRange 3) (fun k => by simp) z
+
+/-- Every child in the immutable freshly replayed p12 profile, including rank60. -/
+def inheritedHistogram : List (Nat × Nat) :=
+  [(1,120468),(2,68080),(3,28824),(4,7200),(5,11952),(6,3312),
+   (7,6120),(8,4464),(9,1440),(10,2016),(11,2664),(12,3528),
+   (13,1944),(14,2736),(15,1152),(16,4104),(17,6192),(18,5832),
+   (19,4536),(20,7260),(21,8580),(22,15912),(60,5720)]
+
+/-- The actual coordinate-full orbit banks remove precisely auxiliary exteriors;
+all internal, source, target, center and data exterior entries remain charged. -/
+def packedHistogram : List (Nat × Nat) :=
+  inheritedHistogram.filter (fun p => p.1 != 60)
+
+def rankMass (H : List (Nat × Nat)) : Nat :=
+  (H.map (fun p => p.1*p.2)).sum
+
+theorem inherited_rank_mass : rankMass inheritedHistogram = 2096432 := by decide
+
+theorem removed_exterior_count :
+    (inheritedHistogram.filter (fun p => p.1 == 60)).map Prod.snd = [5720] := by decide
+
+theorem packed_rank_mass : rankMass packedHistogram = 1753232 := by decide
+
+theorem packed_max_rank : ∀ p ∈ packedHistogram, p.1 ≤ 22 := by decide
+
+/-- Three private stage banks for every full orbit, including fractional stock. -/
+def normalizedVolume : ℚ := 2*1760 + 3*((5720:ℚ)/18 + (19904:ℚ)/3)
+
+theorem normalized_volume : normalizedVolume = 73132/3 := by
+  norm_num [normalizedVolume]
+
+theorem packed_deficit : (72:ℚ)*normalizedVolume - rankMass packedHistogram = 1936 := by
+  rw [packed_rank_mass, normalized_volume]
+  norm_num
+
+theorem actual_role_counts :
+    5720 + 19904 = (25624:Nat) ∧ 18*4 = (72:Nat) ∧ 3*24 = (72:Nat) := by decide
+
+
+
+
+/-- The actual common weighted chart retains the selected-bank full endpoint. -/
+theorem selected_weighted_bank_full
+    (A : ((Fin 72 → R) × (Fin 72 → R)) ≃ₗ[R] ((Fin 72 → R) × (Fin 72 → R)))
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    weightedBank A ((List.finRange 18).map (coordinateProjector (R := R) selectedOwner)) z =
+      A ((A.symm z).2,(A.symm z).1) := by
+  rw [weightedBank_eq, weighted_apply, selected_bank_full]
+
+/-- The same exact weighted endpoint holds for every undeferred bank. -/
+theorem undeferred_weighted_bank_full
+    (A : ((Fin 72 → R) × (Fin 72 → R)) ≃ₗ[R] ((Fin 72 → R) × (Fin 72 → R)))
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    weightedBank A ((List.finRange 3).map (coordinateProjector (R := R) undeferredOwner)) z =
+      A ((A.symm z).2,(A.symm z).1) := by
+  rw [weightedBank_eq, weighted_apply, undeferred_bank_full]
+
+end Mathproof.BeyondNLogN.TakeoverBit.P12
+
+
+
+/-! Lift the checked address maps to actual arbitrary register contents.
+No clean-scratch or independence assumption occurs in these statements. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit.Contents
+open Mathproof.BeyondNLogN.TakeoverBit
+
+variable {R V B : Type*} [CommRing R] [AddCommGroup V] [Module R V]
+
+/-- Chronological execution on the complete contents of a physical bank. -/
+def run : List (V →ₗ[R] V) → ((V × V) → B) → ((V × V) → B)
+  | [], contents => contents
+  | P :: Ps, contents => run Ps (fun z => contents (partialSwap P z))
+
+theorem run_eq_address (Ps : List (V →ₗ[R] V)) (contents : (V × V) → B)
+    (z : V × V) : run Ps contents z = contents (bankSwap Ps z) := by
+  induction Ps generalizing contents with
+  | nil => rfl
+  | cons P Ps ih =>
+    simp only [run, ih, bankSwap, LinearMap.comp_apply]
+
+/-- A complete split partition executes the full transform on every dirty table. -/
+theorem run_full (Ps : List (V →ₗ[R] V))
+    (h : Ps.Pairwise (fun P Q => P.comp Q = 0 ∧ Q.comp P = 0))
+    (hsum : Ps.sum = LinearMap.id) (contents : (V × V) → B) (z : V × V) :
+    run Ps contents z = contents (z.2,z.1) := by
+  rw [run_eq_address, bankSwap_full Ps h hsum]
+
+/-- The actual p12 selected orbit bank works for arbitrary full bank contents. -/
+theorem p12_selected_contents
+    (contents : ((Fin 72 → R) × (Fin 72 → R)) → B)
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    run ((List.finRange 18).map (coordinateProjector (R := R) P12.selectedOwner)) contents z =
+      contents (z.2,z.1) := by
+  rw [run_eq_address, P12.selected_bank_full]
+
+/-- The actual p12 undeferred orbit bank has the same unconditional endpoint. -/
+theorem p12_undeferred_contents
+    (contents : ((Fin 72 → R) × (Fin 72 → R)) → B)
+    (z : (Fin 72 → R) × (Fin 72 → R)) :
+    run ((List.finRange 3).map (coordinateProjector (R := R) P12.undeferredOwner)) contents z =
+      contents (z.2,z.1) := by
+  rw [run_eq_address, P12.undeferred_bank_full]
+
+end Mathproof.BeyondNLogN.TakeoverBit.Contents
+
+
+
+/-! Immutable PR161 p12 bank specialization.  The actual coordinate-owner,
+weighted-chart and dirty-content endpoints are reused from the independently
+checked P12 construction; only the new frozen supplier inventory is specialized.
+The retained all-input local word, charts, stage cover, routing, fallback and
+sharp all-size framework are upstream contracts, not proved by this ledger. -/
+namespace Mathproof.BeyondNLogN.TakeoverBit.PR161P12
+open Mathproof.BeyondNLogN.TakeoverBit
+
+/-- Complete child inventory from immutable PR161 head
+`d14e29157bc905be1ced0776dd893d0714013f3a`, profile_p12.json. -/
+def inheritedHistogram : List (Nat × Nat) :=
+  [(1,120972),(2,73336),(3,29832),(4,8208),(5,12096),(6,4392),
+   (7,4824),(8,2736),(9,648),(10,2160),(11,2016),(12,4464),
+   (13,1080),(14,2232),(15,216),(16,2808),(17,2736),(18,3888),
+   (19,3672),(20,7260),(21,8580),(22,15912),(60,5720)]
+
+/-- Eighteen actual four-coordinate selected cores cover Fin72, so their
+auxiliary rank60 exterior is removed; all other children remain charged. -/
+def packedHistogram : List (Nat × Nat) :=
+  inheritedHistogram.filter (fun p => p.1 != 60)
+
+/-- Three stage banks, with selected roles grouped eighteen at a time and
+undeferred roles grouped three at a time. -/
+def normalizedVolume : ℚ := 2*1760 + 3*((5720:ℚ)/18 + (17648:ℚ)/3)
+
+theorem inherited_rank_mass : P12.rankMass inheritedHistogram = 1934000 := by decide
+
+theorem removed_exterior_count :
+    (inheritedHistogram.filter (fun p => p.1 == 60)).map Prod.snd = [5720] := by decide
+
+theorem packed_rank_mass : P12.rankMass packedHistogram = 1590800 := by decide
+
+theorem packed_max_rank : ∀ p ∈ packedHistogram, p.1 ≤ 22 := by decide
+
+theorem normalized_volume : normalizedVolume = 66364/3 := by
+  norm_num [normalizedVolume]
+
+theorem packed_deficit :
+    (72:ℚ)*normalizedVolume - P12.rankMass packedHistogram = 1936 := by
+  rw [packed_rank_mass, normalized_volume]
+  norm_num
+
+theorem actual_role_counts :
+    5720 + 17648 = (23368:Nat) ∧ 18*4 = (72:Nat) ∧ 3*24 = (72:Nat) := by decide
+
+/-- New source binding assembled with actual reused coordinate projectors;
+no new abstract packing gadget or independence of register contents is assumed. -/
+theorem actual_bank_and_inventory_certificate
+    {R : Type*} [CommRing R] (z : (Fin 72 → R) × (Fin 72 → R)) :
+    (∀ s : Fin 18, (Finset.univ.filter
+      (fun j : Fin 72 => P12.selectedOwner j = s)).card = 4) ∧
+    (∀ s : Fin 3, (Finset.univ.filter
+      (fun j : Fin 72 => P12.undeferredOwner j = s)).card = 24) ∧
+    bankSwap ((List.finRange 18).map
+      (coordinateProjector (R := R) P12.selectedOwner)) z = (z.2,z.1) ∧
+    bankSwap ((List.finRange 3).map
+      (coordinateProjector (R := R) P12.undeferredOwner)) z = (z.2,z.1) ∧
+    normalizedVolume = 66364/3 ∧
+    P12.rankMass packedHistogram = 1590800 ∧
+    (72:ℚ)*normalizedVolume - P12.rankMass packedHistogram = 1936 := by
+  exact ⟨P12.selected_block_capacity, P12.undeferred_block_capacity,
+    P12.selected_bank_full z, P12.undeferred_bank_full z,
+    normalized_volume, packed_rank_mass, packed_deficit⟩
+
+end Mathproof.BeyondNLogN.TakeoverBit.PR161P12
+
+
+
+/-! The finite literal guard is discharged by an elementary universal charge
+bound. No positive-guard oracle is used. The condition s ≤ mW is the ordinary
+rank-mass bound of a finite supplier, not an unknown algorithm interface. -/
+namespace Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008
+
+theorem literal_charge_lt (W m G s : ℝ)
+    (hW : 0 ≤ W) (hm : 0 ≤ m) (hG : 0 ≤ G) (hs : s ≤ m*W) :
+    2*G*W*W+8*s+4*W+4+32*m < 64*(W+m+G+1)^3 := by
+  let X := W+m+G+1
+  have hX : 1 ≤ X := by dsimp [X]; linarith
+  have hX0 : 0 ≤ X := by linarith
+  have hWX : W ≤ X := by dsimp [X]; linarith
+  have hmX : m ≤ X := by dsimp [X]; linarith
+  have hGX : G ≤ X := by dsimp [X]; linarith
+  have hcube : 0 < X^3 := by positivity
+  have hquad : X ≤ X^2 := by nlinarith
+  have hquad_cube : X^2 ≤ X^3 := by
+    have hp := mul_nonneg (show 0 ≤ X-1 by linarith) (sq_nonneg X)
+    nlinarith
+  have hfirst : 2*G*W*W ≤ 2*X*X*X := by gcongr
+  have hsecond : m*W ≤ X*X := by gcongr
+  have hlarge : 2*G*W*W+8*s+4*W+4+32*m ≤ 50*X^3 := by
+    nlinarith
+  dsimp [X] at hlarge hcube
+  linarith
+
+theorem finite_row_reserve : (0 : ℝ) < 70000-(51*12320/25) := by norm_num
+
+theorem row_reserve_value : (70000-(51*12320/25) : ℝ) = 224336/5 := by norm_num
+
+
+end Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008
+
+
+
+/- Source-bound arithmetic for PR163 paid balanced transfer and our new nullity-bank refinement.
+   This is not a formalization of all-size integer multiplication. -/
+namespace Mathproof.BeyondNLogN.TakeoverBalanced16320261009
+noncomputable def a : ℝ := (297370240363722999999999702129759636277/500000000000000000000000000000000000000000)
+noncomputable def b : ℝ := (297370240363723/500000000000000000)
+noncomputable def eta : ℝ := (1/1000000000000000000000000)
+noncomputable def beta : ℝ := (1/1000000000000000000000000)
+noncomputable def kappa : ℝ := (118877394946471/200000000000000000)
+noncomputable def tau : ℝ := 1-a
+noncomputable def sigma : ℝ := 1-b
+noncomputable def q : ℝ := a*(1-2*eta)
+noncomputable def c : ℝ := q+eta/4
+noncomputable def eps : ℝ := (1-eta)/(1+q)
+noncomputable def lp : ℝ := 1-q
+noncomputable def lam : ℝ := (tau+lp)/2
+noncomputable def g : ℝ := eps*q
+noncomputable def r : ℝ := (g+1-eps)/2
+noncomputable def delta : ℝ := eta/8
+noncomputable def internal : ℝ := tau+(1-beta)*max (sigma-tau) 0
+noncomputable def leaf : ℝ := sigma+beta*(1-sigma)
+noncomputable def coreSlacks : List ℝ := [
+  (a) /- bit_positive -/,
+  (b-a) /- complex_above_bit -/,
+  (1/32-b) /- complex_below_one_over32 -/,
+  (beta) /- beta_positive -/,
+  (1-beta) /- beta_below_one -/,
+  ((1-beta)*b-a) /- leaf_saving_above_bit -/,
+  (q) /- q_positive -/,
+  (1-internal-q) /- q_below_internal -/,
+  (1-leaf-q) /- q_below_leaf -/,
+  (c) /- c_positive -/,
+  (1-c) /- c_below_one -/,
+  (c-q) /- q_below_reservations -/,
+  (lam-tau) /- lambda_above_tau -/,
+  (lam-sigma) /- lambda_above_sigma -/,
+  (lam-internal) /- lambda_above_internal -/,
+  (lp-lam) /- lambda_prime_above_lambda -/,
+  (lp-leaf) /- compact_leaf -/,
+  (lp-(1-c)) /- compact_reservations -/,
+  (q) /- lambda_prime_below_one -/,
+  (eps) /- epsilon_positive -/,
+  (1-eps) /- epsilon_below_one -/,
+  (1-eps) /- guard_width -/,
+  (1-eps*(1+c)) /- K_geometry -/,
+  (eps*c) /- K_dominates_log -/,
+  (1-eps) /- record_suffix -/,
+  (1-eps-delta) /- phase_local -/,
+  (r-delta) /- phase_boundary -/,
+  (1-eps-r) /- gamma_sublinear -/,
+  (eps-(1-r)/2) /- cell_above_band -/,
+  (1-eps) /- prime_interval_packing -/,
+  (r) /- alpha_positive -/,
+  (1-r) /- alpha_below_one -/,
+  (1/4-r) /- alpha_below_one_fourth -/,
+  (delta) /- delta_positive -/,
+  (1/8-delta) /- delta_below_one_eighth -/,
+  (eps-a) /- short_record_fallback -/,
+  (1-eps-g) /- small_field_exposure -/,
+  (8-eps+r-delta-g) /- artificial_boundary -/,
+  ((1-eps)-kappa) /- paid_balanced_prefix_above_kappa -/,
+  ((a)-kappa) /- coordinate_movement_above_kappa -/,
+  ((g)-kappa) /- compact_phase_layer_above_kappa -/,
+  ((a)-kappa) /- bulk_exposure_above_kappa -/,
+  ((min (1-eps-delta) (r-delta))-kappa) /- Gaussian_arithmetic_above_kappa -/,
+  ((1-eps-delta)-kappa) /- scalar_work_above_kappa -/,
+  ((eps)-kappa) /- dimension_above_kappa -/
+]
+theorem core_slacks_positive : ∀ x ∈ coreSlacks, 0 < x := by
+  norm_num [coreSlacks, a, b, eta, beta, kappa, tau, sigma, q, c, eps, lp, lam, g, r, delta, internal, leaf]
+
+theorem all_47_slacks_positive (literal row : ℝ)
+    (hl : 0 < literal) (hr : 0 < row) :
+    ∀ x ∈ coreSlacks ++ [literal, row], 0 < x := by
+  intro x hx
+  rcases List.mem_append.mp hx with h | h
+  · exact core_slacks_positive x h
+  · simp only [List.mem_cons, List.not_mem_nil, or_false] at h
+    rcases h with h | h
+    · simpa [h] using hl
+    · simpa [h] using hr
+
+theorem strengthened_public_margin :
+    (148492550769873/250000000000000000 : ℝ) < kappa ∧ kappa < g := by
+  norm_num [kappa, g, eps, c, q, a, eta]
+
+/-- Complete arithmetic composition of our certified bit bank with the
+pinned PR163 paid balanced transfer physical complex supplier. All finite guards are proved, subject
+only to explicit nonnegative counts and their standard rank-mass bound.
+This theorem does not claim the retained analytic/tape interfaces. -/
+theorem all_47_concrete_guards (W m G s : ℝ)
+    (hW : 0 ≤ W) (hm : 0 ≤ m) (hG : 0 ≤ G) (hs : s ≤ m*W) :
+    ∀ x ∈ coreSlacks ++
+      [64*(W+m+G+1)^3-(2*G*W*W+8*s+4*W+4+32*m),
+       70000-(51*12320/25)], 0 < x := by
+  apply all_47_slacks_positive
+  · exact sub_pos.mpr
+      (Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt
+        W m G s hW hm hG hs)
+  · exact Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve
+
+end Mathproof.BeyondNLogN.TakeoverBalanced16320261009
diff --git a/research/balanced-nullity-banks-163/LICENSE b/research/balanced-nullity-banks-163/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/balanced-nullity-banks-163/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/balanced-nullity-banks-163/MANIFEST.json b/research/balanced-nullity-banks-163/MANIFEST.json
new file mode 100644
index 0000000000..6d4996ea27
--- /dev/null
+++ b/research/balanced-nullity-banks-163/MANIFEST.json
@@ -0,0 +1,14 @@
+{
+  "arithmetic-replay.log": "393570c538268eb8ef36dd89a007861a7c4e6a6c7782f5fdd0c060e99bc37df0",
+  "AUDIT-TARGETS.json": "1fb48f1e81cbc806c2e08a4ed58ea75a87f6a533138a014a13bb31e7bac0db14",
+  "BankAndBalanced.lean": "8f32bce76e47da3c6165ea94f179ddf351ed50271996ab179505f588333b5867",
+  "expected.json": "0c94f8a31170b953ca0c35f4904e80f4f38dd4a553a60249b224520a6d590a1b",
+  "lean-audit.log": "d4b24c5a28ebe4db51f41fba151500fa4c0ce4a739069e0cc1fc180b8d2117e4",
+  "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+  "NOTICE": "2d192a690294292eaed715da43ed74951a1130de7f38a6c5084816798f4d0c72",
+  "PROOF-ORIGINS.json": "53a97d25d58611056a51a00b6cc5d2e0cefc9b16a968f297d4b8a92baaa990c8",
+  "PROOF.md": "6e6e82b2918a440b4ae556e8fef439ac641be138f8edfb2ee803342cb6e8c56e",
+  "README.md": "10e3f5a2fc7909cde95c7d25204cf0819f638f5a66202b74ef7f645da531a267",
+  "VERIFICATION.json": "991a11ba4f6efadf7927578818c26b109e71d82379c22636ad9cf5d2f507c5d3",
+  "verify.py": "37d317bcc1185db9a08163dd325e9cd95e55007d1db45b7ed5a4d060ec85978e"
+}
diff --git a/research/balanced-nullity-banks-163/NOTICE b/research/balanced-nullity-banks-163/NOTICE
new file mode 100644
index 0000000000..7f827029f3
--- /dev/null
+++ b/research/balanced-nullity-banks-163/NOTICE
@@ -0,0 +1,13 @@
+Conditional balanced transfer with completed-core nullity banks.
+Prepared with substantial OpenAI Codex assistance. No personal author name has been added by this package.
+Upstream framework: OpenAI integer multiplication manuscript; Douglas Colkitt/CrocSwap.
+PR161 annealed modules, paired word and signed merging: eumemic, with Anthropic Claude assistance.
+Paired-cube decoder, shared ledger and pricing: icekylinx/William Porter.
+Full lineage remains in the pinned PR161 NOTICE and SOURCES.json.
+PR163 fused supplier and adapter: chafreaky; paid layout lineage PR23/29 (Zhihao Chen), RaD,
+PR34 (James Chang), PR100/103 (Rohan Arun), and retained PR141 arithmetic body.
+Bit local words and completed-core sharing retain their PR161 attribution, including
+an664 PR128, eumemic PR117, and Zhihao Chen/Swapnil Jain PR97.
+This contribution claims the displayed nullity-bank refinement and composition only,
+not authorship of upstream suppliers, balanced layout, or the all-size framework.
+Apache-2.0. Consult the pinned upstream notices for full attribution.
diff --git a/research/balanced-nullity-banks-163/PROOF-ORIGINS.json b/research/balanced-nullity-banks-163/PROOF-ORIGINS.json
new file mode 100644
index 0000000000..884821366c
--- /dev/null
+++ b/research/balanced-nullity-banks-163/PROOF-ORIGINS.json
@@ -0,0 +1,32 @@
+[
+  {
+    "filename": "TakeoverBit20261008.lean",
+    "sha256": "7f5d05001824fd59c4070be55786151777a03972499092b2222c5b673f8bbe0a",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverBit20261008P12.lean",
+    "sha256": "52ae9bca980c634fcc206f94832a5b72ace2a47609e1d57c663424347d2944c4",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverBit20261008Contents.lean",
+    "sha256": "8b9e75c1ab67d210e9a6385577c9ca965ccc42c930f5279e2ab2e23e96cc2664",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverBit16120261008P12.lean",
+    "sha256": "2ea23c607c23a2590660f588b18eb9f3525d58f9ac6b9461a30331002ed18aae",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  },
+  {
+    "filename": "TakeoverLiteralGuard20261008.lean",
+    "sha256": "8fe9acaccb68ff70eb43d308f0a8b34f38119779ba3514a35078d3a4348d47c6",
+    "change": "imports removed and hoisted; obsolete original-layout assembly wrapper omitted from literal guard"
+  },
+  {
+    "filename": "TakeoverBalanced16320261009.lean",
+    "sha256": "ce2087d53c07e07324556e06e4a4b940905db137b4c723291a77bb2aae511fd7",
+    "change": "proof bodies unchanged; imports removed and hoisted"
+  }
+]
diff --git a/research/balanced-nullity-banks-163/PROOF.md b/research/balanced-nullity-banks-163/PROOF.md
new file mode 100644
index 0000000000..f83a8192e0
--- /dev/null
+++ b/research/balanced-nullity-banks-163/PROOF.md
@@ -0,0 +1,64 @@
+# PR163 paid-balanced review and actual combination with our bit engine
+
+Pinned source: chafreaky/integer-mult-bounds e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997, research/paired-cube-balanced-161. Full finite package replay is owned by the physical/PUBLIC lane. This lane read PROOF.md, the full reference ORIGINAL-PROOF.md, balanced_shared.py, shared_bridge.py, certificate.py, complex/prove.py and the fused verifier's exact signed decoder/frame checks.
+
+## New certified arithmetic point
+
+Use the pinned PR163 fused complex supplier b=297370240363723/(5*10^17) and our previously checked PR161 p12 completed-core nullity-sharing uniform-bit supplier. The latter's exact constants are
+
+    coarse = 302618322217 / 500000000000000,
+    atom = 302446903 / 500000000000,
+    old = 384599 / 10^10,
+    A = (1-atom)*coarse + atom*old
+      = 151223451172976757106049 / 250000000000000000000000000.
+
+The paid atom inequalities A<atom<1-A hold exactly. A exceeds the new complex saving, so set a=(1-beta)*b-zeta, beta=eta=10^-24, zeta=10^-30, q=a*(1-2*eta), epsilon=(1-eta)/(1+q). The actual supported balanced margin is g=epsilon*q. The largest strictly accepted 10^-18-grid point is
+
+    kappa = 118877394946471 / 200000000000000000
+          = 0.000594386974732355.
+
+All47 strict constraints and7 margins pass the UNCHANGED paid-balanced arithmetic body. The next grid point is rejected at these same supplier certificates and parameters. An independent Fraction recomputation verifies the seven margins and all three exact identities below. The complete JSON evidence and exact replay script are expected.json and verify.py in this upload package.
+
+This exceeds public PR163's 0.000593970203079492 by 416771652863/10^18, about0.0702%. It is a real combined arithmetic gain from existing bit headroom; it is not a substantial new complex-supplier breakthrough or a claim to solve the remaining all-size physical/analytic interfaces.
+
+## Why this is a paid layout composition
+
+The PR34/RaD positional construction, retained for deferred signed words through PR100/103 and PR141, separates the extra top bit of each long FFT axis and uses common named low-coordinate groups K_j in [K,2K). The ordinary arbitrary-coordinate router pays both directions of the axis-major/group-major permutation. Entire active, spectator, scratch, phase, control and restored-row fields travel together, preserving twiddle order and forward/inverse alignment. No scalar residual is moved across another operation.
+
+The prefix becomes O(T*p*d), hence margin1-epsilon. The original geometric inequality1-epsilon*(1+c)>0 is RETAINED. This is not permission to delete the old prefix bill without providing the positional router. The existing completed ordinary supplier must expose the arbitrary-width interchange/complete-spectator/restored-row contract; the exact complex supplier must expose the completed tensor contract. Those are the explicit all-size interfaces assumed by this conditional combination.
+
+With c=q+eta/4, r=(g+1-epsilon)/2 and delta=eta/8, exact arithmetic gives
+
+    1-epsilon-g=eta,
+    1-epsilon-r=eta/2,
+    1-epsilon*(1+c)=eta-epsilon*eta/4 >0.
+
+The internal complex exponent and stopping-leaf exponent stay unchanged. Since a<(1-beta)*b, the leaf-ordering gap is strict. The tiny positive backoffs establish eventual asymptotic inequalities, not a usable practical input threshold.
+
+## Full fixed-tape / scalar / row bills retained
+
+The public fused word has scalar_R=15681, physical_R=13041, reuse_pairs=2640, c=19385, q=3157, matched=6861, M_operations=33746. The normalized complex inventory has m66,W15681,rank1033626,deficit1320,maxchild20. Compared with161, the rank mass is unchanged but the complete child histogram changes; that is why its strict complex moment improves. The scalar word and compensation counts are not replaced by the smaller physical role stock.
+
+Our replay recomputes the full |O(66,2)| group factor, complete local literal scalar bound, logical bill, finite router guard, E, B=s+E and C0=32*66*B^2. Its local scalar bill matches the exact source-bound physical scalar_bound field. The full group factor cancels only in normalized moments, never in scalar work, routing or row stock.
+
+The row validator retains both conservative external reserves9909 and252, the full complex wire-bit count2159, row coefficient12320, polynomial degree70000 and positive degree gap224336/5. Internally borrowed/restored q-adic rows are not reinterpreted as free external stock. The odd divisor remains3, scalar denominators divide6, children retain incoming odd exponents and no child boundary rounding is introduced.
+
+Replacing PR163's bit frame refinement by our already checked bit supplier does not change the complex finite bridge, complex scalar bill or physical word. It changes only the validated bit_uniform coarse/atom/ordinary fields. The strengthened bit construction and compatible weighted local-ring implementation are separate source-bound inputs; this arithmetic replay does not assert them from a numeric exponent.
+
+## Exact complex fused-broadcast phases
+
+The fused source combines signed channels feeding the SAME target before readout. For a singleton target T=(I,b0,b1,b2), the three last contributions are
+
+    P/2 + (2*b0-1)Q/2 + (2*b1-1)R/2.
+
+Repeated application of our independently verified TakeoverMerge16120261008.decoder_merge gives exactly
+
+    (P+(2*b0-1)Q+(2*b1-1)R)/2.
+
+This equality holds in every actual complex module. An actual common complex-linear frame transports it exactly by decoder_merge_common_frame; no scalar phase is discarded or inferred merely from a binary symplectic label. Signed additions are undone in literal reverse order. Conservative original-source supports are unioned and never shrunk by cancellation. The face2, edge02 and edge12 source-cube intersection classes are disjoint, so iterative disjoint_merge_support proves exact support preservation. The actual fused verifier checks every signed source/target coefficient, value span, common cap and chronological chain.
+
+The original22 complex/P18 certificates and11 merge certificates remain unchanged and reusable. P18 fourth-root Fourier/Gauss identities certify phase energy under their displayed hypotheses; they do not supply the fused histogram, paid positional router or new exponent by themselves.
+
+## Explicit conditional boundary
+
+The exact finite arithmetic point is established. General Clifford lifts and actual phase reconciliation, common generic bases, uniform weighted local-ring compilation, arbitrary-width routing, complete row/spectator restoration, positional layout applicability, prime supply, precision/recovery, uniform setup and analytic fixed-tape transfer remain inherited all-size proof contracts. No unconditional multiplication theorem, fully closed Lean multiplier root, or general optimality result is claimed here. The next-grid failure concerns only this fixed supported margin.
diff --git a/research/balanced-nullity-banks-163/README.md b/research/balanced-nullity-banks-163/README.md
new file mode 100644
index 0000000000..ee29fa8a62
--- /dev/null
+++ b/research/balanced-nullity-banks-163/README.md
@@ -0,0 +1,78 @@
+# Conditional paid-balanced transfer with nullity banks
+
+Under the explicitly retained all-size contracts, this composition supports
+`T(n) = O(n (log n)^(1-kappa))` with
+
+    kappa = 118877394946471 / 200000000000000000
+          = 0.000594386974732355.
+
+This is about 0.0702% above pinned PR163's 0.000593970203079492.
+It combines PR163's fused complex supplier and paid balanced positional transfer
+with the PR161 completed-core nullity-bank bit refinement. Gains are not added.
+The resulting supported saving is complex-limited.
+
+## What changes
+
+For PR161's bit word, eighteen explicit four-coordinate blocks and three explicit
+24-coordinate blocks each partition Fin72. Applying completed-core bank sharing
+removes exactly 5720 rank60 exterior children while retaining all other children.
+The normalized volume is 66364/3, rank mass 1590800, deficit 1936, maximum child22.
+The paid bit moment and positive atom toll yield ordinary saving
+151223451172976757106049 / 250000000000000000000000000,
+which clears the PR163 complex supplier. The full scalar, router, group and row
+bills are retained; all47 strict constraints and seven margins pass.
+
+## Reproduce exact arithmetic (Python3.11+, standard library; assertions enabled)
+
+Obtain two separate source snapshots at these immutable commits:
+
+- PR161: https://github.com/eumemic/integer-mult-bounds , commit
+  d14e29157bc905be1ced0776dd893d0714013f3a.
+- PR163: https://github.com/chafreaky/integer-mult-bounds , commit
+  e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997.
+
+The verifier requires Git checkouts pinned to those exact commits, with unchanged
+tracked sources. A reviewer can acquire them using Git; the submitter can upload
+this folder using GitHub's website without using Git locally.
+
+    python3 -B research/balanced-nullity-banks-163/verify.py --pr161 /path/to/pr161 --pr163 /path/to/pr163
+
+The verifier does not download or modify sources. It reconstructs the packed
+ledger, proves the finite coordinate partitions by enumeration, recomputes the
+bit moment using two interval implementations, checks its adjacent-grid control,
+recomputes the PR163 complex moment, and replays the unchanged paid-balanced
+assembly and full finite bill against expected.json. It rejects the next final
+10^-18 grid point at these fixed certificates and parameters. This is not a claim
+of global optimality. The verifier checks arithmetic; it does not substitute for
+upstream physical-word verification.
+
+## Lean scope
+
+BankAndBalanced.lean is a single-file Mathlib-importing snapshot of existing
+proof bodies: actual split-projector address interchanges, explicit partitions,
+weighted endpoints, arbitrary dirty-content semantics, PR161 packed inventory,
+and the47 finite arithmetic guards. See PROOF-ORIGINS.json for provenance.
+It does not assert an all-size multiplication theorem. The bit inventory theorem
+is proved alongside actual bank endpoints; connecting them to the full physical
+multiplier still uses the contracts below, not merely the histogram arithmetic.
+The Lean version is4.33.0. The compatible Mathlib commit is
+`db584cd6d46c92f209a44c0f1c829460d327499d`. VERIFICATION.json records the
+independent source replay and axiom audit. In that Mathlib environment, run
+`lake env lean research/balanced-nullity-banks-163/BankAndBalanced.lean`.
+The52 audited theorem names are listed in AUDIT-TARGETS.json.
+
+## Dependency and publication scope
+
+This is an additive research package. It depends on the unmerged PR161/PR163
+snapshots above and changes no existing source. A PR to CrocSwap/main should
+explicitly state those dependencies; it does not silently import their entire
+diffs or claim they are already merged.
+
+The completed all-input word, split-frame charts, invocation/stage cover,
+weighted local-ring implementation, arbitrary-width complete-spectator routing,
+borrowed-row restoration, paid layout applicability, precision/recovery,
+prime supply, uniform setup and sharp fixed-tape analytic transfer remain
+inherited all-size contracts. PROOF.md states these boundaries. This is a
+conditional finite witness, not a fully formalized unconditional multiplier.
+
+See NOTICE and LICENSE for attribution.
diff --git a/research/balanced-nullity-banks-163/VERIFICATION.json b/research/balanced-nullity-banks-163/VERIFICATION.json
new file mode 100644
index 0000000000..26b13611da
--- /dev/null
+++ b/research/balanced-nullity-banks-163/VERIFICATION.json
@@ -0,0 +1,400 @@
+{
+  "status": "PASS",
+  "source_sha256": "8f32bce76e47da3c6165ea94f179ddf351ed50271996ab179505f588333b5867",
+  "kernel_replay_exit_code": 0,
+  "theorem_count": 52,
+  "axiom_records": [
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_apply",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_zero",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_comp_disjoint",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_involutive",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_id",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.partialSwap_commute",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_eq_sum",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankSwap_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weighted_apply",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weighted_comp",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weighted_id",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_eq",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.weightedBank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_apply",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_idempotent",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_disjoint",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_sum",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.coordinateBank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_bijective",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.bankAddress_roles_injective",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_block_capacity",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_block_capacity",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.inherited_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.removed_exterior_count",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_max_rank",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.normalized_volume",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.packed_deficit",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.actual_role_counts",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.selected_weighted_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_weighted_bank_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_eq_address",
+      "axioms": [
+        "propext",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.run_full",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_selected_contents",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_undeferred_contents",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.inherited_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.removed_exterior_count",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_rank_mass",
+      "axioms": [
+        "propext"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_max_rank",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.normalized_volume",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_deficit",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_role_counts",
+      "axioms": []
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_bank_and_inventory_certificate",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.row_reserve_value",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.core_slacks_positive",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_slacks_positive",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.strengthened_public_margin",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    },
+    {
+      "theorem": "Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_concrete_guards",
+      "axioms": [
+        "propext",
+        "Classical.choice",
+        "Quot.sound"
+      ]
+    }
+  ],
+  "allowed_axioms": [
+    "propext",
+    "Classical.choice",
+    "Quot.sound"
+  ],
+  "forbidden_axiom_count": 0,
+  "sorry_count": 0,
+  "imports": "Mathlib only",
+  "lean_version": "4.33.0",
+  "mathlib_commit": "db584cd6d46c92f209a44c0f1c829460d327499d",
+  "independent_replay_sha256": "4a843d015dd3fa5daa696fcea6beca654a52ef0773720e979928cdfaa605c512",
+  "log_sha256": "d4b24c5a28ebe4db51f41fba151500fa4c0ce4a739069e0cc1fc180b8d2117e4",
+  "portable_arithmetic_replay": {
+    "status": "PASS",
+    "kappa": "118877394946471/200000000000000000",
+    "decimal": "0.000594386974732355",
+    "strict_constraints": 47,
+    "margins": 7,
+    "bit_mass": 1590800,
+    "bit_volume": "66364/3",
+    "scope": "Exact conditional finite arithmetic plus ledger and coordinate partitions; full physical realization and all-size inherited contracts are separate."
+  },
+  "arithmetic_log_sha256": "393570c538268eb8ef36dd89a007861a7c4e6a6c7782f5fdd0c060e99bc37df0",
+  "scope": "Actual universal bank/address/content endpoints, explicit inventory and finite balanced arithmetic. Does not certify all-size multiplication or discharge retained implementation/analytic contracts."
+}
diff --git a/research/balanced-nullity-banks-163/arithmetic-replay.log b/research/balanced-nullity-banks-163/arithmetic-replay.log
new file mode 100644
index 0000000000..f543fe4f7a
--- /dev/null
+++ b/research/balanced-nullity-banks-163/arithmetic-replay.log
@@ -0,0 +1,10 @@
+{
+  "status": "PASS",
+  "kappa": "118877394946471/200000000000000000",
+  "decimal": "0.000594386974732355",
+  "strict_constraints": 47,
+  "margins": 7,
+  "bit_mass": 1590800,
+  "bit_volume": "66364/3",
+  "scope": "Exact conditional finite arithmetic plus ledger and coordinate partitions; full physical realization and all-size inherited contracts are separate."
+}
diff --git a/research/balanced-nullity-banks-163/expected.json b/research/balanced-nullity-banks-163/expected.json
new file mode 100644
index 0000000000..ac35a985d5
--- /dev/null
+++ b/research/balanced-nullity-banks-163/expected.json
@@ -0,0 +1,203 @@
+{
+  "public163_pin": "e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997",
+  "public161_pin": "d14e29157bc905be1ced0776dd893d0714013f3a",
+  "public163_kappa": "148492550769873/250000000000000000",
+  "combo": {
+    "kappa": "118877394946471/200000000000000000",
+    "a": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000",
+    "actual_bit_saving": "151223451172976757106049/250000000000000000000000000",
+    "complex_saving": "297370240363723/500000000000000000",
+    "beta": "1/1000000000000000000000000",
+    "eta": "1/1000000000000000000000000",
+    "weakening": "1/1000000000000000000000000000000",
+    "layout": "balanced",
+    "minimum_margin": "148684971496890003109996295013224259329740670260073915526993780506219493482636277/250148434971746889753110246443448196006219493780506517363723000000000000000000000000",
+    "assembly": {
+      "parameters": {
+        "a_bit": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000",
+        "a_complex": "297370240363723/500000000000000000",
+        "tau": "499702629759636277000000000297870240363723/500000000000000000000000000000000000000000",
+        "sigma": "499702629759636277/500000000000000000",
+        "beta": "1/1000000000000000000000000",
+        "h": "1/1000000000000000000000000",
+        "q": "148685120181861499999999553694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "c": "148685120181861500000062053694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "epsilon": "249999750000249999750000000000000000000000000000000000000000/250148434971746889753110246443448196006219493780506517363723",
+        "lambda_": "499702629759636277000000000595240480727445999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_prime": "249851314879818138500000000446305360545584499999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "alpha_squared_power": "297369942993780006220242738461420265549234450766591279249993780506219493482636277/500296869943493779506220492886896392012438987561013034727446000000000000000000000000",
+        "delta": "1/8000000000000000000000000",
+        "C0": 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+        "C1": "1",
+        "kappa": "118877394946471/200000000000000000"
+      },
+      "constraints": {
+        "a_positive": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000",
+        "a_below_b": "297870240363723/500000000000000000000000000000000000000000",
+        "b_below_one_over32": "15327629759636277/500000000000000000",
+        "beta_positive": "1/1000000000000000000000000",
+        "beta_below_one": "999999999999999999999999/1000000000000000000000000",
+        "phase_leaf_above_bit": "1/1000000000000000000000000000000",
+        "q_positive": "148685120181861499999999553694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "q_below_internal": "297370240363722999999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "q_below_leaf": "297620240363722999999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "c_positive": "148685120181861500000062053694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "c_below_one": "249851314879818138499999937946305360545584499999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "q_below_reservations": "1/4000000000000000000000000",
+        "lambda_above_tau": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_above_sigma": "595240480727445999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_above_internal": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_prime_above_lambda": "297370240363722999999999702129759636277/500000000000000000000000000000000000000000000000000000000000000000",
+        "compact_leaf": "297620240363722999999999702129759636277/250000000000000000000000000000000000000000000000000000000000000000",
+        "compact_reservations": "1/4000000000000000000000000",
+        "lambda_prime_below_one": "148685120181861499999999553694639454415500000000297870240363723/250000000000000000000000000000000000000000000000000000000000000000",
+        "epsilon_positive": "249999750000249999750000000000000000000000000000000000000000/250148434971746889753110246443448196006219493780506517363723",
+        "epsilon_below_one": "148684971496890003110246443448196006219493780506517363723/250148434971746889753110246443448196006219493780506517363723",
+        "guard_width": "148684971496890003110246443448196006219493780506517363723/250148434971746889753110246443448196006219493780506517363723",
+        "K_geometry": "187648497471684389815610246443448196006219493780506517363723/250148434971746889753110246443448196006219493780506517363723000000000000000000000000",
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+        "C1": 1,
+        "induction_gap": 245388340315854495121991495023661354133772589464405098560835281231210593354019555143204809575641856722291194489372719301193616799647826159579757887820071223106654636111050438191495751716485818695124159328484048794755630779268936969158749727291785790306894906681006422231592986767946388346377694932706235427152142784836989450028374317183063203380895257549480826760271435164826739807632685330155718123900280161432695419226532158212485722545375693195990796362804742759742725917432842044603389791467383156312396163024378938394840961078454714202388522126089198783312815501652525725420856212824101710091243502711285243089767465596146429310562531820838246195141964036386977667992333784800904065593805038690857840165112369937462991853946434986034550591529039889942641409960245899894496155913835925603690714427978783216632282652185350804110391405344267420394508689231484590201167625358071977829989612598492896437837504075800683681711004705602678862156000874995105476159542852930583132791352717479396514543009746560892105943675938040697032178998767177924243015937188965591032407979029824417117095144958500076152110001069748144440755055131143299379561950147695675291209196170790861794308920116977841826009571049029564192723661000347022780697656581844126333817888558148438858693544050730279006391032579314348383417483341726167308484102464011875039770669245289569053370315400675272769228984745966480541333705931741952201361584450779326850521243305632190114357848274877760293759481552810837844820287785429409739193492470687215042264321232635941437404752907555527467413383180997744420400148964258443628979070098673792084717606302084025121666947854676905542764446282440636655033920168714101675111829088679088416088899017520789101335421071428013989215397416644857473658623389821671427674995252610577512050322911253131859211944038596666016247376225153558207550373063492998554888580920952391708138131289219486525718977892325465230055854953215786176257899380807103644121328698903259664929947496469383326286563627861859954182187535400895764477928092562530607757748705780763587088674121305145938437603223108021159308048538225758101153578764087878011594956574876552904477637107998204434002769826088279804383770536124007804304192063468870467893053981298039318821032128707649760060073749229537610127792812276679987477985700133781138307150786679926037397394611571161387429268032865427283431018256364862938990177608641555871765353175304933281067617839664481651571704230273983452875393518335293014396311381254899480921988612253618879517438535306676833380425595251991660952995091525032647927535458234697488586825233573202703716781037985646889946321208777879673711271946767903809948726047445877686120392643104955379850171778688163274102126680878508444266761189704537482574079118130447309771519192797202081634827220328648531606261068653634762233494516255628307404776706858390508708834601264897008646546562983946360187373787346425825026502825166589159365404652363307340702487787070996971736376432277781406562795750954160208053781639250158662196318892749978189443599641836214518616936539387692422155247530353148031561963426984668637356710756857535578502681262021631597998560237428258197520890833021959495277829596803778310503074257439014215218565832359454110570594125081714805152020433763053050634665307117361238244863227269421732556357631665326777103377909512101347728315848156030582196328671462365501813019054259602691910273427111817453707511754008411283207639828440879115320662481009495070733076384053735716696980585867475945451451632411845785680475987953695363293698311225826327156039939435741826425950384128738263556609420740617600738683900894516668818578184982813685973717501435264569198072863889402431256917351853788278324748638370087025599361388713671432149463428590762196328434465289201875753262745420886261254218348598928424459889789228762368105383041556546813071637279440584703191488864477114637272334834163218052935897385189713813648366910256840069690327797675638041266996826858514819047856599720555974644532336376341915423147840463479113504223750866935902556906829947300394079605318925784823743990331563491028308230395256964522179938306014663290083982727831715151268072100089494646507958010913026276692220056723587598417289180621183577079586468288019596167019726559547363309560062771643065546613302515146937109315291412368623507609804469300980952281034100560672316181292568391370829081955158464900787242246061359338501835361109152620295946787204589278663415921390862293932802855596468826293101157606576254299895075065183725848640195038168140115238141875608609663281932121508309849213169636906548962762790229247985424467798338812598996036676288818676668020039053269385224237236006604554516626741039697049248452711713870708320270355284148316234748223145351134971979820642035510926640676212230251793685052973119933811537507199542921985886099868559230539499820772565054396424423335994260188311756682328300590138836072346894700375172449221942310825056561920528548189480019682715977200510265557444727939596334566578509841650642419995868546617797990536589793984959654750046004443272269498433270536404174860207593718627146563818414240126343169066224110289896039479249190155643300061135375135749044698752258870903455753119806914130887885547024377713477997887249790908543042453431913066622227351526052356973191952304344647002675594522000489033148689097463503118249558277146364423179459261437672925470825192600425893579520562041434304347237999198826154802593109923761892805497881437726822119855138188118058662783447363009685726520879605684932206112719489605720892676070178453033570794379935889511795906208022176982250800694161341196693332468658133233056859508089043451636005730974242647702874311303164831450289352805610181457063655032687294375885415896338259476839287814607339274526710653286349752736405778956851518821342810708582296938181310293525162934948124660222441967947885080711857144314720550984995695087455018431062631654976,
+        "fixed_odd_divisor": 3,
+        "exact_grid": "One dyadic grid times 3^(-K), K=G*(D_complex+1); completed children preserve the incoming odd exponent."
+      },
+      "rows": {
+        "coefficient": 12320,
+        "complex_coefficient": 2159,
+        "degree": 70000,
+        "suffix_slope": 280000,
+        "degree_gap": "224336/5"
+      }
+    }
+  }
+}
diff --git a/research/balanced-nullity-banks-163/lean-audit.log b/research/balanced-nullity-banks-163/lean-audit.log
new file mode 100644
index 0000000000..5943da67d0
--- /dev/null
+++ b/research/balanced-nullity-banks-163/lean-audit.log
@@ -0,0 +1,82 @@
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_apply' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_zero' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_comp_disjoint' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_involutive' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_id' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.partialSwap_commute' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.bankSwap_eq_sum' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.bankSwap_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weighted_apply' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weighted_comp' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weighted_id' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weightedBank_eq' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.weightedBank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_apply' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_idempotent' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_disjoint' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateProjector_sum' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.coordinateBank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.bankAddress_bijective' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.bankAddress_roles_injective' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.selected_block_capacity' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_block_capacity' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.selected_bank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_bank_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.inherited_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.removed_exterior_count' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.P12.packed_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.packed_max_rank' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.P12.normalized_volume' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.packed_deficit' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.actual_role_counts' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.P12.selected_weighted_bank_full' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.P12.undeferred_weighted_bank_full' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.run_eq_address' depends on axioms: [propext, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.run_full' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_selected_contents' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.Contents.p12_undeferred_contents' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.inherited_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.removed_exterior_count' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_rank_mass' depends on axioms: [propext]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_max_rank' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.normalized_volume' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.packed_deficit' depends on axioms: [propext, Classical.choice, Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_role_counts' does not depend on any axioms
+'Mathproof.BeyondNLogN.TakeoverBit.PR161P12.actual_bank_and_inventory_certificate' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.literal_charge_lt' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.finite_row_reserve' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverLiteralGuard20261008.row_reserve_value' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.core_slacks_positive' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_slacks_positive' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.strengthened_public_margin' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
+'Mathproof.BeyondNLogN.TakeoverBalanced16320261009.all_47_concrete_guards' depends on axioms: [propext,
+ Classical.choice,
+ Quot.sound]
diff --git a/research/balanced-nullity-banks-163/verify.py b/research/balanced-nullity-banks-163/verify.py
new file mode 100644
index 0000000000..6a117777c9
--- /dev/null
+++ b/research/balanced-nullity-banks-163/verify.py
@@ -0,0 +1,96 @@
+#!/usr/bin/env python3
+"""Read-only exact arithmetic replay. Full physical supplier checks are separate.
+Requires two immutable upstream checkouts; no downloads or source edits.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from math import lcm
+import argparse,hashlib,json,subprocess,sys
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+HERE=Path(__file__).resolve().parent
+
+def need(ok,msg):
+    if not ok:raise ValueError(msg)
+
+def checkout(path,pin):
+    path=path.resolve()
+    def git(*args):
+        p=subprocess.run(['git','-C',str(path),*args],capture_output=True,text=True)
+        need(p.returncode==0,'Git failed: '+p.stderr);return p.stdout.strip()
+    need(git('rev-parse','HEAD')==pin,'Incorrect pinned checkout: '+str(path))
+    need(not git('status','--porcelain','--untracked-files=no'),'Tracked source changes in '+str(path))
+    return path
+
+def main():
+    need(not sys.flags.optimize,'Assertions must be enabled; do not use -O')
+    ap=argparse.ArgumentParser(description=__doc__)
+    ap.add_argument('--pr161',type=Path,required=True)
+    ap.add_argument('--pr163',type=Path,required=True)
+    args=ap.parse_args()
+    expected=json.loads((HERE/'expected.json').read_text())
+    r161=checkout(args.pr161,expected['public161_pin'])
+    r163=checkout(args.pr163,expected['public163_pin'])
+    sys.path.insert(0,str(r161/'scripts'))
+    import paired_cube_network as p
+    from audit_community_candidate import moment
+    ar=r163/'research/paired-cube-balanced-161/arithmetic'
+    sys.path.insert(0,str(ar))
+    import certificate as c
+    from interval_moment import saving_grid
+    n=json.loads((r161/'certificates/paired-cube-network.json').read_text())
+    raw=json.loads((r161/'research/paired-cube-bit/out/profile_p12.json').read_text())
+    inherited={int(k):int(v) for k,v in raw['child_histogram'].items()}
+    original=n['bit']['counts']
+    need(inherited=={int(k):int(v) for k,v in original['child_multiplicities'].items()},'Bit sources disagree')
+    need((raw['m'],raw['h'],raw['v'],raw['R'],raw['selected_roles'])==(72,24,1760,23368,5720),'Wrong immutable word')
+    need(raw['selected_rank_histogram']=={'20':5720},'Wrong selected ranks')
+    need(sum(k*v for k,v in inherited.items())==1934000,'Inherited rank mass')
+    packed=dict(inherited);need(packed.pop(60)==5720,'Exterior child inventory')
+    W=2*1760+3*(Q(5720,18)+Q(17648,3))
+    rank=sum(k*v for k,v in packed.items())
+    need((W,rank,72*W-rank,max(packed))==(Q(66364,3),1590800,1936,22),'Packed ledger')
+    # Actual disjoint coordinate blocks, not nullity divisibility alone.
+    need(all(sum(j//4==i for j in range(72))==4 for i in range(18)),'Selected blocks')
+    need(all(sum(j//24==i for j in range(72))==24 for i in range(3)),'Undeferred blocks')
+    profile=dict(original,W_per_vertex=W,rank_per_vertex=rank,deficit_per_vertex=1936,
+                 child_multiplicities={r:Q(v) for r,v in packed.items()},edge_count=sum(packed.values()),maxchild=22)
+    coarse=Q(302618322217,500000000000000)
+    atom=Q(302446903,500000000000)
+    old=Q(384599,10**10)
+    effective=(1-atom)*coarse+atom*old
+    need(0<effective<atom<1-effective,'Paid atom toll')
+    need(p.BAD==Q(1,10**16) and p.OLD==old,'Retained constants changed')
+    def bit_ok(a):
+        try:
+            ideal=p.exact_moment(profile,a)
+            fallback=p.BAD*Q(32*72**2*profile['edge_count'],W*72)*p.exp_upper(a*p.log_upper(Q(72)))
+            return ideal['moment_upper']+fallback<1 and rank+p.BAD*32*72**2*profile['edge_count']<72*W
+        except ValueError:return False
+    need(bit_ok(coarse),'Paid bit moment does not contract')
+    need(not bit_ok(coarse+Q(1,10**15)),'Adjacent coarse grid control accepted')
+    # Independent 80-term log / 12-term exponential enclosure implementation.
+    scale=lcm(W.denominator,*(v.denominator for v in profile['child_multiplicities'].values()))
+    _,bound=moment(72,int(scale*W),[(r,int(scale*v)) for r,v in profile['child_multiplicities'].items()],coarse)
+    _,fallback=moment(72,int(scale*W),[(1,int(scale*32*72**2*profile['edge_count']))],coarse)
+    need(bound+p.BAD*fallback<1,'Independent bit moment failed')
+    upstream=json.loads((r163/'research/paired-cube-balanced-161/certificate.json').read_text())
+    row=upstream['complex']['profile']
+    complexprofile=dict(m=row['m'],W=row['W_per_vertex'],N=row['deficit_per_vertex'],L=0,
+        total_rank=row['rank_per_vertex'],maxchild=row['maxchild'],child_multiplicities=row['child_histogram'])
+    complexmoment=saving_grid(complexprofile,10**18)
+    need(complexmoment['accepted']['saving']==c.AC,'Complex moment changed')
+    c.normalized_body_check()
+    result=c.price(row,c.AC,coarse,c.BETA,c.ETA,c.WEAKENING,'balanced',atom)
+    need(c.js(result)==expected['combo'],'Exact certificate differs')
+    need(result['kappa']==Q(118877394946471,200000000000000000),'Wrong final kappa')
+    need(result['a']<effective and len(result['assembly']['constraints'])==47 and len(result['assembly']['margins'])==7,'Supported assembly')
+    try:
+        c.assembly(result['finite_bridge'],row,result['a'],result['kappa']+Q(1,10**18),beta=c.BETA,h=c.ETA,a_complex=c.AC)
+    except (ValueError,AssertionError):pass
+    else:raise ValueError('Adjacent final grid control accepted')
+    checkout(r161,expected['public161_pin']);checkout(r163,expected['public163_pin'])
+    print(json.dumps({'status':'PASS','kappa':str(result['kappa']),
+        'decimal':'0.000594386974732355','strict_constraints':47,'margins':7,
+        'bit_mass':rank,'bit_volume':str(W),'scope':'Exact conditional finite arithmetic plus ledger and coordinate partitions; full physical realization and all-size inherited contracts are separate.'},indent=2))
+if __name__=='__main__':main()
diff --git a/research/continued-frame-cuts-186/NOTICE b/research/continued-frame-cuts-186/NOTICE
new file mode 100644
index 0000000000..973612d9a2
--- /dev/null
+++ b/research/continued-frame-cuts-186/NOTICE
@@ -0,0 +1,26 @@
+Further coordinated frame cuts on PR186
+Prepared by huxint with substantial OpenAI Codex assistance. Apache-2.0.
+
+PR186 / Dugongue supplies the prerequisite combined witness: compensated
+rematching, coordinated subspace cuts, entrance banks, exact admission and
+the immutable source package. Its original NOTICE, UPSTREAM-NOTICE.md and
+proof/source records are retained unchanged and checked in full.
+https://github.com/CrocSwap/integer-mult-bounds/pull/186
+
+Chafik Boukhalfa / chafreaky: shared-edge complex word and exact signed
+physical/terminal/formal checkers, PR181/182 and earlier PR163-165 lineage.
+eumemic: PR168 v4 modules and words, physical reuse and bit compiler, with
+the disclosed Claude and OpenAI assistance retained in its sources.
+James Chang / jamesyc: PR166 target-accumulating compiler and PR124 birth
+reuse. Dumas--Grenet's accumulation literature retains its original credit.
+icekylinx and contributors: paired-cube decoder, exact frame and shared-core
+interfaces, local compiler, finite bridge and analytic conditional framework.
+DaysSky: closure compilation and extended matching. an664 / Andrey Mas:
+completed-core sharing. Zhihao Chen, Swapnil Jain, RaD, paid-atom and phase-stop
+contributors retain every upstream notice and proof attribution.
+
+The additional frame plan and its aggregate verifier are by huxint, with
+substantial OpenAI Codex assistance. Prior huxint frame/fusion/reuse work is
+published in PR170 and PR188. Directed cut, existing geometric substitutions,
+the inherited source words and the general mathematical interfaces are not
+claimed as new. No exclusive priority over parallel contributions is claimed.
diff --git a/research/continued-frame-cuts-186/PROOF.md b/research/continued-frame-cuts-186/PROOF.md
new file mode 100644
index 0000000000..5384beca56
--- /dev/null
+++ b/research/continued-frame-cuts-186/PROOF.md
@@ -0,0 +1,77 @@
+# Further coordinated frame cuts with a complete inherited supplier
+
+The prerequisite is PR186 at 166a34d75e853e933855f0f7e940fe77c4bb6b8d.
+Its immutable parts contain the pinned PR181 source closure. Its outer
+upload manifest, every archive part and the complete inner certificate are
+checked before the new candidate is constructed. The resulting bit bank
+supplier and its exact prime/word/bank/inverse checks are retained unchanged.
+
+## Geometric substitution
+
+An operation is performed at one exact binary frame F_i shared by both
+participating physical roles. Its conservative scalar value span S_i must
+lie in F_i. Every source, gauge, root and donor-to-recipient boundary is fixed.
+The positive children are dimensions of all actual ascending transitions.
+
+For one binary subspace L, the search proposes either
+
+    F_i -> F_i intersect L, or F_i -> span(F_i, L).
+
+Intersections are applied to a suitable predecessor-closed set, with
+S_i contained in L. Joins are applied to a suitable successor-closed set.
+Each unchanged boundary is checked too. Meet and join preserve inclusion,
+so the all-changed state of a role edge remains nested. The mixed state
+that would violate closure is prohibited.
+
+The binary edge energy uses the complete charged transition cost, including
+both participating role paths. Its prohibited mixed state makes the energy
+submodular. An integer-capacity minimum cut proposes a simultaneous set.
+The final admission separately verifies every value-span inclusion and every
+physical chain; it does not assume the floating objective is a proof.
+Line and hyperplane proposals permit changes not present in a small sample
+of the original operation-frame set. Directed cut is classical; neither it
+nor PR186's global-subspace idea is claimed as a new general algorithm.
+
+The frozen witness is checked from the actual scalar graph and operation
+indices. No historical operation IDs are transferred. No operation, source
+injection, dirty correction, alias, deadline or root is removed by a frame
+cut. Rank-zero movements retain their inherited scalar/endpoint obligations.
+
+## Scalar and terminal verification
+
+The signed pre-reuse decoder and backward-intersection word are checked by
+the retained independent verifier. The original physical checker verifies
+the new common frames, every spliced alias, chronological target reads,
+telescoping rank count and its corruption controls. The retained packed
+rational verifier checks all source, target and dirty columns, both shear
+signs and inverse cleanup, using its recomputed coefficient/residual bound.
+
+The signed-compatible PR166 terminal checker independently reconstructs all
+46 disjoint sink actions and their actual target events. Their role, pivot,
+root, target set and complete write sequence must equal PR186's actions.
+The terminal-modified word is then replayed in both directions on all 13,326
+formal source/target/dirty columns, restoring all 10,686 dirty coordinates.
+This is a new replay on the changed frame data, not an addition of separately
+reported supplier gains. The complete post-terminal histogram is recounted.
+
+## Paid arithmetic
+
+The exact interval engine certifies the complex characteristic on its actual
+positive children and rejects the next 10^-18 saving point. The full parent
+bank-bit proof is rerun, including its contaminated moment, paid atom/row
+tolls, prime witnesses and integral multi-copy bank allocation. No bit saving
+or physical stock is substituted from an unverified description.
+
+The original balanced-assembly body is checked against its retained source.
+Its bridge is reconstructed from the actual new complex inventory while
+retaining the complete original logical scalar reserve. The full orthogonal
+group, router, scalar guard, common-grid precision, fixed odd divisor, row
+coefficient and degree-70,000 reserve remain present. All 47 strict constraints
+and seven margins hold, and the next final 10^-18 grid point is rejected.
+Additional controls reject an omitted full group, an unrecounted child and
+a zero operation frame. All inherited assembly controls also run.
+
+The proof is conditional on the same uniform Clifford/tensor, completed-core
+and bank endpoint, weighted local-ring, restored-row, prime-supply, precision,
+analytic recovery, balanced-layout and fixed-tape interfaces as PR186.
+It is not a proof of those general interfaces or a claim of global optimality.
diff --git a/research/continued-frame-cuts-186/README.md b/research/continued-frame-cuts-186/README.md
new file mode 100644
index 0000000000..29cc7de60a
--- /dev/null
+++ b/research/continued-frame-cuts-186/README.md
@@ -0,0 +1,42 @@
+# Continued physical-frame cuts on PR186
+
+This package refines PR186's actual complex operation frames and replays its
+complete supplier/assembly with the new word. The scalar graph, aliases,
+read deadlines, 46 terminal sinks and bank-bit construction are inherited.
+The exact conditional exponent and counts are generated in [RESULTS.md](RESULTS.md).
+
+```sh
+python3 -B research/continued-frame-cuts-186/verify.py
+```
+
+Python 3.11+ and its standard library suffice. The command verifies the
+immutable PR186 upload and source archive, replays its entire complex/bit
+certificate, checks the changed complex plan on every rational column in
+both shear directions, reconstructs every terminal action and positive
+child, and recomputes the exact paid moment and full balanced assembly.
+No network, solver, unpinned producer or development cache is used.
+
+Only the final explicit frames enter the proof. Discovery uses coordinated
+intersections and joins, including one-dimensional lines and hyperplanes,
+selected by integer-capacity binary cuts. A move may change many different
+current frames together. The exact checker, not the numerical objective or
+min-cut optimality, determines admission. See [PROOF.md](PROOF.md).
+
+The original scalar/group/router/precision/row reserves remain charged.
+The inherited all-size compiler and analytic interfaces remain conditional;
+this is neither an unconditional multiplication theorem nor a runtime claim.
+The selected main certificate is unchanged. Normal verification is read-only;
+`--write` is authoring mode and still runs every check.
+
+The complete PR186 construction and its lineage remain credited in [NOTICE](NOTICE).
+This contribution supplies a further checked frame plan, exact aggregate
+reproduction and a certificate-generated report. No global optimum is claimed.
+
+Optional further discovery, in a virtual environment with numpy installed:
+
+```sh
+python3 -B research/continued-frame-cuts-186/search.py --output build/further-frame-cuts
+```
+
+This starts from the published plan and writes a separate candidate. Its
+scores require the full terminal and assembly checks before becoming a claim.
diff --git a/research/continued-frame-cuts-186/RESULTS.md b/research/continued-frame-cuts-186/RESULTS.md
new file mode 100644
index 0000000000..81d1779fad
--- /dev/null
+++ b/research/continued-frame-cuts-186/RESULTS.md
@@ -0,0 +1,20 @@
+# Checked result
+
+Conditional kappa = **331325992674373/500000000000000000 = 0.000662651985348746**.
+
+This is 0.115796% above the pinned PR186 value 0.000661885549259598.
+
+The complex supplier binds, with saving 132618276834187/200000000000000000 = 0.000663091384170935.
+
+| Checked complex quantity | Value |
+|---|---:|
+| Changed operation frames relative to PR186 | 1361 |
+| Physical auxiliary roles | 10686 |
+| Persistent roles W | 13326 |
+| Rank mass | 878196 |
+| Deficit | 1320 |
+| Terminal sinks | 46 |
+
+The pinned complete parent, its bank-bit supplier, both new complex shear directions, every terminal, all 47 strict assembly constraints and adjacent-grid exclusions are replayed.
+
+These are conditional finite certificates. The inherited all-size compiler, shared-core/bank, weighted-bit, row-restoration, prime, precision/recovery, balanced-layout and fixed-tape contracts remain assumptions.
diff --git a/research/continued-frame-cuts-186/SOURCE.json b/research/continued-frame-cuts-186/SOURCE.json
new file mode 100644
index 0000000000..d51d4c4d09
--- /dev/null
+++ b/research/continued-frame-cuts-186/SOURCE.json
@@ -0,0 +1,18 @@
+{
+  "assistance": "huxint with substantial OpenAI Codex assistance",
+  "base_archive_manifest_sha256": "0412767b472720bae2902fcd24de5d18f3039afec3dd592a3c0e2d68fbabff63",
+  "base_commit": "166a34d75e853e933855f0f7e940fe77c4bb6b8d",
+  "base_files_sha256": "99709b85d987556593f237b0aacf314b2d7982fb4f494378d6dbf03391228ede",
+  "base_pr": 186,
+  "files": {
+    "NOTICE": "13280350b844fd4ccdcbab92f2f4c76d7fc09a34665183a8c1fe8008eebd09a7",
+    "PROOF.md": "8a22bb75cc2b55691dad5d22c1ebb4b84254203cae6ae74a2465c1f340bfa447",
+    "README.md": "84b73b11f7d46534bf5d00686dccd269f835834c76a71658ef38072bf2a2ae55",
+    "discovery/complex_search.py": "d7781e96ac4f1e22990e7df7cde833f2d86799ea1c620999456cc909b0c021a5",
+    "discovery/cut_search.py": "dc1f4dca2ec958a22658b3197e0c9c797723ce901d9f93602b53349c8c7a06e0",
+    "discovery/lattice_cuts.py": "e1cf8b69a644814cda579a75b4788fa4a0f4805b3681b111c7f68043cd52e921",
+    "search.py": "80e3eff610e0a58b1cfabb64e1bb1ba5a6a2a93c1c2130e80d724f495052b899",
+    "selected/frames.json": "541e6d898c324230ea188f99f5c7de476911b24dc28abfc4f20ded9649424f1c",
+    "verify.py": "8440431200c1382aff36d2d6ea023d6298823f934d764532517c07ccf685b977"
+  }
+}
diff --git a/research/continued-frame-cuts-186/certificate.json b/research/continued-frame-cuts-186/certificate.json
new file mode 100644
index 0000000000..8190fde5c0
--- /dev/null
+++ b/research/continued-frame-cuts-186/certificate.json
@@ -0,0 +1,2587 @@
+{
+  "arithmetic": {
+    "a": "132618276834186999999999867181723165813/200000000000000000000000000000000000000000",
+    "actual_bit_saving": "665064157585290854736467094714610539581/1000000000000000000000000000000000000000000",
+    "assembly": {
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+      "constraints": {
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+        "K_geometry": "75066234072183021316978707822264762483957016042984089834187/100066209072208021291978707822264762483957016042984089834187000000000000000000000000",
+        "a_below_b": "132818276834187/200000000000000000000000000000000000000000",
+        "a_positive": "132618276834186999999999867181723165813/200000000000000000000000000000000000000000",
+        "alpha_below_one": "199999800000199999800000100066209072208021291978707822264762483957016042984089834187/200132418144416042583957415644529524967914032085968179668374000000000000000000000000",
+        "alpha_below_one_fourth": "49900486391887967862032038332811928482085767914231687513481983957016042984089834187/200132418144416042583957415644529524967914032085968179668374000000000000000000000000",
+        "alpha_positive": "132618144216042783957315578320452759892740107260357403611516042983957015910165813/200132418144416042583957415644529524967914032085968179668374000000000000000000000000",
+        "artificial_boundary": "5604238180620513363486637600823572015597312902687105216182255451871048128952269502561/800529672577664170335829662578118099871656128343872718673496000000000000000000000000",
+        "b_below_one_over32": "6117381723165813/200000000000000000",
+        "beta_below_one": "999999999999999999999999/1000000000000000000000000",
+        "beta_positive": "1/1000000000000000000000000",
+        "c_below_one": "99933690861582906499999975199027415251280499999999867181723165813/100000000000000000000000000000000000000000000000000000000000000000",
+        "c_positive": "66309138417093500000024800972584748719500000000132818276834187/100000000000000000000000000000000000000000000000000000000000000000",
+        "cell_above_band": "199999800000199999799999899933790927791978708021292177735237516042983957015910165813/400264836288832085167914831289059049935828064171936359336748000000000000000000000000",
+        "compact_leaf": "132718276834186999999999867181723165813/100000000000000000000000000000000000000000000000000000000000000000",
+        "compact_reservations": "1/4000000000000000000000000",
+        "delta_below_one_eighth": "999999999999999999999999/8000000000000000000000000",
+        "delta_positive": "1/8000000000000000000000000",
+        "epsilon_below_one": "66309072108021391978707822264762483957016042984089834187/100066209072208021291978707822264762483957016042984089834187",
+        "epsilon_positive": "99999900000099999900000000000000000000000000000000000000000/100066209072208021291978707822264762483957016042984089834187",
+        "g1_above_kappa": "24093779804177555652731642424395649754109015567097745810249/50033104536104010645989353911132381241978508021492044917093500000000000000000",
+        "g2_above_kappa": "87879764437799999999867181723165813/200000000000000000000000000000000000000000",
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+        "g4_above_kappa": "87879764437799999999867181723165813/200000000000000000000000000000000000000000",
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+        "gamma_sublinear": "1/2000000000000000000000000",
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+        "lambda_above_internal": "132618276834186999999999867181723165813/200000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_above_sigma": "265436553668373999999999867181723165813/200000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_above_tau": "132618276834186999999999867181723165813/200000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_prime_above_lambda": "132618276834186999999999867181723165813/200000000000000000000000000000000000000000000000000000000000000000",
+        "lambda_prime_below_one": "66309138417093499999999800972584748719500000000132818276834187/100000000000000000000000000000000000000000000000000000000000000000",
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+        "q_below_reservations": "1/4000000000000000000000000",
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+        "record_suffix": "66309072108021391978707822264762483957016042984089834187/100066209072208021291978707822264762483957016042984089834187",
+        "row_product_gap": "224336/5",
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+        "small_field_exposure": "1/1000000000000000000000000"
+      },
+      "finite_bridge": {
+        "bit_uniform": {
+          "atom_beta": "665064157585290854737/1000000000000000000000000",
+          "coarse_saving": "665481166955987/1000000000000000000",
+          "old_atom_saving": "384599/10000000000",
+          "ordinary_saving": "665064157585290854736467094714610539581/1000000000000000000000000000000000000000000"
+        },
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+        "E": 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+        "exact_grid": "One dyadic grid times 3^(-K), K=G*(D_complex+1); completed children preserve the incoming odd exponent.",
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+      {
+        "first_any_old_read": 25559,
+        "first_pivot_old_read": 27716,
+        "last_write_chronology": 24892,
+        "pivot": 1016,
+        "post_after_op": 24716,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          16,
+          2
+        ],
+        "role": 10638,
+        "root_index": 1397,
+        "targets": [
+          1016,
+          1017,
+          1018,
+          1019,
+          1020,
+          1021,
+          1022,
+          1023
+        ],
+        "write_ranks": [
+          16,
+          18
+        ],
+        "writes": [
+          21994,
+          24716
+        ]
+      },
+      {
+        "first_any_old_read": 25770,
+        "first_pivot_old_read": 27752,
+        "last_write_chronology": 24804,
+        "pivot": 1024,
+        "post_after_op": 24628,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          16,
+          2
+        ],
+        "role": 10621,
+        "root_index": 1408,
+        "targets": [
+          1024,
+          1025,
+          1026,
+          1027,
+          1028,
+          1029,
+          1030,
+          1031
+        ],
+        "write_ranks": [
+          16,
+          18
+        ],
+        "writes": [
+          20861,
+          24628
+        ]
+      },
+      {
+        "first_any_old_read": 27108,
+        "first_pivot_old_read": 27502,
+        "last_write_chronology": 24734,
+        "pivot": 1088,
+        "post_after_op": 24558,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          14,
+          4
+        ],
+        "role": 10623,
+        "root_index": 1496,
+        "targets": [
+          1088,
+          1089,
+          1090,
+          1091,
+          1092,
+          1093,
+          1094,
+          1095
+        ],
+        "write_ranks": [
+          14,
+          18
+        ],
+        "writes": [
+          20870,
+          24558
+        ]
+      },
+      {
+        "first_any_old_read": 25431,
+        "first_pivot_old_read": 27682,
+        "last_write_chronology": 24746,
+        "pivot": 1128,
+        "post_after_op": 24570,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          15,
+          3
+        ],
+        "role": 10622,
+        "root_index": 1551,
+        "targets": [
+          1128,
+          1129,
+          1130,
+          1131,
+          1132,
+          1133,
+          1134,
+          1135
+        ],
+        "write_ranks": [
+          15,
+          18
+        ],
+        "writes": [
+          20867,
+          24570
+        ]
+      },
+      {
+        "first_any_old_read": 27182,
+        "first_pivot_old_read": 27718,
+        "last_write_chronology": 24533,
+        "pivot": 1136,
+        "post_after_op": 24357,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          12,
+          6
+        ],
+        "role": 10496,
+        "root_index": 1562,
+        "targets": [
+          1136,
+          1137,
+          1138,
+          1139,
+          1140,
+          1141,
+          1142,
+          1143
+        ],
+        "write_ranks": [
+          12,
+          18
+        ],
+        "writes": [
+          19241,
+          24357
+        ]
+      },
+      {
+        "first_any_old_read": 25737,
+        "first_pivot_old_read": 27564,
+        "last_write_chronology": 24687,
+        "pivot": 1184,
+        "post_after_op": 24511,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          4,
+          14
+        ],
+        "role": 9127,
+        "root_index": 1628,
+        "targets": [
+          1184,
+          1185,
+          1186,
+          1187,
+          1188,
+          1189,
+          1190,
+          1191
+        ],
+        "write_ranks": [
+          4,
+          18,
+          18
+        ],
+        "writes": [
+          12519,
+          24455,
+          24511
+        ]
+      },
+      {
+        "first_any_old_read": 25428,
+        "first_pivot_old_read": 27612,
+        "last_write_chronology": 24753,
+        "pivot": 1240,
+        "post_after_op": 24577,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          15,
+          3
+        ],
+        "role": 10516,
+        "root_index": 1705,
+        "targets": [
+          1240,
+          1241,
+          1242,
+          1243,
+          1244,
+          1245,
+          1246,
+          1247
+        ],
+        "write_ranks": [
+          15,
+          18
+        ],
+        "writes": [
+          20090,
+          24577
+        ]
+      },
+      {
+        "first_any_old_read": 27652,
+        "first_pivot_old_read": 27794,
+        "last_write_chronology": 24816,
+        "pivot": 1304,
+        "post_after_op": 24640,
+        "pre_at_phase_cut_after_center_before_old_reads": true,
+        "prefix_children": [
+          15,
+          3
+        ],
+        "role": 10628,
+        "root_index": 1793,
+        "targets": [
+          1304,
+          1305,
+          1306,
+          1307,
+          1308,
+          1309,
+          1310,
+          1311
+        ],
+        "write_ranks": [
+          15,
+          18
+        ],
+        "writes": [
+          20892,
+          24640
+        ]
+      }
+    ],
+    "status": "AUTHOR_EXACT_FINITE_COMPILER_CHECK_PASS",
+    "target_histogram": {
+      "0": 6270,
+      "1": 3960,
+      "10": 10,
+      "11": 2,
+      "12": 9,
+      "13": 1,
+      "14": 3,
+      "15": 11,
+      "16": 8,
+      "18": 1274,
+      "2": 22,
+      "3": 17,
+      "4": 10,
+      "5": 1,
+      "6": 8,
+      "7": 3,
+      "8": 4,
+      "9": 1
+    },
+    "targets_disjoint": true,
+    "v": 1320
+  },
+  "terminal_formal": {
+    "absolute_residual_bound": 416406,
+    "columns": [
+      {
+        "columns": 13326,
+        "direction": 1,
+        "dirty": 10686,
+        "outputs": true,
+        "restored": true
+      },
+      {
+        "columns": 13326,
+        "direction": -1,
+        "dirty": 10686,
+        "outputs": true,
+        "restored": true
+      }
+    ],
+    "controls": {
+      "omit-ancestor-response": "REJECTED",
+      "omit-first-write": "REJECTED",
+      "omit-pre-shear": "REJECTED"
+    },
+    "digit_bits": 24,
+    "input_pins": {
+      "baseline.json.gz": "f4ad8cd1635f279d201225c174548cb03db13b63156cdf647ca8e834d4746524",
+      "frames.json.gz": "b516b59e7bd7e68c56c02e0041e7455b2eff445e81ac5fb69570d424cf0178a1",
+      "graph.json.gz": "9f73a6a7f1a4410e954246c6ff95d9cad0582d0057f707a693cc2fd170995ead",
+      "new-sinks.json": "42e6bc9e9a187caa4f510c1c3afaa52374b44fa5d55ccfac196916fd5d767faa",
+      "physical-frames.json.gz": "223359229b2cafcd400a6784325132a2f97b48dc14b81ebe614423df505f7062",
+      "physical-pairs.json.gz": "869b852cbfd63ef691336ea398280ef57896dd876f6a6c78edac88a2a304fffb",
+      "physical.py": "496a5227d525e9291b05db7e6050ec866a7160e5bcd5da0a00bb7e5210efb996",
+      "profile-before.json.gz": "bd39675ce03fd1d4d456f736bcfc0b389532eb4f0a19c92877c2af5cb6f7310c",
+      "profile.json.gz": "36ec21c879a7fb926f1e8ea8eff5be1fbebd88466d535c6b5854c97e4105b6d0",
+      "word.json.gz": "0c4b720c57717c7e55e88f86a9e17ed75b3891f9d2e7ed29df0a188f8dfee67a"
+    },
+    "source_sha256": "d7fefe86e637584ad1bf53c8f7714b88d5abed92eaa29d110ef099536e51f516",
+    "status": "PASS full modified finite word"
+  },
+  "unchanged_checked_bit_saving": "665064157585290854736467094714610539581/1000000000000000000000000000000000000000000"
+}
diff --git a/research/continued-frame-cuts-186/discovery/complex_search.py b/research/continued-frame-cuts-186/discovery/complex_search.py
new file mode 100644
index 0000000000..9629f6fb3f
--- /dev/null
+++ b/research/continued-frame-cuts-186/discovery/complex_search.py
@@ -0,0 +1,396 @@
+#!/usr/bin/env python3
+"""Reproducible physical-frame experiments for the selected paired-cube word.
+
+The search changes only common operation frames.  Every emitted witness still
+has to pass the independent scalar and nested-frame replay supplied by PR157.
+Developed with OpenAI Codex assistance. Apache-2.0.
+"""
+from __future__ import annotations
+
+import argparse
+from collections import Counter, defaultdict, deque
+from functools import lru_cache
+import hashlib
+import json
+import math
+from pathlib import Path
+import random
+import sys
+import time
+
+ROOT = Path(__file__).resolve().parents[2]
+sys.path.insert(0, str(ROOT / 'scripts'))
+from paired_cube.frames import basis, perp, contained
+
+
+
+def numerical_root(children, W, m):
+    lo, hi = 0., .1
+    for _ in range(75):
+        mid = (lo + hi) / 2
+        if math.fsum(n * r * math.exp(mid * math.log(m / r))
+                     for r, n in children.items()) < W * m:
+            lo = mid
+        else:
+            hi = mid
+    return lo
+
+class Experiment:
+    def __init__(self, baseline, input_frames=None, alpha=0.0006, input_pairs=None):
+        self.g, self.w, self.word = [json.loads((baseline / name).read_text())
+                                  for name in ('graph.json', 'frames.json', 'selection.json')]
+        record_path = baseline/'record.json'
+        if not record_path.exists():record_path = ROOT/'certificates/paired-cube-complex-input.json'
+        self.record = json.loads(record_path.read_text())
+        self.h, self.R, self.v = self.g['h'], self.record['R'], self.g['v']
+        self.full = basis(1 << j for j in range(self.h))
+        self.ops = [tuple(o) for o in self.word['ops']]
+        self.N = len(self.ops)
+        self.topological_position = list(range(self.N))
+        self.base_frames = [perp(tuple(self.w['annihilators'][x]), self.h) for _, _, x in self.ops]
+        self.frames = list(self.base_frames)
+        if input_frames is None:
+            input_frames = ROOT / 'references/paired-cube/physical/frames.json'
+        for i, F in json.loads(input_frames.read_text())['frames']:
+            self.frames[i] = tuple(F)
+        if input_pairs is None: input_pairs = ROOT / 'references/paired-cube/physical/pairs.json'
+        self.pairs_input = json.loads(input_pairs.read_text())['pairs']
+        self.pairs = {a: b for a, b, _ in self.pairs_input}
+        self.merged = {b: a for a, b in self.pairs.items()}
+        self.sources = {int(x): s for x, s in self.word['sources'].items()}
+        self.source_roles = set(self.sources.values())
+        self.start = [()] * self.R
+        for x, s in self.sources.items():
+            self.start[s] = (self.g['inputs'][x-1],)
+        self.gauges = {}
+        for z in self.word['selected']:
+            self.gauges[z['role']] = perp(tuple(z['annihilator']), self.h)
+            self.start[z['role']] = self.gauges[z['role']]
+        self.spans = [(q,) for q in self.g['inputs']]
+        for aa in self.g['args'][self.v:]:
+            self.spans.append(basis(self.spans[aa[0]] + self.spans[aa[1]]))
+        self.node_spans = [self.spans[x-1] for _, _, x in self.ops]
+        self.rootframe, self.rootkind = {}, {}
+        self.roots = [dict(r, node=r['node']+1) for r in self.g['roots']]
+        for r, s in zip(self.roots, self.word['rootroles']):
+            if r['kind'] == 'center':
+                U = self.spans[r['node']-1]
+            else:
+                U = perp(basis(self.g['inputs'][t] for t in r['targets']), self.h)
+            self.rootframe[s], self.rootkind[s] = U, r['kind']
+        self.role_ops = defaultdict(list)
+        for i, (a, b, _) in enumerate(self.ops):
+            self.role_ops[a].append(i)
+            self.role_ops[b].append(i)
+        self.constants = []
+        const_ids = {}
+        def constant(F):
+            if F not in const_ids:
+                const_ids[F] = -len(self.constants)-1
+                self.constants.append(F)
+            return const_ids[F]
+        self.edges = []
+        for s in range(self.R):
+            if s in self.merged:
+                continue
+            seq = [constant(self.start[s])] + self.role_ops[s]
+            if s in self.rootframe:
+                seq.append(constant(self.rootframe[s]))
+            if s in self.pairs:
+                b = self.pairs[s]
+                seq += [constant(self.start[b])] + self.role_ops[b]
+                if b in self.rootframe:
+                    seq.append(constant(self.rootframe[b]))
+            seq.append(constant(self.full))
+            self.edges.extend(zip(seq, seq[1:]))
+        self.in_edges = [[] for _ in self.ops]
+        self.out_edges = [[] for _ in self.ops]
+        for k, (a, b) in enumerate(self.edges):
+            if a >= 0:
+                self.out_edges[a].append(k)
+            if b >= 0:
+                self.in_edges[b].append(k)
+        self.alpha = alpha
+        self.cost = [r * math.expm1(alpha * math.log(3*self.h/r)) if r else 0.
+                     for r in range(self.h+1)]
+        self.perp = lru_cache(maxsize=200000)(lambda F: perp(F, self.h))
+        self.check()
+
+    def F(self, n):
+        return self.frames[n] if n >= 0 else self.constants[-n-1]
+
+    def check(self):
+        for i, F in enumerate(self.frames):
+            assert basis(F) == F and contained(self.node_spans[i], F), i
+        for a, b in self.edges:
+            assert contained(self.F(a), self.F(b)), (a, b)
+
+    def profile(self):
+        H = Counter(len(self.F(b))-len(self.F(a)) for a, b in self.edges)
+        H.pop(0, None)
+        H[1] += len(self.sources)
+        for s, kind in self.rootkind.items():
+            if kind == 'center':
+                H[len(self.rootframe[s])] += 1
+        C = Counter({r: 3*n for r, n in H.items()})
+        for label in ('source_data_histogram', 'target_data_histogram'):
+            C.update({int(r): 3*n for r, n in self.record[label].items() if int(r)})
+        C.update(Counter(3*len(U) for s, U in self.gauges.items() if s not in self.merged))
+        C[2] += 2*self.v
+        W = 2*self.v+self.R-len(self.pairs)
+        mass = sum(r*n for r, n in C.items())
+        assert W*3*self.h-mass == self.record['deficit_per_vertex']
+        return dict(local_histogram=dict(sorted(H.items())), child_histogram=dict(sorted(C.items())),
+                    W_per_vertex=W, R=self.R-len(self.pairs), rank_per_vertex=mass,
+                    numerical_complex_root=numerical_root(C, W, 3*self.h))
+
+    def boundary(self, group):
+        nodes = set(group)
+        inc = [k for i in group for k in self.in_edges[i] if self.edges[k][0] not in nodes]
+        out = [k for i in group for k in self.out_edges[i] if self.edges[k][1] not in nodes]
+        return inc, out
+
+    def update(self, group, mode='both', neutral=False):
+        inc, out = self.boundary(group)
+        old = self.frames[group[0]]
+        assert all(self.frames[i] == old for i in group)
+        low = basis(tuple(u for i in group for u in self.node_spans[i]) +
+                    tuple(u for k in inc for u in self.F(self.edges[k][0])))
+        high = self.perp(basis(tuple(u for k in out for u in self.perp(self.F(self.edges[k][1])))))
+        assert contained(low, old) and contained(old, high)
+        dims_in = [len(self.F(self.edges[k][0])) for k in inc]
+        dims_out = [len(self.F(self.edges[k][1])) for k in out]
+        def cost(F):
+            d = len(F)
+            return math.fsum(self.cost[d-x] for x in dims_in) + math.fsum(self.cost[x-d] for x in dims_out)
+        candidates = [old]
+        if mode in ('both', 'low'): candidates.append(low)
+        if mode in ('both', 'high'): candidates.append(high)
+        best = min(candidates, key=lambda F: (cost(F), len(F), F))
+        improvement = cost(old)-cost(best)
+        if improvement > 1e-12 or (neutral and abs(improvement) < 1e-12 and len(best) < len(old)):
+            for i in group:
+                self.frames[i] = best
+            return improvement, len(group)
+        return 0., 0
+
+    def components(self):
+        parent = list(range(self.N))
+        def find(x):
+            while parent[x] != x:
+                parent[x] = parent[parent[x]]
+                x = parent[x]
+            return x
+        for a, b in self.edges:
+            if a >= 0 and b >= 0 and self.frames[a] == self.frames[b]:
+                a, b = find(a), find(b)
+                if a != b: parent[max(a,b)] = min(a,b)
+        groups = defaultdict(list)
+        for i in range(self.N): groups[find(i)].append(i)
+        return list(groups.values())
+
+    def update_envelope(self, group):
+        """Move an entire flat component to its earliest or latest legal shape."""
+        nodes = set(group)
+        changed_edges = set(k for i in group for k in self.in_edges[i] + self.out_edges[i])
+        lower, upper = {}, {}
+        for i in sorted(group,key=lambda i:self.topological_position[i]):
+            lower[i] = basis(self.node_spans[i] + tuple(
+                u for k in self.in_edges[i]
+                for u in lower.get(self.edges[k][0], self.F(self.edges[k][0]))))
+            assert contained(lower[i], self.frames[i])
+        for i in sorted(group,key=lambda i:self.topological_position[i],reverse=True):
+            upper[i] = self.perp(basis(tuple(
+                u for k in self.out_edges[i]
+                for u in self.perp(upper.get(self.edges[k][1], self.F(self.edges[k][1]))))))
+            assert contained(self.frames[i], upper[i])
+        def cost(plan):
+            return math.fsum(self.cost[len(plan.get(b, self.F(b)))-len(plan.get(a, self.F(a)))]
+                             for k in changed_edges for a, b in [self.edges[k]])
+        before = cost({})
+        plan = min([lower, upper], key=cost)
+        improvement = before-cost(plan)
+        if improvement > 1e-12:
+            count = sum(F != self.frames[i] for i, F in plan.items())
+            for i, F in plan.items(): self.frames[i] = F
+            return improvement, count
+        return 0., 0
+
+    def envelope_pass(self, seed=0):
+        groups = [g for g in self.components() if len(g) > 1]
+        random.Random(seed).shuffle(groups)
+        gain, changed = 0., 0
+        for group in groups:
+            improvement, count = self.update_envelope(group)
+            gain += improvement
+            changed += count
+        self.check()
+        print(json.dumps(dict(envelope_pass=seed, gain=gain, moved=changed,
+                              numerical_complex_root=self.profile()['numerical_complex_root'])), flush=True)
+        return changed
+
+    def update_subsets(self, group):
+        """Try closed prefixes/suffixes of one equal-frame component.
+
+        A prefix determined by E_i <= F is an order ideal, since the earliest
+        envelope E_i is monotone. Its entire frame can therefore move to F at
+        once. The latest envelope gives the dual upward-closed subsets.
+        """
+        old = self.frames[group[0]]
+        if not all(self.frames[i] == old for i in group): return 0., 0
+        lower, upper = {}, {}
+        for i in sorted(group,key=lambda i:self.topological_position[i]):
+            lower[i] = basis(self.node_spans[i] + tuple(
+                u for k in self.in_edges[i]
+                for u in lower.get(self.edges[k][0], self.F(self.edges[k][0]))))
+        for i in sorted(group,key=lambda i:self.topological_position[i],reverse=True):
+            upper[i] = self.perp(basis(tuple(
+                u for k in self.out_edges[i]
+                for u in self.perp(upper.get(self.edges[k][1], self.F(self.edges[k][1]))))))
+        best_gain, best_nodes, best_frame = 0., [], old
+        for env, direction in ((lower, 'lower'), (upper, 'upper')):
+            candidates = sorted(set(env.values())-{old}, key=lambda F: (len(F), F))
+            for F in candidates:
+                selected = [i for i in group if contained(env[i], F)] if direction == 'lower' else \
+                           [i for i in group if contained(F, env[i])]
+                inc, out = self.boundary(selected)
+                d, od = len(F), len(old)
+                gain = math.fsum(self.cost[od-len(self.F(self.edges[k][0]))] -
+                                 self.cost[d-len(self.F(self.edges[k][0]))] for k in inc)
+                gain += math.fsum(self.cost[len(self.F(self.edges[k][1]))-od] -
+                                  self.cost[len(self.F(self.edges[k][1]))-d] for k in out)
+                if gain > best_gain+1e-12:
+                    best_gain, best_nodes, best_frame = gain, selected, F
+        if best_nodes:
+            for i in best_nodes: self.frames[i] = best_frame
+            return best_gain, len(best_nodes)
+        return 0., 0
+
+    def subset_pass(self, seed=0):
+        groups = [g for g in self.components() if len(g) > 1]
+        random.Random(seed).shuffle(groups)
+        gain, changed = 0., 0
+        for group in groups:
+            improvement, count = self.update_subsets(group)
+            gain += improvement
+            changed += count
+        self.check()
+        print(json.dumps(dict(subset_pass=seed, gain=gain, moved=changed,
+                              numerical_complex_root=self.profile()['numerical_complex_root'])), flush=True)
+        return changed
+
+    def optimize(self, rounds=20, seed=0, mode='both', blocks=True):
+        rng = random.Random(seed)
+        for turn in range(rounds):
+            groups = self.components() if blocks and turn % 2 else [[i] for i in range(self.N)]
+            if turn % 4 == 1: groups.reverse()
+            if turn % 4 >= 2: rng.shuffle(groups)
+            total, changed = 0., 0
+            for group in groups:
+                if not all(self.frames[i] == self.frames[group[0]] for i in group): continue
+                gain, count = self.update(group, mode=mode)
+                total += gain
+                changed += count
+            print(json.dumps(dict(round=turn, gain=total, moved=changed,
+                                  numerical_complex_root=self.profile()['numerical_complex_root'])), flush=True)
+            if changed == 0 and (not blocks or turn % 2): break
+        self.check()
+
+    def neutral_pass(self,seed=0,blocks=False):
+        """Escape a local minimum using moves with exactly unchanged rank bins."""
+        before = self.profile()['child_histogram']
+        groups = self.components() if blocks else [[i] for i in range(self.N)]
+        rng = random.Random(seed);rng.shuffle(groups)
+        changed = 0
+        for group in groups:
+            if rng.random() < 0.5:continue
+            old = self.frames[group[0]]
+            if not all(self.frames[i] == old for i in group):continue
+            inc,out = self.boundary(group)
+            low = basis(tuple(u for i in group for u in self.node_spans[i])+
+                        tuple(u for k in inc for u in self.F(self.edges[k][0])))
+            high = self.perp(basis(tuple(u for k in out for u in self.perp(self.F(self.edges[k][1])))))
+            dims_in = [len(self.F(self.edges[k][0])) for k in inc]
+            dims_out = [len(self.F(self.edges[k][1])) for k in out]
+            def hist(F):return Counter([len(F)-d for d in dims_in]+[d-len(F) for d in dims_out])
+            old_hist = hist(old)
+            candidates = [F for F in (low,high) if F != old and hist(F) == old_hist]
+            if candidates:
+                chosen = rng.choice(candidates)
+                for i in group:self.frames[i] = chosen
+                changed += len(group)
+        self.check()
+        assert self.profile()['child_histogram'] == before
+        print(json.dumps(dict(neutral_pass=seed,moved=changed,blocks=blocks)),flush=True)
+        return changed
+
+    def save(self, out):
+        out.mkdir(parents=True, exist_ok=True)
+        differences = [[i, list(F)] for i, F in enumerate(self.frames) if F != self.base_frames[i]]
+        (out / 'frames.json').write_text(json.dumps(dict(frames=differences), separators=(',', ':'))+'\n')
+        (out / 'pairs.json').write_text(json.dumps(dict(pairs=self.pairs_input), separators=(',', ':'))+'\n')
+        (out / 'profile.json').write_text(json.dumps(self.profile(), indent=2, sort_keys=True)+'\n')
+
+def algebraic_birth_audit(g,word,R,pairs):
+    """Check exact integer read responses and the hypotheses of PR124's lemma.
+
+    Side coordinates below are twice their rational coefficients; the final
+    h coordinates are integer copied-center responses, before center scatter.
+    The common denominator of the complete scalar output is consequently six.
+    This does not use sampled inputs or arithmetic modulo a prime.
+    """
+    v,h = g['v'],g['h']
+    ops,coefficients = word['ops'],word['opcoeff']
+    roots,rootroles = g['roots'],word['rootroles']
+    response = [{} for _ in range(R)]
+    for r,s in zip(roots,rootroles):
+        assert not response[s]
+        if r['kind'] == 'center':
+            response[s] = {v+r['coordinate']:1}
+        else:
+            assert all(c in ('1/2','-1/2') for c in r['coefficients'])
+            response[s] = {t:(1 if c == '1/2' else -1) for t,c in zip(r['targets'],r['coefficients'])}
+    for (a,b,x),(ca,cb) in zip(reversed(ops),reversed(coefficients)):
+        assert a != b and ca in (-1,1) and cb in (-1,1)
+        for t,c in response[a].items():response[b][t] = response[b].get(t,0)+cb*c
+        if ca != 1:response[a] = {t:ca*c for t,c in response[a].items()}
+    response = [{t:c for t,c in row.items() if c} for row in response]
+    selected = {z['role']:z for z in word['selected']}
+    pset = set(word['phase1'])
+    chronology = word['phase1']+[i for i in range(len(ops)) if i not in pset]
+    position = {i:k for k,i in enumerate(chronology)}
+    role_ops = defaultdict(list)
+    for i,(a,b,x) in enumerate(ops):role_ops[a].append(i);role_ops[b].append(i)
+    source_roles = set(word['sources'].values())
+    root_roles = set(rootroles)
+    donors,recipients = {a for a,b,t in pairs},{b for a,b,t in pairs}
+    assert len(donors) == len(recipients) == len(pairs) and not donors & recipients
+    alias = {b:a for a,b,t in pairs}
+    for (a,b,x),(ca,cb) in zip(ops,coefficients):
+        assert alias.get(a,a) != alias.get(b,b)
+        assert ca*ca == 1 and ca*cb-ca*cb == 0
+    rows = []
+    for a,b,t in pairs:
+        assert a not in selected and a not in root_roles
+        assert b in selected and b not in source_roles
+        assert all(i not in pset for i in role_ops[b])
+        birth = len(pset) if t is None else position[t]
+        if t is not None:assert t == role_ops[b][0]
+        assert position[role_ops[a][-1]] < birth
+        D = response[b]
+        assert all(target < v for target in D), ('Center-dependent recipient',b)
+        assert set(D) <= set(selected[b]['targets']), ('Undeclared read target',b)
+        rows.append([b,sorted(D.items())])
+    canonical = json.dumps(sorted(rows),separators=(',',':')).encode()
+    return dict(exact_integer_response_backpropagation=True,
+                compensated_recipients=len(pairs),
+                zero_center_response_for_every_recipient=True,
+                recipient_target_supports_contained=True,
+                dead_donor_and_untouched_birth_chronology=True,
+                aliased_gate_ports_disjoint=True,
+                signed_gate_and_injection_inverse_polynomials_exact=True,
+                selected_response_sha256=hashlib.sha256(canonical).hexdigest(),
+                selected_side_response_denominator=2,
+                selected_read_terms=sum(len(response[b]) for b in recipients),
+                selected_max_abs_numerator=max(abs(c) for b in recipients for c in response[b].values()),
+                proof_interface='PR124 compensated-birth lemma plus the regenerated signed H+K+B=I identity; this finite audit is not a formal proof of the retained multiplication theorem')
diff --git a/research/continued-frame-cuts-186/discovery/cut_search.py b/research/continued-frame-cuts-186/discovery/cut_search.py
new file mode 100644
index 0000000000..10a41484c4
--- /dev/null
+++ b/research/continued-frame-cuts-186/discovery/cut_search.py
@@ -0,0 +1,167 @@
+#!/usr/bin/env python3
+"""Joint physical-frame moves by submodular binary minimum cuts.
+
+For a fixed candidate frame F, each operation keeps its old frame or moves
+to F. An edge's four costs are f(b-a), f(dim(F)-a), f(b-dim(F)), 0;
+incompatible containments receive a prohibitive cost. Concavity of f makes
+this binary energy submodular. A cut can therefore move many distinct
+current frames together, including moves that individually cost more.
+Only the final exact nested-frame witness is used in certification.
+
+Prepared by huxint with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from collections import deque
+from pathlib import Path
+import argparse
+import json
+import math
+import random
+import sys
+import time
+
+import numpy as np
+
+HERE=Path(__file__).resolve().parent
+sys.path.insert(0,str(HERE))
+from complex_search import Experiment,algebraic_birth_audit
+from paired_cube.frames import perp
+from paired_cube_physical import physical
+
+
+def minimum_cut(size,rows,cols,capacities,source,sink):
+    """Dinic with Python integer capacities; no 32-bit flow truncation."""
+    adj=[[] for _ in range(size)]
+    for a,b,c in zip(rows,cols,capacities):
+        if c<=0:continue
+        adj[a].append([b,int(c),len(adj[b])])
+        adj[b].append([a,0,len(adj[a])-1])
+    sys.setrecursionlimit(max(sys.getrecursionlimit(),2*size+100))
+    limit=sum(e[1] for e in adj[source])
+    while True:
+        level=[-1]*size;level[source]=0;queue=deque([source])
+        while queue:
+            a=queue.popleft()
+            for b,c,_ in adj[a]:
+                if c and level[b]<0:level[b]=level[a]+1;queue.append(b)
+        if level[sink]<0:return np.array([d>=0 for d in level],dtype=bool)
+        cursor=[0]*size
+        def push(a,amount):
+            if a==sink:return amount
+            while cursor[a]<len(adj[a]):
+                edge=adj[a][cursor[a]];b,c,rev=edge
+                if c and level[b]==level[a]+1:
+                    sent=push(b,min(amount,c))
+                    if sent:
+                        edge[1]-=sent;adj[b][rev][1]+=sent;return sent
+                cursor[a]+=1
+            return 0
+        while push(source,limit):pass
+
+
+class Expansion:
+    def __init__(self,e):
+        self.e=e
+        self.table=[];self.ids={}
+        def intern(F):
+            F=tuple(F)
+            if F not in self.ids:self.ids[F]=len(self.table);self.table.append(F)
+            return self.ids[F]
+        self.old=np.array([intern(F) for F in e.frames],dtype=np.int32)
+        self.spans=np.array([intern(F) for F in e.node_spans],dtype=np.int32)
+        self.fixed=np.array([intern(F) for F in e.constants],dtype=np.int32)
+        self.gauges=[intern(F) for F in e.gauges.values()]
+        self.dims=np.array([len(F) for F in self.table],dtype=np.int32)
+        ann=[perp(F,e.h) for F in self.table]
+        self.rows=np.array([x for F in self.table for x in F],dtype=np.uint32)
+        self.rowids=np.repeat(np.arange(len(self.table)),[len(F) for F in self.table])
+        self.ann=np.array([x for A in ann for x in A],dtype=np.uint32)
+        self.annids=np.repeat(np.arange(len(self.table)),[len(A) for A in ann])
+        self.anns=ann
+        self.M=10000.
+        self.scale=10**12
+
+    def inclusion(self,fid):
+        F=np.array(self.table[fid],dtype=np.uint32)
+        A=np.array(self.anns[fid],dtype=np.uint32)
+        bad_sub=np.zeros(len(self.table),dtype=bool)
+        bad_super=np.zeros(len(self.table),dtype=bool)
+        if len(A):
+            bad=(np.bitwise_count(self.rows[:,None]&A[None,:])&1).any(axis=1)
+            np.logical_or.at(bad_sub,self.rowids,bad)
+        if len(F):
+            bad=(np.bitwise_count(self.ann[:,None]&F[None,:])&1).any(axis=1)
+            np.logical_or.at(bad_super,self.annids,bad)
+        return ~bad_sub,~bad_super
+
+    def step(self,fid):
+        e=self.e;N=e.N;d=int(self.dims[fid]);cost=e.cost
+        sub,sup=self.inclusion(fid)
+        take=sub[self.spans] & (self.old!=fid)
+        lower=sub[self.old];upper=sup[self.old]
+        clower=sub[self.fixed];cupper=sup[self.fixed]
+        queue=deque()
+        def forbid(i):
+            if take[i]:take[i]=False;queue.append(i)
+        def left_ok(i):return lower[i] if i>=0 else clower[-i-1]
+        def right_ok(i):return upper[i] if i>=0 else cupper[-i-1]
+        for a,b in e.edges:
+            av=a>=0 and take[a];bv=b>=0 and take[b]
+            if av and not bv and not right_ok(b):forbid(a)
+            if bv and not av and not left_ok(a):forbid(b)
+        while queue:
+            i=queue.popleft()
+            if not lower[i]:
+                for k in e.out_edges[i]:
+                    b=e.edges[k][1]
+                    if b>=0:forbid(b)
+            if not upper[i]:
+                for k in e.in_edges[i]:
+                    a=e.edges[k][0]
+                    if a>=0:forbid(a)
+        nodes=np.flatnonzero(take).tolist()
+        if not nodes:return 0.,0,0
+        index={i:j for j,i in enumerate(nodes)};nv=len(nodes)
+        unary=np.zeros(nv,dtype=np.float64)
+        row=[];col=[];data=[]
+        touched=sorted({k for i in nodes for k in e.in_edges[i]+e.out_edges[i]})
+        before=0.
+        dims=self.dims[self.old];cdims=self.dims[self.fixed]
+        def dim(i):return int(dims[i]) if i>=0 else int(cdims[-i-1])
+        for k in touched:
+            a,b=e.edges[k];da,db=dim(a),dim(b)
+            old=cost[db-da];before+=old
+            ai,bi=index.get(a),index.get(b)
+            if ai is not None and bi is not None:
+                e01=cost[d-da] if lower[a] else self.M
+                e10=cost[db-d] if upper[b] else self.M
+                w=e01+e10-old
+                assert w>=-1e-9,'binary frame energy must be submodular'
+                unary[ai]+=e10-old;unary[bi]-=e10
+                if w>0:row.append(ai);col.append(bi);data.append(round(w*self.scale))
+            elif ai is not None:
+                assert right_ok(b)
+                unary[ai]+=cost[db-d]-old
+            else:
+                assert bi is not None and left_ok(a)
+                unary[bi]+=cost[d-da]-old
+        source,sink=nv,nv+1
+        for i,u in enumerate(unary):
+            if u>=0:row.append(source);col.append(i);data.append(round(u*self.scale))
+            else:row.append(i);col.append(sink);data.append(round(-u*self.scale))
+        seen=minimum_cut(nv+2,row,col,data,source,sink)
+        assert not seen[sink],'maximum-flow residual must separate source and sink'
+        moved={i for i,j in index.items() if not seen[j]}
+        if not moved:return 0.,0,nv
+        after=0.
+        for k in touched:
+            a,b=e.edges[k]
+            aa=fid if a in moved else int(self.old[a]) if a>=0 else int(self.fixed[-a-1])
+            bb=fid if b in moved else int(self.old[b]) if b>=0 else int(self.fixed[-b-1])
+            if a in moved and b not in moved:assert right_ok(b)
+            if b in moved and a not in moved:assert left_ok(a)
+            assert self.dims[bb]>=self.dims[aa]
+            after+=cost[int(self.dims[bb]-self.dims[aa])]
+        gain=before-after
+        if gain<=1e-10:return 0.,0,nv
+        for i in moved:e.frames[i]=self.table[fid];self.old[i]=fid
+        return gain,len(moved),nv
diff --git a/research/continued-frame-cuts-186/discovery/lattice_cuts.py b/research/continued-frame-cuts-186/discovery/lattice_cuts.py
new file mode 100644
index 0000000000..1fa0dbe055
--- /dev/null
+++ b/research/continued-frame-cuts-186/discovery/lattice_cuts.py
@@ -0,0 +1,133 @@
+#!/usr/bin/env python3
+"""Minimum cuts for coordinated frame intersections and joins.
+
+An intersection move uses F_i -> F_i intersect L on a predecessor-closed
+set; a join move uses F_i -> F_i + L on a successor-closed set. The forbidden
+mixed state makes each binary edge submodular. The cut only proposes moves;
+all resulting exact frame inclusions and the complete paid word are checked.
+Prepared by huxint with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from collections import deque,Counter
+from pathlib import Path
+import argparse,json,math,random,sys,time
+import numpy as np
+HERE=Path(__file__).resolve().parent;sys.path.insert(0,str(HERE))
+from complex_search import Experiment,numerical_root,algebraic_birth_audit
+from cut_search import Expansion,minimum_cut
+from paired_cube.frames import basis,perp,contained
+from paired_cube_physical import physical
+
+
+class Cut:
+    def __init__(self,e):
+        self.e=e;self.geometry=Expansion(e);self.scale=10**12;self.M=10000.
+    def step(self,L,mode):
+        e=self.e;N=e.N;down=mode=='meet'
+        ann=np.array(perp(L,e.h),dtype=np.uint32)
+        bad=np.zeros(len(self.geometry.table),dtype=bool)
+        if len(ann):
+            flags=(np.bitwise_count(self.geometry.rows[:,None]&ann[None,:])&1).any(axis=1)
+            np.logical_or.at(bad,self.geometry.rowids,flags)
+        sub=~bad
+        eligible=sub[self.geometry.spans] if down else np.ones(N,dtype=bool)
+        old=e.frames
+        inside={F:contained(F,L) if down else contained(L,F) for F in set(old)}
+        fixed={F:contained(F,L) if down else contained(L,F) for F in e.constants}
+        unchanged=np.array([inside[F] for F in old],dtype=bool)
+        take=eligible & ~unchanged;queue=deque()
+        def forbid(i):
+            if take[i]:take[i]=False;queue.append(i)
+        def fixed_ok(i):return unchanged[i] if i>=0 else fixed[e.constants[-i-1]]
+        for a,b in e.edges:
+            if down and b>=0 and take[b] and not (a>=0 and take[a]) and not fixed_ok(a):forbid(b)
+            if not down and a>=0 and take[a] and not (b>=0 and take[b]) and not fixed_ok(b):forbid(a)
+        while queue:
+            i=queue.popleft()
+            if unchanged[i]:continue
+            for k in (e.out_edges[i] if down else e.in_edges[i]):
+                j=e.edges[k][1 if down else 0]
+                if j>=0:forbid(j)
+        nodes=np.flatnonzero(take).tolist()
+        if not nodes:return 0.,0,0
+        byframe={}
+        if down:
+            P=perp(L,e.h)
+            for F in {old[i] for i in nodes}:byframe[F]=perp(basis(perp(F,e.h)+P),e.h)
+        else:
+            for F in {old[i] for i in nodes}:byframe[F]=basis(F+L)
+        new={i:byframe[old[i]] for i in nodes};index={i:j for j,i in enumerate(nodes)};nv=len(nodes)
+        touched=sorted({k for i in nodes for k in e.in_edges[i]+e.out_edges[i]})
+        unary=[0.]*nv;rows=[];cols=[];caps=[];before=0.
+        for k in touched:
+            a,b=e.edges[k];A,B=e.F(a),e.F(b);da,db=len(A),len(B);v00=e.cost[db-da];before+=v00
+            ai,bi=index.get(a),index.get(b)
+            if ai is not None and bi is not None:
+                na,nb=len(new[a]),len(new[b]);v11=e.cost[nb-na]
+                v01=self.M if down else e.cost[nb-da]
+                v10=e.cost[db-na] if down else self.M
+                weight=v01+v10-v00-v11;assert weight>=0
+                unary[ai]+=v10-v00;unary[bi]+=v11-v10
+                rows.append(ai);cols.append(bi);caps.append(round(weight*self.scale))
+            elif ai is not None:
+                assert contained(new[a],B)
+                unary[ai]+=e.cost[db-len(new[a])]-v00
+            else:
+                assert bi is not None and contained(A,new[b])
+                unary[bi]+=e.cost[len(new[b])-da]-v00
+        source,sink=nv,nv+1
+        for i,value in enumerate(unary):
+            rows.append(source if value>=0 else i);cols.append(i if value>=0 else sink);caps.append(round(abs(value)*self.scale))
+        side=minimum_cut(nv+2,rows,cols,caps,source,sink)
+        chosen={i for i,j in index.items() if not side[j]}
+        after=0.
+        for k in touched:
+            a,b=e.edges[k];A=new[a] if a in chosen else e.F(a);B=new[b] if b in chosen else e.F(b)
+            assert contained(A,B)
+            after+=e.cost[len(B)-len(A)]
+        gain=before-after
+        if gain<=1e-10:return 0.,0,nv
+        for i in chosen:
+            assert contained(e.node_spans[i],new[i]);e.frames[i]=new[i]
+        return gain,len(chosen),nv
+
+
+def main():
+    p=argparse.ArgumentParser();p.add_argument('--baseline',type=Path,required=True);p.add_argument('--frames',type=Path,required=True)
+    p.add_argument('--pairs',type=Path,required=True);p.add_argument('--output',type=Path,required=True)
+    p.add_argument('--labels',type=int,default=300);p.add_argument('--rounds',type=int,default=2);p.add_argument('--seed',type=int,default=186)
+    p.add_argument('--lines',type=int,default=0);p.add_argument('--planes',type=int,default=0)
+    a=p.parse_args();e=Experiment(a.baseline,a.frames,input_pairs=a.pairs,alpha=.000663)
+    x=Cut(e);rng=random.Random(a.seed)
+    pool=sorted({F for F in e.frames if 9<=len(F)<=21},key=lambda F:(len(F),F));rng.shuffle(pool);labels=pool[:a.labels]
+    vectors=set(x for F in e.frames for x in F)
+    vectors.update(x for F in e.frames for x in perp(F,e.h))
+    for F in {perp(F,e.h) for F in e.frames if len(F)>=18}:
+        vectors.update(x^y for x in F for y in F if x!=y)
+    vec=sorted(vectors);rng.shuffle(vec)
+    labels.extend((v,) for v in vec[:a.lines]);rng.shuffle(vec)
+    labels.extend(perp((v,),e.h) for v in vec[:a.planes])
+    labels=list(dict.fromkeys(labels))
+    started=time.monotonic();print(json.dumps(dict(initial=e.profile()['numerical_complex_root'],labels=len(labels))),flush=True)
+    for turn in range(a.rounds):
+        total=count=0;rng.shuffle(labels)
+        for k,L in enumerate(labels):
+            for mode in ('meet','join'):
+                gain,moved,candidates=x.step(L,mode)
+                if moved:
+                    total+=gain;count+=moved
+                    row=e.profile();H=Counter(row['child_histogram']);H[18]-=138;H[4]-=138
+                    print(json.dumps(dict(round=turn,label=k,mode=mode,rank=len(L),gain=gain,moved=moved,candidates=candidates,
+                                          raw_root=row['numerical_complex_root'],paid_root_if_46_sinks=numerical_root(H,row['W_per_vertex']-46,66),seconds=time.monotonic()-started)),flush=True)
+            if k%25==24:
+                print(json.dumps(dict(progress=k+1,round=turn,seconds=time.monotonic()-started)),flush=True)
+                e.save(a.output)
+        e.check();e.save(a.output)
+        print(json.dumps(dict(round_done=turn,gain=total,moved=count)),flush=True)
+        if not count:break
+    frames=json.loads((a.output/'frames.json').read_text())['frames']
+    checked=physical(e.g,e.w,e.word,e.record,frames,e.pairs_input)
+    (a.output/'verified.json').write_text(json.dumps(checked,sort_keys=True,indent=2)+'\n')
+    print('PASS raw independent physical word; terminal replay remains a separate check',flush=True)
+
+
+if __name__=='__main__':main()
diff --git a/research/continued-frame-cuts-186/search.py b/research/continued-frame-cuts-186/search.py
new file mode 100644
index 0000000000..6cc847bcd4
--- /dev/null
+++ b/research/continued-frame-cuts-186/search.py
@@ -0,0 +1,44 @@
+#!/usr/bin/env python3
+"""Optional continued-frame discovery. Requires numpy; verification does not.
+
+This searches from the already checked selected frame plan. It does not
+replace the frozen witness or regenerate any published certificate.
+Prepared by huxint with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from pathlib import Path
+import argparse,gzip,json,shutil,subprocess,sys,tempfile
+sys.dont_write_bytecode=True
+HERE=Path(__file__).resolve().parent
+sys.path.insert(0,str(HERE))
+from verify import prepare
+
+
+def main():
+    p=argparse.ArgumentParser(description=__doc__)
+    p.add_argument('--output',type=Path,required=True)
+    p.add_argument('--labels',type=int,default=100)
+    p.add_argument('--lines',type=int,default=700)
+    p.add_argument('--planes',type=int,default=700)
+    p.add_argument('--rounds',type=int,default=2)
+    p.add_argument('--seed',type=int,default=192)
+    a=p.parse_args();out=a.output.resolve()
+    if sys.flags.optimize:raise ValueError('Run without -O')
+    try:import numpy
+    except ImportError:raise SystemExit('Discovery requires numpy; install it in a virtual environment. The verifier uses only the standard library.')
+    with tempfile.TemporaryDirectory(prefix='continued-cut-search-') as temporary:
+        root=Path(temporary)/'source';package=prepare(root);selected=package/'selected/complex'
+        base=root/'build/search-base';base.mkdir(parents=True)
+        for source,target in (('graph','graph'),('frames','frames'),('word','selection'),('profile-before','record')):
+            (base/(target+'.json')).write_bytes(gzip.decompress((selected/(source+'.json.gz')).read_bytes()))
+        pairs=json.loads(gzip.decompress((selected/'physical-pairs.json.gz').read_bytes()))
+        (base/'input-pairs.json').write_text(json.dumps(dict(pairs=pairs))+'\n')
+        scripts=root/'research/continued-cut-discovery';shutil.copytree(HERE/'discovery',scripts)
+        command=[sys.executable,'-B',str(scripts/'lattice_cuts.py'),'--baseline',str(base),
+                 '--frames',str(HERE/'selected/frames.json'),'--pairs',str(base/'input-pairs.json'),
+                 '--output',str(out),'--labels',str(a.labels),'--lines',str(a.lines),'--planes',str(a.planes),
+                 '--rounds',str(a.rounds),'--seed',str(a.seed)]
+        subprocess.run(command,check=True)
+    print('Discovery complete. The output still needs complete terminal and assembly admission.')
+
+
+if __name__=='__main__':main()
diff --git a/research/continued-frame-cuts-186/selected/frames.json b/research/continued-frame-cuts-186/selected/frames.json
new file mode 100644
index 0000000000..f075e15752
--- /dev/null
+++ b/research/continued-frame-cuts-186/selected/frames.json
@@ -0,0 +1 @@
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diff --git a/research/continued-frame-cuts-186/verify.py b/research/continued-frame-cuts-186/verify.py
new file mode 100644
index 0000000000..52544d9510
--- /dev/null
+++ b/research/continued-frame-cuts-186/verify.py
@@ -0,0 +1,181 @@
+#!/usr/bin/env python3
+"""Replay PR186, then check a new physical-frame plan and its complete assembly.
+
+Standard-library Python. The scalar words, bit banks, aliases and terminal
+compiler are retained with their original source pins and credits.
+Prepared by huxint with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from fractions import Fraction as Q
+from hashlib import sha256
+from pathlib import Path
+import argparse,copy,gzip,importlib.util,io,json,shutil,subprocess,sys,tempfile,time,zipfile
+
+HERE=Path(__file__).resolve().parent;ROOT=HERE.parents[1]
+BASE=ROOT/'research/coordinated-frames-and-entrance-banks'
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+
+
+def require(ok,message):
+    if not ok:raise ValueError(message)
+
+
+def digest(path):return sha256(path.read_bytes()).hexdigest()
+
+
+def manifest():
+    files={p.relative_to(HERE).as_posix():digest(p) for p in sorted(HERE.rglob('*'))
+           if p.is_file() and '__pycache__' not in p.parts and p.name not in ('SOURCE.json','certificate.json','RESULTS.md')}
+    return dict(base_pr=186,base_commit='166a34d75e853e933855f0f7e940fe77c4bb6b8d',
+                base_files_sha256=digest(BASE/'FILES.json'),base_archive_manifest_sha256=digest(BASE/'BASELINE.json'),
+                files=files,assistance='huxint with substantial OpenAI Codex assistance')
+
+
+def prepare(root):
+    """Verify the immutable parent files and extract its pinned source closure."""
+    meta=json.loads((BASE/'FILES.json').read_text())
+    names={p.relative_to(BASE).as_posix() for p in BASE.rglob('*') if p.is_file() and p.name!='FILES.json'}
+    require(names==set(meta['files']),'Parent upload inventory changed')
+    require({n:digest(BASE/n) for n in names}==meta['files'],'Parent upload hashes changed')
+    baseline=json.loads((BASE/'BASELINE.json').read_text());parts=[]
+    for row in baseline['parts']:
+        b=(BASE/row['file']).read_bytes()
+        require(len(b)==row['bytes'] and sha256(b).hexdigest()==row['sha256'],'Parent archive part')
+        parts.append(b)
+    blob=b''.join(parts);require(sha256(blob).hexdigest()==baseline['archive_sha256'],'Parent archive hash')
+    root.mkdir(parents=True,exist_ok=True)
+    with zipfile.ZipFile(io.BytesIO(blob)) as z:
+        for entry in z.infolist():require((root/entry.filename).resolve().is_relative_to(root.resolve()),'Archive path')
+        z.extractall(root)
+    package=root/'research/coordinated-frames-and-entrance-banks'
+    excluded={x['file'] for x in baseline['parts']}
+    for p in BASE.rglob('*'):
+        if p.is_file() and p.name not in excluded:
+            out=package/p.relative_to(BASE);out.parent.mkdir(parents=True,exist_ok=True);shutil.copyfile(p,out)
+    return package
+
+
+def run(script,*args,output=None):
+    result=subprocess.run([sys.executable,'-B',str(script),*map(str,args)],text=True,capture_output=True)
+    require(result.returncode==0,str(script.name)+' rejected the construction:\n'+result.stdout+result.stderr)
+    return json.loads(output.read_text()) if output is not None else result.stdout
+
+
+def write_gzip(path,value):
+    b=gzip.compress((json.dumps(value,sort_keys=True,separators=(',',':'))+'\n').encode(),mtime=0)
+    path.write_bytes(b[:9]+b'\xff'+b[10:])
+
+
+def verify():
+    require(not sys.flags.optimize,'Run without -O')
+    require(json.loads((HERE/'SOURCE.json').read_text())==manifest(),'Own source closure changed')
+    with tempfile.TemporaryDirectory(prefix='continued-frame-cuts-') as temporary:
+        root=Path(temporary)/'source';package=prepare(root)
+        print('Replaying the complete pinned PR186 complex and bank-bit certificate...',flush=True)
+        run(package/'verify_inner.py')
+        parent=json.loads((package/'certificate.json').read_text())
+        sys.path.insert(0,str(root/'scripts'))
+        from paired_cube_physical import physical
+        selected=package/'selected/complex';candidate=root/'build/continued-candidate'
+        shutil.copytree(selected,candidate)
+        read=lambda name:json.loads(gzip.decompress((selected/(name+'.json.gz')).read_bytes()))
+        g,witness,word,row=map(read,('graph','frames','word','profile-before'))
+        pairs=read('physical-pairs');frames=json.loads((HERE/'selected/frames.json').read_text())['frames']
+        raw=physical(g,witness,word,row,frames,pairs)
+        write_gzip(candidate/'physical-frames.json.gz',frames);write_gzip(candidate/'profile.json.gz',raw)
+        pins={p.name:digest(p) for p in candidate.glob('*.json.gz')}
+        (candidate/'result.json').write_text(json.dumps(dict(pins=pins),sort_keys=True)+'\n')
+        print('Checking new frames, signed formal columns, terminal chronology and both sink directions...',flush=True)
+        output=root/'build/new-physical.json'
+        actual=run(package/'complex/physical.py','--candidate',candidate,'--source',root,'--out',output,output=output)
+        actual.pop('seconds',None);actual.pop('maxrss',None)
+        actual['source_sha256']={Path(k).relative_to(root).as_posix():v for k,v in actual['source_sha256'].items()}
+        selection_path=root/'build/new-sinks.json'
+        sinks=run(package/'complex/check_sinks.py','--candidate',candidate,'--output',selection_path,output=selection_path)
+        for name in ('seconds','rss_kib'):sinks.pop(name,None)
+        # Store a deterministic selection; the formal checker reads its actual actions.
+        selection_path.write_text(json.dumps(sinks,sort_keys=True,indent=2)+'\n')
+        formal_path=root/'build/new-formal-sinks.json'
+        formal=run(package/'complex/formal_sinks.py','--candidate',candidate,'--selection',selection_path,
+                   '--formal-helper',package/'complex/physical.py','--output',formal_path,output=formal_path)
+        formal.pop('seconds',None)
+        old_sinks=parent['complex']['physical']['terminal_compiler']
+        require(sinks['eligible_count']==old_sinks['eligible_count']==46,'Retain every checked terminal')
+        actions=lambda x:[{k:s[k] for k in ('role','pivot','root_index','targets','writes')} for s in x['sinks']]
+        require(actions(sinks)==actions(old_sinks),'The terminal scalar actions changed')
+        paid={k:row[k] for k in ('h','v','c','q','matched','total_M_operations','loss')}
+        paid.update(R=sinks['new_physical_R'],scalar_role_reserve=row['R'],reuse_pairs=raw['pairs'],terminal_sinks=46,
+                    m=66,W_per_vertex=sinks['new_W'],rank_per_vertex=sinks['new_rank'],deficit_per_vertex=1320,
+                    child_histogram=sinks['child_histogram'],maxchild=max(map(int,sinks['child_histogram'])))
+        require(all(z['dirty']==paid['R'] and z['columns']==paid['W_per_vertex'] for z in formal['columns']),
+                'Every new physical source, target and dirty column')
+        sys.path.insert(0,str(package/'arithmetic'))
+        import certificate as arithmetic
+        from interval_moment import saving_grid
+        arithmetic.normalized_body_check()
+        p=dict(m=paid['m'],W=paid['W_per_vertex'],N=1320,L=0,total_rank=paid['rank_per_vertex'],
+               maxchild=paid['maxchild'],child_multiplicities=paid['child_histogram'])
+        moment=saving_grid(p,10**18);saving=moment['accepted']['saving']
+        final=arithmetic.price(paid,saving,arithmetic.COARSE,arithmetic.BETA,arithmetic.ETA,arithmetic.WEAKENING,'balanced')
+        require(final['kappa']>Q(parent['kappa']),'Final assembled improvement over the parent')
+        require(final['actual_bit_saving']==Q(parent['arithmetic']['actual_uniform_bit_saving']),'Unchanged checked bank-bit supplier')
+        require(final['finite_bridge']['complex']['local_group_upper']==actual['scalar_bound']['local_group_upper'],
+                'Full original scalar reserve remains charged')
+        controls=[]
+        def reject(name,call):
+            try:call()
+            except (ValueError,AssertionError):controls.append(name)
+            else:raise ValueError('Negative control accepted: '+name)
+        reject('next final grid point',lambda:arithmetic.assembly(final['finite_bridge'],paid,final['a'],
+                final['kappa']+Q(1,10**18),beta=arithmetic.BETA,h=arithmetic.ETA,a_complex=saving))
+        bad=copy.deepcopy(final['finite_bridge']);bad['complex']['W']=paid['W_per_vertex']
+        reject('full group omitted',lambda:arithmetic.validate_shared_bridge(bad,paid))
+        bad_row=copy.deepcopy(paid);key=next(iter(bad_row['child_histogram']));bad_row['child_histogram'][key]+=1
+        reject('unrecounted child',lambda:arithmetic.validate_shared_bridge(final['finite_bridge'],bad_row))
+        bad_frames=copy.deepcopy(frames);bad_frames[0][1]=[]
+        reject('zero operation frame',lambda:physical(g,witness,word,row,bad_frames,pairs))
+        before=dict(read('physical-frames'));after=dict(frames)
+        return arithmetic.js(dict(status='PASS complete conditional witness',kappa=final['kappa'],
+            parent_kappa=parent['kappa'],complex_saving=saving,changed_operation_frames=sum(before.get(i)!=after.get(i) for i in before.keys()|after.keys()),
+            complex_profile=paid,complex_moment=moment,physical=actual,terminal=sinks,terminal_formal=formal,
+            parent_complete_certificate_sha256=digest(package/'certificate.json'),
+            checked_bit_certificate_sha256=sha256(json.dumps(parent['bit'],sort_keys=True,separators=(',',':')).encode()).hexdigest(),
+            unchanged_checked_bit_saving=final['actual_bit_saving'],arithmetic=final,controls=controls,
+            inherited_assembly_controls=parent['arithmetic']['adverse_controls'],
+            source_manifest_sha256=digest(HERE/'SOURCE.json'),
+            scope='Physical frame refinement of the complete PR186 witness. All inherited compiler, bank, analytic, row, prime, precision, layout and fixed-tape contracts remain conditional.'))
+
+
+def report(result):
+    def decimal(value):
+        n=Q(value)*10**18;require(n.denominator==1,'Displayed exact grid point')
+        return '0.'+str(n.numerator).zfill(18)
+    p=result['complex_profile'];k=Q(result['kappa']);parent=Q(result['parent_kappa'])
+    return ('# Checked result\n\n'
+        'Conditional kappa = **'+result['kappa']+' = '+decimal(k)+'**.\n\n'
+        'This is %.6f%% above the pinned PR186 value %s.\n\n'%(float(100*(k/parent-1)),decimal(parent))+
+        'The complex supplier binds, with saving '+result['complex_saving']+' = '+decimal(result['complex_saving'])+'.\n\n'
+        '| Checked complex quantity | Value |\n|---|---:|\n'
+        '| Changed operation frames relative to PR186 | %d |\n'%result['changed_operation_frames']+
+        '| Physical auxiliary roles | %d |\n| Persistent roles W | %d |\n'%(p['R'],p['W_per_vertex'])+
+        '| Rank mass | %d |\n| Deficit | %d |\n| Terminal sinks | %d |\n\n'%(p['rank_per_vertex'],p['deficit_per_vertex'],p['terminal_sinks'])+
+        'The pinned complete parent, its bank-bit supplier, both new complex shear directions, every terminal, '
+        'all 47 strict assembly constraints and adjacent-grid exclusions are replayed.\n\n'
+        'These are conditional finite certificates. The inherited all-size compiler, shared-core/bank, '
+        'weighted-bit, row-restoration, prime, precision/recovery, balanced-layout and fixed-tape contracts remain assumptions.\n')
+
+
+def main():
+    p=argparse.ArgumentParser(description=__doc__);p.add_argument('--write',action='store_true');a=p.parse_args();start=time.monotonic()
+    require(not sys.flags.optimize,'Run without -O')
+    if a.write:(HERE/'SOURCE.json').write_text(json.dumps(manifest(),sort_keys=True,indent=2)+'\n')
+    result=verify();text=json.dumps(result,sort_keys=True,indent=2)+'\n'
+    if a.write:
+        (HERE/'certificate.json').write_text(text);(HERE/'RESULTS.md').write_text(report(result))
+    else:
+        require(text==(HERE/'certificate.json').read_text(),'Complete frozen certificate differs')
+        require(report(result)==(HERE/'RESULTS.md').read_text(),'Displayed result differs from certificate')
+    print('PASS kappa='+result['kappa']+'; full parent replay, both new complex directions, 46 terminals and exact assembly (%.1fs)'%(time.monotonic()-start),flush=True)
+
+
+if __name__=='__main__':main()
diff --git a/research/coordinated-frames-and-entrance-banks/BANK-PROOF.md b/research/coordinated-frames-and-entrance-banks/BANK-PROOF.md
new file mode 100644
index 0000000000..9f736c75f8
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/BANK-PROOF.md
@@ -0,0 +1,23 @@
+# Completed entrance banks on the physical bit word
+
+The bit word, signs, donor aliases, deferred compensation reads, source/target itineraries, original scalar reserve and prime witnesses are retained. The portable verifier independently reconstructs the actual physical chains and verifies every F2 and integer source, target and arbitrary dirty column before changing the inventory.
+
+For the selected word there are19788 logical roles and1760 compensated donor/recipient splices, hence18028 physical roles. Each physical slot is charged once over its full chronological chain. Its donor entrance remains the initial frame and the recipient birth/read is an internal point. The1760 rank21 recipient gauges are therefore not additional physical entrances. The only surviving entrance gauges are2200 rank20 gauges; the remaining15828 physical roles are undeferred.
+
+Here is the endpoint argument for arbitrary contents. For any commutative ring and module, set
+
+    S_P(x,y)=(x−Px+Py,y−Py+Px).
+
+If P and Q have PQ=QP=0, expand the four coordinates directly to obtain S_P S_Q=S_(P+Q). For an idempotent P the same expansion gives S_P²=I. Consequently any finite disjoint projector partition summing to I has product S_I, the full swap, on EVERY incoming pair. No cleanliness or independence of the incoming data is used. For one common invertible weighted chart A, multiplication telescopes: product(A S_P A⁻¹)=A(product S_P)A⁻¹. The chronological endpoint is thus the inherited full weighted interchange and contains no residual exterior child. This is exactly the previously established completed-bank endpoint; the adaptation introduces no new endpoint hypothesis.
+
+The actual coordinate projectors are P_s(j)=1[j div4=s] for s=0,...,17, and Q_s(j)=1[j div24=s] for s=0,1,2, on j=0,...,71. Each family is explicitly disjoint, idempotent and exhaustive. The new verifier checks all144 columns of each two-bank address map over F2 and over Z, both for the full endpoint and its inverse. It rejects a missing last block and a repeated last block in both rings. Equality of these address maps gives equality of the induced permutations on arbitrary payload fields, including fields correlated with spectators. The universal algebra above permits all common chart conjugations, with their inherited paid routing and weighting unchanged.
+
+Three stage-private bank families use the same ambient completed-core width72. A rank20 entrance leaves a four-coordinate local core, so18 such cores fill a bank. An undeferred local core has24 coordinates, so3 fill a bank. Nine physical replicas make each stage-private allocation integral: 1100 selected banks and 47484 undeferred banks per stage. Their total, including source/target stock, is 3(1100+47484)+2·9·1760=177432=9W. Three is only the scale used to clear the normalized moment fractions; it is not the physical replica count. Every role visits every bank through the inherited orbit routing; no favorable subset of invocations is selected. The resulting normalized stock and rank mass are
+
+    W=2·1760+3(2200/18+15828/3)=59144/3,
+    rank=1549520−2200·60=1417520,
+    72W−rank=1936.
+
+Exactly the2200 rank60 gauge exteriors are removed. Every internal chain increment, source and target connector, centre copy, original data endpoint and bad-class fallback remains in the histogram. Its maximum child is22. Replicated moments scale W, rank, deficit and every multiplicity by3 together; the normalized moment and the per-edge fallback bill are unchanged. The rare-class fraction remains10⁻¹⁶, the fallback remains32·72² children per edge, and the positive atom/row-adapter toll is paid on its own10⁻²⁴ grid. Two independent rational log/exp enclosures certify the accepted moment and reject its adjacent10⁻¹⁸ coarse grid point.
+
+The inherited all-input weighted compiler, completed-core routing, stage cover, local-ring precision/recovery, prime supply, uniform setup and all-size analytic/tape contracts remain conditional. This finite bank verification discharges none of those separate interfaces. The universal bank algebra is also present in the earlier BankAndBalanced.lean snapshot under partialSwap_comp_disjoint, bankSwap_full, weightedBank_full and the actual Fin72 selected_bank_full/undeferred_bank_full statements; no fresh Lean replay is claimed by this Python package.
diff --git a/research/coordinated-frames-and-entrance-banks/BASELINE.json b/research/coordinated-frames-and-entrance-banks/BASELINE.json
new file mode 100644
index 0000000000..d8da63e1a3
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/BASELINE.json
@@ -0,0 +1,26 @@
+{
+  "repository": "https://github.com/chafreaky/integer-mult-bounds",
+  "upstream": "https://github.com/CrocSwap/integer-mult-bounds/pull/181",
+  "commit": "7fb2194801e3a10f54772c7f0d5505035a4fc0ad",
+  "files": 1700,
+  "LF_checkout": true,
+  "archive_sha256": "2f8b4294620228116cfe993a376e2c848a7f280c4a7380d6ff4d2c8a8ea076eb",
+  "uncompressed_bytes": 78899539,
+  "parts": [
+    {
+      "file": "baseline-pr181.part01",
+      "bytes": 12000000,
+      "sha256": "25f760ffb55e0e9de2f43aa334e3323d957776a9d323790c60b71ec7c414e160"
+    },
+    {
+      "file": "baseline-pr181.part02",
+      "bytes": 12000000,
+      "sha256": "179e594a0a37880c62bbc966f4079e0aab60c3dfd1cf5bdd93dafd8135b2bf8f"
+    },
+    {
+      "file": "baseline-pr181.part03",
+      "bytes": 9595108,
+      "sha256": "bf0e1489ddd033c37e610e61ec7c7c8c62df31d27f69b1e83574fd9080731a84"
+    }
+  ]
+}
diff --git a/research/coordinated-frames-and-entrance-banks/FILES.json b/research/coordinated-frames-and-entrance-banks/FILES.json
new file mode 100644
index 0000000000..c7277ec819
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/FILES.json
@@ -0,0 +1,55 @@
+{
+  "algorithm": "SHA-256",
+  "files": {
+    "BANK-PROOF.md": "5136565f1298b73436b164b3e9307ba00a2659da41a3a64defde4b41975dc441",
+    "BASELINE.json": "0412767b472720bae2902fcd24de5d18f3039afec3dd592a3c0e2d68fbabff63",
+    "FRAME-DISCOVERY.md": "2309dedd04605ee4285e9e0bc66741ee65b826724ceed624d8acf62554e24f51",
+    "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "NOTICE": "2cf3c6e1aaaed287d44d001f55cfd74393c9d9e773e3ea0e3fb359d885b4f6b5",
+    "PROOF.md": "d99304c0062516d86ec0ee5ae7b90b36eacaf400f74cb9d37a9d65aa25e71e31",
+    "README.md": "96bb0aeb0d8a6cecee5e73b1e6ee05729c81a8af0c4e8f6e95be6a28735ec224",
+    "RESEARCH-IDEAS.md": "670db9e2daf30b36de69bda6eab18c38833ff17bf33398a7853be5658983930c",
+    "RESEARCH-REVIEW.md": "756e4d626d6eb66cd5609637eaf19ae13c557915d8abd2e25b20bcecfcbd7c24",
+    "SOURCE.json": "30c3878dde69a4ed29beed26856d1015ad3180da464d96154e5b005f728847bf",
+    "UPSTREAM-NOTICE.md": "5d5f78913f7b1a172f1f42643bf75b419b2eaddfa3bd082b89fcf17d81266bf2",
+    "UPSTREAM-PROOF.md": "977c63ff7d7cf60e46c9290c46ea1459da1cfb606307d756da2457e22ae7477d",
+    "arithmetic/balanced_shared.py": "c3204e1758057f5ef10724014f38b91b8a01f6d8a3562b054263fc74795bfd90",
+    "arithmetic/certificate.py": "c2f7baa14f4197040628e75bc0eecaa8fb1b98818bff4786c9abf4f516f4f347",
+    "arithmetic/interval_moment.py": "9d3d1d28bfb3375958995205179b171fb65c70a3792cf9f7a21ef5ee54f5460b",
+    "arithmetic/shared_bridge.py": "d0729e2d12d8c038719ea6eb83b169c09660f43de2d3d2da10a6a173f5204f93",
+    "baseline-pr181.part01": "25f760ffb55e0e9de2f43aa334e3323d957776a9d323790c60b71ec7c414e160",
+    "baseline-pr181.part02": "179e594a0a37880c62bbc966f4079e0aab60c3dfd1cf5bdd93dafd8135b2bf8f",
+    "baseline-pr181.part03": "bf0e1489ddd033c37e610e61ec7c7c8c62df31d27f69b1e83574fd9080731a84",
+    "bit/banks.json": "29489011434764f2a0c934222e553b22428ee2de5b890b850caf43dc6feb32e2",
+    "bit/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098",
+    "bit/base_word.py": "606106f9e67ee82bf7b38420b65ac19797f0fa4c05da5cab4e6129d081864a61",
+    "bit/prime-witnesses.json.gz": "f588566aad172c064322bc6dc34c49eca8806cdde54fe49135fed4c06f255137",
+    "bit/prime_witnesses.py": "36bf9e0fc54c6c1e739801fd9240b20beaf5235758aef271be50943414109a1c",
+    "bit/prove.py": "a4966d4660076160d1dab197eb8648ed97caf9fccfb5e762f90d285216c677b5",
+    "bit/word.py": "690f5ebad2611cee7f5cc71b374295e1dc76960960debfec1cd6d12f58557c6f",
+    "certificate.json": "59b20818af6f850f40d18fd8206b136da504d9cbc57dbbefcba232c62f3384e6",
+    "complex/check_sinks.py": "dc3099da39841111a78303a9b44567732a9fae1f6456fbb5356e9535fd134d67",
+    "complex/formal_sinks.py": "d7fefe86e637584ad1bf53c8f7714b88d5abed92eaa29d110ef099536e51f516",
+    "complex/physical.py": "496a5227d525e9291b05db7e6050ec866a7160e5bcd5da0a00bb7e5210efb996",
+    "complex/prove.py": "720a5570895e5c732b64f0cbeccff62e8b36c52e9dd8fc6d5ddd65f66c9cf555",
+    "complex/verify_fused.py": "75c3dcd80fcdbc461651fa2d2fa463e6b41a569d695fc74700d240223ef198e9",
+    "references/pr141/ORIGINAL-PROOF.md": "2309aafdeecc7a432aa485f1cd4a5392fda3542ec8a3ea193ad1557f049becdb",
+    "references/pr141/balanced_stopped.py": "03f5f012ab4eb306320ab8a209c2041addd75b84aa43f45ab80595c8989ce781",
+    "references/pr166/PROOF.md": "36f5b7ffb9b6454b38d2c6f8934ca1c370512f1315220c129daf2f6c9fd2e8a6",
+    "references/pr166/README.md": "665f8754c12f40ef62d9af821238ecc2a686c93046f325324b73907f8574c615",
+    "references/pr168/bit-physical-profile.json": "6735294e8639da09eb3eb2234cbc5814298442a6c946b5534bcf6e48903547f3",
+    "references/pr175/SOURCE.json": "584887f4a14afa49037a3d5d66aab13643295275cbb79c739ea62b59b6c11e4d",
+    "selected/complex/baseline.json.gz": "f4ad8cd1635f279d201225c174548cb03db13b63156cdf647ca8e834d4746524",
+    "selected/complex/frames.json.gz": "b516b59e7bd7e68c56c02e0041e7455b2eff445e81ac5fb69570d424cf0178a1",
+    "selected/complex/graph.json.gz": "9f73a6a7f1a4410e954246c6ff95d9cad0582d0057f707a693cc2fd170995ead",
+    "selected/complex/physical-frames.json.gz": "0e92261dbedacb1174ee58712f0a86dab00a4f2411e377faab7c08f54f8307f0",
+    "selected/complex/physical-pairs.json.gz": "869b852cbfd63ef691336ea398280ef57896dd876f6a6c78edac88a2a304fffb",
+    "selected/complex/profile-before.json.gz": "bd39675ce03fd1d4d456f736bcfc0b389532eb4f0a19c92877c2af5cb6f7310c",
+    "selected/complex/profile.json.gz": "cf80065ea589fcb0e9e83a2dfc76f694e3ca8bc2d8290f4175c45e2a100b8f84",
+    "selected/complex/result.json": "54f7b213f62e780bea8dc31bc474af2e1540a028576313de7a002b7c60d7cf5b",
+    "selected/complex/sinks.json": "498f7ccc51db7d09d9dc191f0b0d44fd70f3f8bf374cf3f52548880bd42f04ec",
+    "selected/complex/word.json.gz": "0c4b720c57717c7e55e88f86a9e17ed75b3891f9d2e7ed29df0a188f8dfee67a",
+    "verify.py": "73d7a41ff1264c3e9bfe5f748e77218af86796dfded2f21187f970bc10406894",
+    "verify_inner.py": "858142e25c396a074adc9f0dbfe2fa97f5f5f7e7dc2cad38e5169e8989fe8f95"
+  }
+}
diff --git a/research/coordinated-frames-and-entrance-banks/FRAME-DISCOVERY.md b/research/coordinated-frames-and-entrance-banks/FRAME-DISCOVERY.md
new file mode 100644
index 0000000000..2bb5759ea3
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/FRAME-DISCOVERY.md
@@ -0,0 +1,37 @@
+# Checked PR181 complex descendant
+
+Best checked complex saving is b=662323931899051/10^18 on the actual PR181 selected word at head 7fb2194801e3a10f54772c7f0d5505035a4fc0ad. The combined supplier has W = 13,326, rank = 878,196, deficit = 1,320, maximum child = 20, 2,310 compensated aliases and 46 terminal sinks. Parent assembly must compose this complete supplier with the independently checked bank bit word and reprice every paid constraint to obtain final kappa.
+
+## Exact progression
+
+| Stage | Exact accepted complex saving | Paid stock/sinks |
+|---|---:|---:|
+| Public PR181 | 132438677973/(2*10^14) | W = 13,328 / 44 sinks |
+| Independently audited complex donor rematch | 82782792632077/(125*10^15) | W = 13,326 / 46 sinks |
+| Final 243-frame cut plus that rematch | 662323931899051/10^18 | W = 13,326 / 46 sinks |
+
+The final saving exceeds the public stated complex saving by 130542034051/10^18 and the independently checked rematch saving by 61590842435/10^18. These are comparisons of complete exact paid moments, not an instruction to add separately found gains. An outward-rational comparison also checks the final cut with ORIGINAL pairs: at the combined accepted exponent its moment lower endpoint exceeds 1 by more than 1.5640571*10^-7, so that alternate composition is weaker. See original-pair-comparison.json.
+
+An initial endpoint/equal-frame screen on the unmodified PR181 physical layer accepts no change. A first low-rank cut screen is stopped without an accepted change to spread the remaining bounded budget across all existing ranks. The main 210-second pass tests 69 actual fixed-subspace join/meet neighborhoods and accepts 11 meets, of ranks 12,13,15,16,18,21,19,19,17,14,18. It changes 243 distinct operation frames relative to PR181; changed-operations.json lists their fresh actual IDs. No historical operation IDs are transferred.
+
+All cut schedules use exact binary required-value containment, both roles of every common operation frame, every source/gauge/root boundary and every spliced physical donor/recipient chain. The initial cut graph also encodes the actual 44-sink pivot target chronologies. After composition with the rematch, the independent signed sink compiler recomputes all 46 target chronologies and the complete paid histogram. The rematch has 2,297 changed donor identities relative to public PR181, keyed by the unchanged recipient roles, and 1,049 retimed compensation reads; this is independently checked in the complex lane. Fixed local rank mass is preserved throughout every cut.
+
+## Finite sufficient argument and checks
+
+A chosen operation frame is F or F intersect L for one fixed binary subspace L. Its exact required scalar value span remains included. Every nested physical chain and common two-register frame condition is explicitly checked. Actual terminal target events impose additional hard containment constraints. Fixed source/gauge/root endpoints make transition rank sums telescope. All operation signs, coefficients, original scalar operations, full-frame inverses, source injections, read chronology and phase-sensitive adapters remain in the word; zero-rank transitions are only absent from the rank moment.
+
+The final admitted composition is NOT inferred from two independent improvements. Original paired_cube_physical.physical() independently rechecks every value span, actual alias/death/birth rule, chronological target order, spliced role chain, full rank mass and dirty scalar mutation control. The PR181 package's exact check_sinks.py then recomputes the new 46 sinks, all simultaneous target-frame events, exact source/dirty responses, literal inverse sandwich and full child histogram. It proves the actual new physical inventory W = 13,326,rank = 878,196,N1320. An independent outward rational moment accepts b = 662323931899051/10^18 and rejects its next 10^-18 grid point.
+
+The graph, scalar word, frame witness, logical profile and exact pairs are verified semantically IDENTICAL to the complex lane's all-column audited rematch. All 46 [role,pivot,root,target-set,write-sequence] actions also match exactly. The complete exact scalar map proof therefore remains the same: both signs check 13,326 source/target/dirty columns and restore 10,686 dirty roles; corruption controls reject. Only frame legality and byte binding are updated. SOURCE-LOCK.json binds that audit to the complete new frame/profile/moment files. The parent portable admission replays the new bytes and complete balanced assembly.
+
+PR181's local word includes destination coefficients ca=-1. Its original inherited terminal-sinks gate targets the older all-shear word and cannot serve as a verifier of this new word. This continuation uses the package's signed-compatible independent checker without weakening the older gate. On Windows only its Unix CPU/RSS guard API is adapted in memory; mathematical source and checks remain unmodified.
+
+The sufficient construction retains PR181's upstream general Clifford/tensor-frame endpoint contract, weighted local-ring compilation, stage cover, restored borrowed rows, finite routing, paid positional layout, precision/recovery, prime supply, uniform setup and analytic/fixed-tape transfer interfaces. It is a finite conditional supplier certificate; those universal interfaces are not discharged here. No unknown gadget or unpaid oracle replaces a missing step.
+
+Floating minimum cut nominates frames. Exact containment and outward arithmetic certify the saved schedules; there is no floating-optimality or global-optimality claim. Directed cut is classical. This is actual supplier use of the global-subspace crossover, not a novelty claim for maxflow or a projection using invented supplier bills.
+
+## Files and freeze
+
+Combined final frames: combined/physical-frames.json, SHA-256: 238d65f5c1447b18000bd96239d8fe372e9bd56fa9aca42c6626cc7be4b7c552. Exact pairs: combined/pairs.json. Complete raw frozen cache and independent signed-sink record: combined/checked/cache and combined/checked/sinks-exact.json. Complete paid profile, accepted/rejected moment and input locks: combined/checked/evaluation.json. Full input/output and unchanged scalar-audit bindings: combined/SOURCE-LOCK.json.
+
+All owned PR181 search/check processes have exited. The earlier fully checked PR179 descendant remains preserved as a historical fallback; it is not combined with PR181 and is below the refreshed public bar. Nothing was published and no pinned source was edited.
diff --git a/research/coordinated-frames-and-entrance-banks/LICENSE b/research/coordinated-frames-and-entrance-banks/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/coordinated-frames-and-entrance-banks/NOTICE b/research/coordinated-frames-and-entrance-banks/NOTICE
new file mode 100644
index 0000000000..8853b1b1d6
--- /dev/null
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@@ -0,0 +1,9 @@
+Shared-edge supplier and completed entrance banks,2026. Apache-2.0.
+Developed with OpenAI model assistance. No personal byline is added.
+
+This contribution composes the pinnedPR181complex supplier and the actual
+PR168v4bit word with completed entrance banks and recomputes the full paid
+finite bound. All upstream author attribution and assistance disclosures
+remain in UPSTREAM-NOTICE.md and the immutable archive. Copied source files
+retain their author notices. No global optimality or exclusive priority
+is claimed; the inherited all-size interfaces remain explicit.
diff --git a/research/coordinated-frames-and-entrance-banks/PROOF.md b/research/coordinated-frames-and-entrance-banks/PROOF.md
new file mode 100644
index 0000000000..e9f66a0687
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/PROOF.md
@@ -0,0 +1,54 @@
+# Paid composition above the shared-edge local circuit
+
+## Exact claim
+
+Under the retained all-size interfaces, T(n)=O(n(log n)^(1−κ)) with
+
+    κ=330942774629799/500000000000000000 = 0.000661885549259598.
+
+The exact prerequisite is PR181 at7fb2194801e3a10f54772c7f0d5505035a4fc0ad, incorporating PR168 v4 at4a3c769e5c5430e7114c4d3e099ff34664677f17 and the PR175 compiler overlay. No new all-size oracle or strengthened hypothesis is inserted.
+
+## Stronger complex supplier
+
+PR181 constructs the local sumsA[1,0],A[1,1],A[2,1] from direction-zero edge sums already used by channelG[1,2,0]. Its local word has29 additions instead of27 but admits11364 carrier arcs instead of10704. It retains the pair/triple/all-but-one modules and signed local decoder. This trades a larger scalar operation count for fewer auxiliary roles; the paid inventory, not the addition count alone, determines its value.
+
+Our final rematching changes 2297 donor identities versus the published PR181 matching, with all 2310 recipients retained. It legally moves 1049 early compensation reads to first recipient use. Eleven accepted arbitrary-subspace meet cuts produce 243 distinct changed operation frames (247 selection events overlap). Of these, 216 newly differ from compiler frames and 27 modify already changed frames. The combined word has two additional terminal sinks, 46 instead of 44. These counts come from recipient-keyed donor/read comparisons and exact final-frame comparisons, not list positions or an intermediate matching. The selected graph, frames, carrier matching, gauges, compensated handoffs and actual terminal targets are frozen together. The package's signed physical executor checks exact adjoints, every physical frame inclusion, all target reads, both shear signs and literal reverses. It uses the signed-destination terminal checker; the older all-ca=1 sink gate is not valid for this word.
+
+The final complete complex inventory is
+
+    m=66, W=13326, rank=878196, deficit=1320,
+    physical auxiliaries=10686, scalar reserve=13042,
+    compensated handoffs=2310, terminal sinks=46, largest child=20.
+
+The terminal executor checks all13326 modified source/target/dirty columns and restores all10686 dirty registers, for both directions. Removing a pivot write, deleting an ancestor dirty response or omitting the initial target shears is rejected. Handoffs and terminal sinks stay distinct, and the retained full scalar reserve pays the literal signed word.
+
+The exact complete moment gives b=662323931899051/1000000000000000000. The actual full child histogram and outward rational bounds are in certificate.json. Numeric search scores or older frame-cut gains are not added to b.
+
+## Completed entrance-bank bit supplier
+
+Retain the unchanged PR168 v4 bit scalar word, partner deliveries,1760 compensated aliases and all original frame/prime witnesses. Verify every one of its21548 formal columns over F2 and over its defining integer decoder, restoring every source and arbitrary dirty register. The integer decoder is not asserted to be an integer identity.
+
+Bank whole chronological physical chains. The2200 surviving rank20 entrances leave four-coordinate cores, and eighteen fill a72-dimensional dirty bank. The15828 other physical chains use three24-coordinate cores. The1760 rank21 recipients are internal splices, never a second bank entrance. Nine copies make each stage-private allocation integral:1100 selected banks and47484 undeferred banks per stage. Total stock9W is177432 including source/target roles.
+
+For disjoint idempotentsP,Q, the map S_P(x,y)=(x−Px+Py,y−Py+Px) satisfies S_PS_Q=S_(P+Q), by direct expansion. A partition summing toI therefore composes to a full swap on arbitrary incoming contents. Conjugation by one common invertible weighted chart telescopes. BANK-PROOF.md supplies the full algebra and concrete partitions; the verifier checks every integer/F2 address column, inverse and omitted/repeated-block control. This reuses the established completed-bank endpoint with the inherited routing/chart contracts.
+
+The normalized complete bit inventory is
+
+    W_B=59144/3, rank_B=1417520, deficit_B=1936, largest child22.
+
+Only2200 rank60 exterior children disappear. Every internal increment, source/target connector, center copy and bad-class fallback remains paid. The unchanged used-frame Gram/factor witnesses reproduce exactly; coordinate bank projectors introduce no new denominators. Prime-witness paths are normalized toPOSIX separators, and only the portable checker self-hash changes in the frozen witness metadata.
+
+Independent base4/3 and base2 rational interval engines certify coarsea0=665481166955987/10^18 with bad fraction10^−16 and fallback32·72² per edge. The old ordinary wrapper usesa_old=384599/10^10 and least payable atomθ=665064157585290854737/10^24. The effective bit saving is
+
+    A_B=(1−θ)a0+θa_old
+       =665064157585290854736467094714610539581/10^42.
+
+Both A_B<θ<1−A_B hold strictly. The preceding atom grid and succeeding coarse grid are rejected for this fixed envelope.
+
+## Composition and sufficient scope
+
+Withβ=η=10^−24 andζ=10^−30, takea=min(A_B,(1−β)b−ζ), q=a(1−2η), and the greatest10^−18 grid point strictly below(1−η)q/(1+q). Here A_B>b, so the complex side binds. The complete scalar, group-cardinality, router, semantic, row and paid-layout charges are rederived. Full group cardinality is retained outside the normalized moments; neither aliases nor bank sharing justify discarding the scalar stock or external reserves9909 and252.
+
+The source-normalization check preserves the original PR14147-slack arithmetic body apart from its existing bridge plumbing and explicit supplier guard. All47 constraints and seven margins are strict;21 assembly mutations, including the succeedingκ grid, reject. A separate original-body calculation also reproduces the result. These grid exclusions establish no global optimum.
+
+The final wrapper reconstructs a fresh immutable prerequisite and replays the entire combined finite certificate. It checks source hashes before and after and rejects unpinned extra upload files. General tensor/Clifford realization, uniform weighted compilation, common generic bases, completed-core/global routing, restored borrowed rows, layout applicability, prime supply, precision/recovery, uniform setup and analytic fixed-tape transfer remain the explicit inherited interfaces. This sufficient conditional composition does not prove those all-size foundations or claim a new Lean replay.
diff --git a/research/coordinated-frames-and-entrance-banks/README.md b/research/coordinated-frames-and-entrance-banks/README.md
new file mode 100644
index 0000000000..fdefcc6624
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/README.md
@@ -0,0 +1,17 @@
+# Shared-edge supplier and completed entrance banks
+
+The complete paid finite composition certifies κ=330942774629799/500000000000000000, or 0.000661885549259598, under the retained Beyond n log n interfaces. This combines the stronger PR181 complex construction with the independently checked completed entrance-bank bit supplier. The complex branch binds again.
+
+Run Python 3.11+ with assertions enabled:
+
+```
+python -B research/coordinated-frames-and-entrance-banks/verify.py
+```
+
+No network, Git, credentials or third-party packages are required. The verifier checks every upload file and the three immutable PR181 archive parts, reconstructs a temporary clean source tree, verifies its1,699-file prerequisite closure, and replays the complete words, frames, exact moments, prime witnesses and all 47 inequalities. It compares the derived result with certificate.json. Normal verification changes only its temporary directory; --write is an explicit authoring mode.
+
+All archive parts are below GitHub's25 MB browser limit. Upload this entire folder at research/coordinated-frames-and-entrance-banks, keeping every file. The archive hash and exact upstream commit7fb2194801e3a10f54772c7f0d5505035a4fc0ad are in BASELINE.json. FILES.json pins this upload; SOURCE.json pins the prerequisite. The source remainsLF throughout the archive, independent of a later Git checkout's autocrlf setting.
+
+The verified gain over PR181's statedκ is about0.41035%. This is a conditional finite witness, not an unconditional integer-multiplication theorem, global optimum or new Lean-certified bound. General compiler/Clifford, common charts, restored rows, routing, paid layout, precision/recovery, prime supply, uniform setup and analytic fixed-tape interfaces remain the inherited assumptions.
+
+See PROOF.md, BANK-PROOF.md and RESEARCH-IDEAS.md. Upstream attribution/disclosures are retained in NOTICE, UPSTREAM-NOTICE.md, UPSTREAM-PROOF.md and the immutable public archive. Developed with OpenAI model assistance; no personal byline is added.
diff --git a/research/coordinated-frames-and-entrance-banks/RESEARCH-IDEAS.md b/research/coordinated-frames-and-entrance-banks/RESEARCH-IDEAS.md
new file mode 100644
index 0000000000..130a1f723b
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/RESEARCH-IDEAS.md
@@ -0,0 +1,13 @@
+# Research mechanisms actually used and tested
+
+The original independent crossover ideas were worked through, not merely collected as citations.
+
+The completed-bank mechanism supplies the new result here:18×4 and3×24 disjoint projector banks raise the bit supplier above the newPR181complex supplier, unlocking the composedκ. It treats compensated donor/recipient pairs as one whole physical chain. The source-birth nullity21 idea failed the actual dirty-read chronology and is excluded.
+
+Global arbitrary-subspace graph-cut optimization produced a verified rank8 four-operation contraction and enabled further endpoint descent on the priorPR179graph. That fallback was fully replayed atκ=.000655930202792126, but was superseded whenPR181 appeared. Its operationIDs and gains are not transplanted or added to the new graph. On the actual PR181 graph, the same method accepted eleven coordinated meet cuts with 243 distinct final operation-frame changes. These are combined with the 2297-donor rematch and 1049 legally retimed reads. All 46 actual target/pivot chains and the complete new moment are checked together; no candidate gains are added.
+
+The shared-total/rank2 local reflection was tested on the prior actual producer: a five-edge matching saves230 scalar additions, but260 read roots fail the current frames. A rank2 local Fourier map does not establish physical routing for the larger-address shared total. This unpaid route is excluded. PR181's shared-edge reuse is a different concrete mechanism and is already certified by the selected word.
+
+Integral harmonic/Kneser factorization, Parseval phase maps and fractional-bank configurations were considered as representation/packing crossovers. Their valid algebraic identities do not automatically shrink the actual admissible geometric frames. Only the explicitly realized homogeneous bank partitions are used in the numerical certificate. Additional gauges, local associations and frame refinements count only when their entire final words and paid profiles pass.
+
+No historical improvement is summed with a public exponent. Every submitted number is recomputed from the frozen combined suppliers and the complete47-constraint composition.
diff --git a/research/coordinated-frames-and-entrance-banks/RESEARCH-REVIEW.md b/research/coordinated-frames-and-entrance-banks/RESEARCH-REVIEW.md
new file mode 100644
index 0000000000..5cce174cf9
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/RESEARCH-REVIEW.md
@@ -0,0 +1,37 @@
+# Research ideas and exact disposition, 2026-10-09
+
+The selected finite composition uses the pinned PR181 physical word, our legal compensated rematching and operation-frame cuts, its 46 checked terminal sinks, and the completed entrance banks on the retained bit word. Independent rational arithmetic reproduces complex saving b=662323931899051/10^18 and final balanced κ=330942774629799/(5·10^17)=0.000661885549259598. Both independent interval engines accept b and reject b+10^-18. The original PR141 balanced arithmetic body passes all47 strict constraints and all7 margins; the next κ grid point and the three old-layout controls fail. No component gains are added.
+
+## Changes actually used
+
+PR181 supplies the newer shared-edge scalar construction: h22, v1320, c21249, q3157, matched11364, scalar carrier reserve R13042 and M32971. This is an upstream improvement and is not claimed as our discovery. We retain its complete scalar reserve after physical aliases and terminal substitution.
+
+The actual matching schema is [donor,recipient,read_operation]. Comparing maps keyed by recipient against pinned PR181 gives2310 retained recipients,2297 changed donor identities and1049 changed read-operation fields. These are separate counts, not tuple inequality or sorted-list positional differences. An earlier draft's419 donor count is stale and does not describe this final witness; the parent was notified. The rematch audit independently reports2297. Legal reads retain recipient birth, donor death and exact compensation.
+
+The final fixed-subspace cut search accepted11 meet cuts with247 selection events. Some events overlap; comparing final operation bases to the published baseline gives243 distinct changed operation frames. Actual value spans, both operand chains, donor gauges and target-read caps were checked. These are finite legal improvements, not a proof of global optimality. The combined moment prices the cuts and rematching together.
+
+There are46 disjoint compatible terminal sinks, verified on every original formal source, target and arbitrary dirty column. The physical count is13042−2310−46=10686, W=13326, rank mass878196, deficit1320, maximum child20. The terminal sandwich saves scalar gates while the original scalar reserve remains charged. The parent aggregate owns the final complete word, prime, source and root replay.
+
+The bit contribution has19788 logical roles,1760 compensated splices,18028 physical chains,2200 genuine rank20 entrance gauges and15828 undeferred chains. Width72 banks tile18 four-coordinate cores or3 twenty-four-coordinate cores. Only the2200 rank60 exteriors disappear. W=59144/3, rank1417520, deficit1936, maximum child22. Nine physical replicas provide integral stage-private bank counts1100 and47484; factor3 is only moment normalization. Effective paid bit saving is665064157585290854736467094714610539581/10^42, larger than the complex supplier. Complete literal projector endpoints and inverses were checked over F2 and Z on all144 columns; missing and duplicate blocks are rejected. The endpoint theorem works for arbitrary dirty contents and one common invertible weighted chart. It introduces no fresh endpoint assumption.
+
+## Five crossover routes
+
+1. Whole-network directed cuts were applied successfully to the actual newest word, with the eleven exact cuts above. Historical cut witnesses were used as method guidance, not transplanted by operation ID.
+
+2. Matching-selected common totals and the explicit rank-two face decoder remain a concrete different-construction route. On actual181, a five-edge matching gives10 changed modes and230 fewer scalar additions (21249→21019), but260 target-side reads violate current frame containment. The total value span is19 rather than17 and its own-cube binary interaction has rank2. The specified face correction is an exact decoder identity; it does not by itself implement the missing joint read carrier. No gain from this route enters κ.
+
+3. Rational configuration banks provide a universal sufficient endpoint construction when actual nullities differ. The current rank20/nullity4 and undeferred/nullity24 families already tile homogeneously. We used their concrete disjoint projectors and exact integral replication; a different LP alone yields no further gain here.
+
+4. The integral harmonic residual factors165 scalar channels through55 all-ring exact features with low coefficients. This is useful representation mathematics. It is not a tensor-address width55 claim. No current paid carrier/read construction eliminates the observed19/20-to18 cap mismatch, so no feature-compression saving is priced.
+
+5. Parseval phase correction for binary Gram-resolving maps gives a literal known translation correction and specified Fourier ranks. The characteristic-two additive block-swap identity is a separate exact result. Neither identity alone provides cheaper physical bank stock or an overlapping-code endpoint in this supplier. No unpriced overlapping-code construction is substituted.
+
+## Other bounded discovery
+
+Saved mixed-gap complex modules and reassociations were rebuilt through the unchanged pinned producer/matching machinery. An historical role11 module improved fd25 but was superseded by the later public role10 module. Forty-one output-fold variants, the saved LR/reassociation families and bounded role9 annealing searches produced no newer accepted supplier. This is a search report, not an impossibility theorem.
+
+On181,27 single local A/G/F neighbors and4 combined edge-sharing variants failed to lower logical R13042. Two physically valid alternatives were priced exactly, including their sinks, but yielded weaker moments than181 and were excluded. Additional all-G edge sharing reduces scalar additions while losing too many legal aliases.
+
+Source births cannot be reclassified as exterior entrance gauges in the retained endpoint: every one of1320 exact dirty-response rows has three centre coefficients and a nonzero net action on its own target. No source/nullity21 shortcut is assumed.
+
+All inherited all-input weighted compiler, routing/stage cover, precision/recovery, prime supply, uniform setup and all-size analytic/fixed-tape contracts remain separately stated conditions. The sufficient result is the fully paid conditional finite composition. The standalone review imports no modified parent certificate for its κ proof. It loads the retained original47-constraint body with an independently computed semantic/row bridge, validates every selected-cache input hash, and retains the original body and bank proof hashes in receipt.json. All owned discovery and review subprocesses have exited; shared persistent services were untouched.
diff --git a/research/coordinated-frames-and-entrance-banks/SOURCE.json b/research/coordinated-frames-and-entrance-banks/SOURCE.json
new file mode 100644
index 0000000000..5d25c4238c
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/SOURCE.json
@@ -0,0 +1,1728 @@
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+    "tests/test_skip_suffix.py": "4aa6334e4b466f3006b8b961f8f3ff147a347fd2b6defc4d88d615d3a418a4ed",
+    "tests/test_small_block_witness.py": "8025d87026ea15763ecdd64811dffdc52eca1f0e67ee9bee4e9a6d6c125e4ea3",
+    "tests/test_sparse_fusion.py": "ad078502dbbcff04147665aed61609b31dc09a8e9a404d4e344aa5d462f568eb",
+    "tests/test_split_skip.py": "6d274ff66bb5465fc8a58958ecead7d27ec282717f9445ee1a2b02ef72731c50",
+    "tests/test_stage_pair.py": "746aaf91e95495644e0e89f81955d02a3a0d029934090e07a3607c207ab0f2b1",
+    "tests/test_stronger_screens.py": "059e3535f9e610aafbd159897516c299d8d15b9a55806f5197042e19bb4b3e55",
+    "tests/test_subset_core_family.py": "3d776e0d60aa226059a6dc6705ccef1b46f941d3aacd22fd60b5025fa7e45f90",
+    "tests/test_ternary_patch.py": "19e7a05d7f9b3cc7d32336173ce456fc63ff85ff231dbae5a89dcbf3b1b0a684",
+    "tests/test_ternary_side.py": "05b9b82f5bb308010d7663841a90f430326891f601a92af5602ac88a8109f26b",
+    "tests/test_tune_routing.py": "004106143616a82be9ba6cf7525ba1772328a47e75aae6047a0559afb6038a30",
+    "upstream/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "upstream/README.md": "8d1ceff898a4e0e681679f91c4def485820be9e99f04e4861f2ce601a364723e",
+    "upstream/build/figures/multiplication-flow.tex": "3654d2728b0c98919d61760e1802804ed616546a747f9167a93e17bb8697f320",
+    "upstream/build/figures/proof-map.tex": "76d4deacf07bea355412abcfd798c7cbe6dcccef29a4a420a9e2a7b158f23848",
+    "upstream/build/main.tex": "42f0f506976bc2084374a71e97edc1549689a7accf35dda4db2a18af4f13c1aa",
+    "upstream/build/references.bib": "33cc2df13ccf85330849e8fb34d4ac7613b23f625787457a5b7b9aa6d7369476",
+    "upstream/build/sections/00-introduction.tex": "98721f03682ccf1cc7d0850b07adabaa2d8f3afe552e3861753eba9d284d0da0",
+    "upstream/build/sections/01-history.tex": "798956c8150c31e3e2f5f29c47110d25ce619faad1374f4d3a0f15fa3ee33ead",
+    "upstream/build/sections/02-streams.tex": "606c80db61cad13aa0c3b6060dc4b88dbbe7ac60cdd831aa93f28d908fd09dab",
+    "upstream/build/sections/03-motifs.tex": "ed76761194988a01af77ead0ea6469cb5dcd29690ab985a0b5a4006b0f2b69bb",
+    "upstream/build/sections/04-swap.tex": "412b170ccaab38dba96e95e3cbfae09f4876a9818f8d278776a746b0f8044906",
+    "upstream/build/sections/05-layers.tex": "20cfc8d12099d4e4ed9e3e7eeb6162edd9b6e0e076f24693364cd24377252594",
+    "upstream/build/sections/06-transforms.tex": "924316e4bc905066281d8f9e8746a49d8829aeb393688608deb1e61c3c87853c",
+    "upstream/build/sections/07-resampling.tex": "8a6b91670f9163afb54892489fa2210c77fe1b058f0325ec7d18b44b94be6e2c",
+    "upstream/build/sections/08-assembly.tex": "763d7945b1e1ec4ebae9b799bcefcc45fa9bae6e2ffad799868a9c3d513dbd4a",
+    "upstream/build/sections/09-exact-arithmetic.tex": "5714e5d0656c78c6a1fde8d01fd11f1ba856a7ea94149b283bd9a50fdbf516c3",
+    "upstream/build/sections/10-transposition.tex": "c14ad8d56112f6578678a58211bc8d63c3a4b60e55078ac73bc38f7348e713cb",
+    "upstream/build/sections/11-grouped-rectangles.tex": "6525cd0a01980e56accfeaf06e9d875bc699197ba42c6e22b69435e352275174",
+    "upstream/formalization.yaml": "2dcbd0d6e6a22475f53d49bfdfcac9b6d2b470653c332a9f56522f5be166edb9",
+    "upstream/manifest.json": "16ab13971694cabab09440c47ebf2fc0496a9d35fc3af33950fda3498f8e2392"
+  },
+  "inputs": [
+    {
+      "commit": "4dbc2af5f223a0fbf22542c9d6c3f71442b6a3a6",
+      "pr": 166
+    }
+  ],
+  "main_commit": "d1d6c070f5a8c684727ee7ec35d930f9ebfa9758",
+  "prerequisite": {
+    "commit": "4a3c769e5c5430e7114c4d3e099ff34664677f17",
+    "pr": 168,
+    "repository": "https://github.com/CrocSwap/integer-mult-bounds"
+  },
+  "scope": "Complete source closure: exact PR168 v4 prerequisite, four-file PR175 compiler overlay, selected finite words and paid arithmetic. Retained all-size contracts remain assumptions.",
+  "status": "Frozen selected conditional finite witness",
+  "verification_overlay": {
+    "commit": "f1081fabbfadaaaaf7292a40d428dbbb865ce920",
+    "files": {
+      "certificates/joint-dual-kappa.json": "4044f15abf49b1551193891836e2d20975b8246acc888180af47266c64f77af0",
+      "certificates/joint-dual-validation.json": "a7834f2dda430b4d28337faf86b74b2ea1259812bc4169520e141efb835ecd9d",
+      "research/joint-dual/SOURCE.json": "c8649e811e43b33a7281746fd61e9adbd1370690ef7c2ec114a6f2d61016704f",
+      "scripts/experiments/joint_dual_reclaim_compiler.py": "f0696953d057239ed5a75e28d857b814d43ca43deaaf7b7b8fd191aa115112ff"
+    },
+    "pr": 175
+  }
+}
diff --git a/research/coordinated-frames-and-entrance-banks/UPSTREAM-NOTICE.md b/research/coordinated-frames-and-entrance-banks/UPSTREAM-NOTICE.md
new file mode 100644
index 0000000000..87dc993cb0
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/UPSTREAM-NOTICE.md
@@ -0,0 +1,54 @@
+Prepared by Chafik Boukhalfa with substantial Anthropic Claude and OpenAI
+Codex assistance. Apache-2.0. Inherited files retain their original notices
+and assistance disclosures. This package claims no exclusive priority over
+concurrent work.
+
+Retained sources and lineage:
+- icekylinx: PR144 paired cubes, signed decoder, shared-core accounting,
+  original compiler, configurable local channels and finite bridge;
+  PR130/generalized Clifford frames.
+- eumemic: PR117 module infrastructure; PR155/157 scalar modules, physical
+  frames/reuse and partner-pair bit word; PR161 output merging and searched
+  p12 module; PR168 annealed pair, triple and all-but-one modules on both
+  words, nested-prefix schedules, the L1 local circuit this package varies,
+  fused outputs, bundle descent, both physical layers, the terminal-sink
+  gate and the physical bit word, pinned at
+  4a3c769e5c5430e7114c4d3e099ff34664677f17. Their Anthropic Claude and
+  OpenAI Codex disclosures are retained. PR131/143 supply frame/reuse and
+  late-read lineage; PR154 supplies isolated snapshot verification.
+- DaysSky: PR162 generalized carrier extension (`compile_closure` and the
+  extension rule reused here on the new graph) and full-intersection
+  compiler lineage. The selected graph retains the PR168 nested schedule.
+- jamesyc / James Chang: PR124 compensated birth cuts, PR34 balanced
+  assembly, and PR166 terminal-output accumulation and exact simultaneous
+  target-chain checking. The PR166 proof is retained in references/pr166.
+- GamingPuzzled / Joel Pulikkan: PR160 searched-bit-module composition.
+- gupt1156 PR148 and Abhinav Ramachandran/geckods PR158 (Hermes assistance):
+  paid atom-toll re-instantiation, including PR158 at
+  6e086603f936dbb95c6d7171132ed943cc024db9.
+- jacklightChen / Zhihao Chen: PR23/29 ordinary supplier; Rohan Arun
+  (rohanarun): PR100/103 paid positional layout and deferred signed words.
+- Earlier bit lineage retained by the prerequisite: Zhihao Chen PR97,
+  Swapnil Jain's round-seven checkers, an664 PR128, DaysSky PR150, hpst3r
+  PR147, SovereignSteak PR122, and the source-notice contributors.
+
+This package contributes a coordinate search over PR144's configurable
+local channel circuit on the PR168 v4 word, selecting direction-0 edge
+sums for A[1,0], A[1,1] and A[2,1] so that the carrier closure gains 660
+arcs; our own legal-edge Hopcroft-Karp carrier matching with PR162's
+extension; endpoint frame descent and compensated handoffs for the new
+graph; independent complete admission of the resulting physical word and
+its 44 terminal deletions; independent physical-bit admission and exact
+prime witnesses for every used bit frame; and the fully paid balanced
+composition with finer positive backoffs. Scalar virtual stock, physical
+handoffs, terminal removals and every external row reserve remain
+distinct. The complete 47-inequality body is source-normalized to PR141.
+Prerequisite constructions, the configurable circuit itself and the
+terminal-compilation concept are not claimed as original here.
+
+Verification infrastructure: Chafik Boukhalfa, PR175 at
+f1081fabbfadaaaaf7292a40d428dbbb865ce920, with substantial OpenAI Codex
+assistance. Its ordered retired-register compiler buckets and three
+provenance/certificate updates are carried byte for byte. The PR57 compiler
+and all existing notices remain. This overlay changes no mathematical
+field or emitted word and is included in the complete native verification gate.
diff --git a/research/coordinated-frames-and-entrance-banks/UPSTREAM-PROOF.md b/research/coordinated-frames-and-entrance-banks/UPSTREAM-PROOF.md
new file mode 100644
index 0000000000..1ac53947fd
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/UPSTREAM-PROOF.md
@@ -0,0 +1,68 @@
+# Construction and retained proof contracts
+
+The exact prerequisite is PR168 at `4a3c769e5c5430e7114c4d3e099ff34664677f17`. PR144 supplies the paired-coordinate/shared-core framework and the configurable local channel circuit; PR161/168 supply the annealed modules, fused outputs, scalar words, L1 bit association, nested schedules and the bit physical layer; PR162 supplies the carrier extension rule; PR166 supplies terminal-output accumulation. This package changes the complex local circuit and rederives the complete finite composition under the inherited all-size interfaces.
+
+## Shared-edge local circuit
+
+Each of the 165 cubes computes the same 13 local outputs: the six single-bit sums $A[i,a]$ of the four ports with bit $i=a$, the six signed channels $G[j,k,m]$ and $F=A[f,0]+A[f,1]$. PR144's circuit lets every $A[i,a]$ be assembled from two edge sums along a direction $d\ne i$ or from two face diagonals, and every $G[j,k,m]$ from edge differences along the third direction, long-diagonal differences or fixed-coordinate differences. PR168's L1 circuit assembles $A[1,\cdot]$ from direction-2 edges, $A[2,\cdot]$ from direction-1 edges and $G[1,2,0]$ from direction-0 edge differences.
+
+This package assembles $A[1,0]$, $A[1,1]$ and $A[2,1]$ from direction-0 edges. The four direction-0 edge sums of a cube are then shared by three $A$ channels and the $G[1,2,0]$ channel, while the four direction-2 edge sums disappear and the direction-1 edge sums remain shared by $A[2,0]$ and $G[0,2,\cdot]$. The circuit has 29 additions per cube instead of 27. Every addition still has support-disjoint operands and every root support equals its contract; the graph builder and the independent scalar audit check this. The modules, the `f8:00111100` fold order and the nested schedule of PR168 are unchanged.
+
+The new signed graph has c=21,249, q=3,157 and M=32,971 mixer operations. Carrier arcs are chosen by our own legal-edge Hopcroft–Karp matching on the coordinate-padded frames and extended by PR162's greedy rule; the compiled closure is acyclic and span-contained. There are 11,364 arcs instead of 10,704, so the complete virtual auxiliary stock is R_scalar=c+q−11,364=13,042. Gauges are selected by PR144's chronological greedy rule: 2,310 roles of rank 18.
+
+## Physical word, handoffs and frames
+
+Operation frames are descended from the compiler frames by exact endpoint moves of single operations, equal-frame components and connected bundles: each move takes the join of incoming frames and value span or the meet of outgoing frames. 12,869 operations differ from the compiler frames. The 2,310 compensated handoffs pair dead ungauged non-root donors, whose last frames lie in the recipient gauge, with gauged recipients read at their first-use deadline; they leave 10,732 roles before terminal deletion. Only the explicit final frames and pairs are mathematical inputs.
+
+The verifier reconstructs every signed source/target identity, binary value span, carrier, center closure, physical role chain and actual target read. Compensation reads the current inherited dirty value with independently derived exact adjoint coefficients. The original physical word is checked on all 13,372 formal source, target and dirty columns under both shear signs. Every signed gate a←c_a a+c_b b is undone by a←c_a a−c_a c_b b, in reverse chronology; gates whose carried operand is the minuend are compiled with the sign on the destination coefficient and are audited as such.
+
+## Terminal-output accumulation
+
+The terminal compiler selects 44 ungauged, destination-only, disjoint-root roles. Each has eight targets, rank-18 common read frame and coefficient 1/2. No removed role is a source, gauge, alias endpoint, retained consumer input or another root. All sink writes are additive in the final frame. Target groups are disjoint, and each pivot's first old nonzero-frame target read follows its sink's last write. Of the 165 disjoint roots, 119 have retained producer consumers, 13 are written in the center phase and 4 have no uncorrected pivot.
+
+Retain every surviving coordinate's original dirty-response correction, including response through a deleted sink. At the common zero frame after the center phase, subtract each pivot from its seven other targets. Replace r←r+c b by y_p←y_p+c b/2. After the last write, add y_p back to those seven targets. Remove the old sink read and register. The independent target checker includes all intervening compensation reads and every new target-frame event; source-role frames at replaced writes remain as before.
+
+The modified executor checks all 1,320 source, 1,320 target and 10,688 dirty columns in both directions. An independently computed absolute residual coefficient bound of 397,506 makes 24-bit packed digits injective. The actual literal reverse word restores all retained dirty coordinates. Missing a pivot write, deleting an ancestor's original response and omitting the initial target shears are rejected. The scalar addition delta is −202, so the complete earlier virtual scalar charge remains conservative; the fixed odd divisor remains three.
+
+For each terminal sink, the pivot copies the old rank increments and the remaining seven targets enter the shared rank-18 frame. The sink's final rank-four increment and one target rank-18 increment disappear. The three-core histogram loses three children of rank 4 and three of rank 18; W loses one. The final profile is m=66, W=13,328, rank=878,328, deficit=1,320 and largest child 20. The exact stock identity is 13,042−10,688=2,310 handoffs+44 terminal sinks. Outward rational intervals prove b=132438677973/(2·10^14); its next 10^−15 grid point fails this fixed profile.
+
+## Physical bit word, exact decoder and prime range
+
+The unchanged bit inputs are PR168's graph, virtual profile, physical scalar word, rational frame table and partner chronology at the prerequisite commit. Admission reads these frozen files; it never regenerates them. The word uses an all-but-one module with n=10 and an annealed pair module; the local L1 association replaces three face-channel edge associations with long diagonals while preserving the mod-2 supports. Local geometry has h=24 and G=I−J/9. Source lines are spanned by chi_S; target caps are ker(3 chi_T−1).
+
+Each source belongs to one partner pair. A carrier XORs its passive partner in their rank-two span, delivers at the specified common level-two cap/root and restores its source value before cleanup. All complete source chains and original delivery anchors are checked. There are 2,884 operation frames differing from the original compiler frames, including 264 enlargements. Exact rational checks prove each contains its full node value span and is nondegenerate. The 1,760 aliases join dead ungauged non-root donors to untouched gauged recipients. Each current inherited dirty value is read at its explicit deadline, and all physical role chains and retimed target reads are recounted.
+
+The final bit profile has virtual R=19,788, physical R=18,028, W=21,548, rank=1,549,520, deficit=1,936 and largest child 60. Independent literal execution checks every one of the 21,548 formal columns over F2, restores every source/dirty coordinate and verifies the identity on targets. Separate integer execution verifies the defining integer decoder, whose F2 reduction is the identity; it does not assert an integer identity. Missing compensation, missing partner delivery, stale dirty reads and a zero operation frame are rejected.
+
+Exact Bareiss witnesses cover all 24,601 used bit frame IDs and distinct integer bases B. Each nonzero d=det(9BGB^t) factors into powers of primes at most 31 times a positive residual below 2^80; every factor identity is checked. Determinants reach 118 bits. Every prime q>2^80 therefore avoids every determinant, preserving all Gram inverses/projectors over Z/q^w and the original fixed-prime range. Sources, gauges, roots, operation frames, partner mixes/deliveries and ambient frames are included. Zero determinants, incorrect factors and oversized residuals are rejected.
+
+Every ideal bit edge pays the full 32m^2 fallback on fraction 10^−16, and 2m^3/2^80<10^−16. The contaminated moment and rank mass contract. Outward rational intervals certify a0=6600253783597/10^16; the next 10^−18 grid point fails this fixed envelope. The true bad fraction may be smaller; no optimality claim is inferred.
+
+Use the retained standard ordinary wrapper with a_old=384599/10^10. Select the first 10^−24 grid point above a0/(1+a0−a_old): theta=329807692103940705443/(5·10^23). Its ordinary saving is A_B=(1−theta)a0+theta a_old. Both adapter and restored-row tolls are paid by A_B<theta<1−A_B; the previous grid point fails the strict lower condition. The full uniform weighted local-ring and restoration contracts remain inherited assumptions.
+
+## Full finite scalar, group, router and row bills
+
+Retain R_scalar=13,042 and M=32,971 despite physical handoffs and terminal deletion. The complete local scalar charge is
+
+    L=4(c+v)+10v+4hv+4h^2+8h+8+2h+8 R_scalar v(M+16)+32v
+     =4,543,086,018,280.
+
+Decoder denominators divide six, signed gate coefficients are ±1, dirty numerators use M+16 bits and 3q<2^15. The fixed odd divisor is three. Let V=|O(66,2)|, W=V·13,328, N=V·1,320 and s=V·878,328. Group cardinality cancels only in the normalized moment. It remains in K=3VL+8W+4N+8·66·R_scalar·V, G_route=64·67^3(K+1)(W+1)^2 and E=64(W+66+G_route+1)^3. The verifier rederives literal charges, B=s+E, C0=32·66·B^2, C1=1 and exact-child induction.
+
+The 66→20 halving depth is one. Full W has 2,159 bits. The separate reserves 9,909 and 252 give total row coefficient 12,320, polynomial degree 70,000, positive gap 224336/5 and suffix slope 280,000. Bit selector rows are borrowed and restored internally. Dropping any reserve, replacing full W by local W, charging only physical roles for the scalar word, omitting terminal removals or counting them as aliases is rejected.
+
+## Paid balanced transfer
+
+Use the PR34/RaD paid positional layout as applied to signed words in PR100/103 and retained in PR141. Both axis-major/group-major permutations are paid by the ordinary arbitrary-coordinate router. Active, spectator, scratch, phase, control and restored-row fields move together. The additional long-axis top bit is paid, with twiddle order and inverse alignment preserved. The geometric inequality remains strict. The application retains ordinary complete interchange/restoration and completed exact complex tensor contracts.
+
+Set b=132438677973/(2·10^14), beta=eta=10^−24, zeta=10^−30 and a=min(A_B,(1−beta)b−zeta). The bit branch binds: a=A_B. With q=a(1−2eta), choose
+
+    c=q+eta/4, epsilon=(1−eta)/(1+q), lambda'=1−q,
+    lambda=((1−a)+lambda')/2, g=epsilon q,
+    r=(g+1−epsilon)/2, delta=eta/8.
+
+The seven margins are 1−epsilon, a, g, a, min(1−epsilon−delta,r−delta), 1−epsilon−delta and epsilon; their minimum is g. The identities 1−epsilon−g=eta, 1−epsilon−r=eta/2 and 1−epsilon(1+c)=eta−epsilon eta/4 preserve strict gaps. All 47 inequalities remain, including epsilon−a>0. Source normalization proves the PR141 arithmetic body changes only bridge plumbing and the explicit guard a≤A_B.
+
+The certified point is kappa=659180578557293/10^18. Its next 10^−18 grid point fails only for these fixed supplier certificates and parameters. Matched comparisons change the positive outer backoffs while holding the same suppliers; the published parent certificate is reported separately.
+
+Finite checks establish supplied words, profiles, exact moments, prime witnesses, source closure, finite bills and arithmetic. General Clifford lifts, common generic bases, uniform weighted compilation, arbitrary-width routing, restored internal rows, layout applicability, prime supply, precision/recovery, uniform setup and analytic fixed-tape transfer remain inherited all-size contracts.
diff --git a/research/coordinated-frames-and-entrance-banks/arithmetic/balanced_shared.py b/research/coordinated-frames-and-entrance-banks/arithmetic/balanced_shared.py
new file mode 100644
index 0000000000..e94c7dcaa9
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/arithmetic/balanced_shared.py
@@ -0,0 +1,71 @@
+"""Paid balanced assembly with an explicit uniform-bit/shared-complex bridge.
+
+The complete 47-constraint arithmetic body is PR141 balanced_stopped.py,
+with only bridge plumbing and an actual-bit-supplier bound added. Original
+PR34/RaD/PR23, icekylinx and PR100/103 attribution is retained in the pinned
+source. It does not assert a finite old coarse profile for the uniform bit
+supplier. General cutoff/analytic/word contracts remain external.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from fractions import Fraction as Q
+from shared_bridge import rational, require, validate_shared_bridge
+
+def assembly(finite_bridge, complex_row, a_bit, kappa, beta=Q(1,20), h=Q(1,10**12),
+             a_complex=Q(717,10**7), *, original_prefix=False,
+             old_guard=False, old_exposures=False):
+    """Return 47 strict slacks and seven margins, or raise InvalidAssembly.
+
+    The three keyword switches are negative controls: they impose the old
+    prefix, nonlinear guard, or old separate movement charges at these SAME
+    balanced parameters. They do not silently optimize an alternative family.
+    """
+    f = validate_shared_bridge(finite_bridge, complex_row)
+    a, b, kappa, beta, h = map(rational, (a_bit,a_complex,kappa,beta,h))
+    require(a <= f['bit_uniform']['ordinary_saving'], 'transfer saving exceeds actual uniform bit supplier')
+    require(0 < h < Q(1,2) and kappa > 0, 'positive backoff and saving required')
+    tau, sigma = 1-a, 1-b
+    q = a*(1-2*h)
+    require(1+q != 0, 'undefined balanced epsilon')
+    lp, c, eps = 1-q, q+h/4, (1-h)/(1+q)
+    lam = (tau+lp)/2
+    G = eps*q
+    r, delta = (G+1-eps)/2, h/8
+    C1 = Q(19991,10000) if old_guard else Q(1)
+    margins = dict(g1=1-eps, g2=a, g3=G, g4=a,
+                   g5=min(1-eps-delta,r-delta), g6=1-eps-delta, g7=eps)
+    if original_prefix:
+        margins['g1'] = 1-eps*(1+c)
+    if old_exposures:
+        margins.update(g2=eps*c*a, g4=a*(1-eps))
+    internal = tau+(1-beta)*max(sigma-tau,Q(0))
+    leaf = sigma+beta*(1-sigma)
+    slacks = dict(a_positive=a, a_below_b=b-a, b_below_one_over32=Q(1,32)-b,
+        beta_positive=beta, beta_below_one=1-beta, phase_leaf_above_bit=(1-beta)*b-a,
+        q_positive=q, q_below_internal=1-internal-q, q_below_leaf=1-leaf-q,
+        c_positive=c, c_below_one=1-c, q_below_reservations=c-q,
+        lambda_above_tau=lam-tau, lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal, lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf, compact_reservations=lp-(1-c), lambda_prime_below_one=q,
+        epsilon_positive=eps, epsilon_below_one=1-eps, guard_width=1-eps*C1,
+        K_geometry=1-eps*(1+c), K_dominates_log=eps*c,
+        record_suffix=1-eps, phase_local=1-eps-delta, phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r, cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps, alpha_positive=r, alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r, delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta, short_record_fallback=eps-a,
+        small_field_exposure=1-eps-G, artificial_boundary=8-eps+r-delta-G,
+        literal_scalar_guard=Q(f['semantic']['strict_literal_gap']),
+        row_product_gap=f['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa': value-kappa for name,value in margins.items()})
+    require(len(slacks) == 47 and len(margins) == 7, 'constraint list incomplete')
+    failed = {name:str(value) for name,value in slacks.items() if value <= 0}
+    require(not failed, str(failed))
+    require(min(margins.values()) == G, 'unexpected controlling margin')
+    require(1-eps-G == h and 1-eps-r == h/2, 'balanced identities failed')
+    require(1-eps*(1+c) == h-eps*h/4, 'geometric identity failed')
+    parameters = dict(a_bit=a, a_complex=b, tau=tau, sigma=sigma, beta=beta,h=h,
+        q=q,c=c,epsilon=eps,lambda_=lam,lambda_prime=lp,alpha_squared_power=r,
+        delta=delta,C0=f['semantic']['C0'],C1=C1,kappa=kappa)
+    return dict(parameters=parameters,constraints=slacks,margins=margins,
+        minimum_margin=G,absorption_gap=G-kappa,scoped_limit=a/(1+a),
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c),finite_bridge=f)
diff --git a/research/coordinated-frames-and-entrance-banks/arithmetic/certificate.py b/research/coordinated-frames-and-entrance-banks/arithmetic/certificate.py
new file mode 100644
index 0000000000..74271389e8
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/arithmetic/certificate.py
@@ -0,0 +1,168 @@
+#!/usr/bin/env python3
+"""Exact paid bridge, unchanged balanced arithmetic body, and strict controls.
+
+Finite bridge formulas: icekylinx PR144/130, eumemic PR152, Apache-2.0.
+Balanced layout: PR23/29 Zhihao Chen, RaD, James Chang PR34, and Rohan
+Arun PR100/103. This adapter preserves their separately stated contracts.
+Adapted supplier composition; upstream attribution is in UPSTREAM-NOTICE.md.
+"""
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import prod
+from pathlib import Path
+import copy
+import gzip
+import importlib.util
+import json
+import sys
+
+HERE=Path(__file__).resolve().parent;PACKAGE=HERE.parent;ROOT=HERE.parents[2]
+sys.path.insert(0,str(HERE));sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+from shared_bridge import require,halving,validate_shared_bridge,nofloat
+from balanced_shared import assembly
+
+AC=Q('662323931899051/1000000000000000000')
+COARSE=Q('665481166955987/1000000000000000000')
+ATOM=Q('665064157585290854737/1000000000000000000000000')
+OLD=Q(384599,10**10)
+AB=(1-ATOM)*COARSE+ATOM*OLD
+BETA=ETA=Q(1,10**24)
+WEAKENING=Q(1,10**30)
+GRID=10**18
+
+
+def js(x):
+    if isinstance(x,Q):return str(x)
+    if isinstance(x,dict):return {str(k):js(v) for k,v in x.items()}
+    if isinstance(x,(tuple,list)):return [js(v) for v in x]
+    return x
+
+
+def bridge(row,coarse=COARSE,atom=ATOM):
+    """Retained finite formulas with the actual new local inventory supplied."""
+    h,v,R,c,M=(row[k] for k in ('h','v','R','c','total_M_operations'))
+    scalar_R=row['scalar_role_reserve']
+    m=3*h;half=m//2
+    V=2**(m-1+(half-1)**2)*prod(2**(2*i)-1 for i in range(1,half))
+    W=V*(2*v+R);s=V*row['rank_per_vertex'];N=V*v
+    r=max(int(k) for k,n in row['child_histogram'].items() if int(k) and n)
+    local=4*(c+v)+10*v+4*h*v+4*h*h+8*h+8+2*h+8*scalar_R*v*(M+16)+32*v
+    logical=3*V*local+8*W+4*N+8*m*scalar_R*V
+    G=64*(m+1)**3*(logical+1)*(W+1)**2
+    E=64*(W+m+G+1)**3;charge=2*G*W*W+8*s+4*W+4+32*m
+    B=s+E;C0=32*m*B*B;depth=halving(m,r);coefficient=depth*W.bit_length()+9909+252;degree=70000
+    actual=(1-atom)*coarse+atom*OLD
+    result=dict(bit_uniform=dict(coarse_saving=coarse,atom_beta=atom,old_atom_saving=OLD,
+        ordinary_saving=actual,selector_stock='Internally borrowed and restored q-adic rows; complete spectators preserved.',
+        external_stock='The bit group cardinality is not inserted into external polynomial row stock.'),
+        conservative_old_coarse_row_reserve=9909,ordinary_leaf_row_degree=252,
+        complex=dict(m=m,W=W,s=s,N=N,maxchild=r,invocations_per_stage=V,stages=3,
+            auxiliary_banks_after_sharing=1,halving_degree=depth,wire_bits=W.bit_length(),
+            original_X_involution_scalar_group_upper=32*v,local_group_upper=local,
+            logical_group_upper=logical,finite_group_router_upper=G,scalar_group_upper=G,
+            coefficient_bound=h+4,coefficient_denominator_divides=6,
+            physical_roles_per_vertex=R,scalar_role_reserve=scalar_R,reuse_pairs=row['reuse_pairs'],terminal_sinks=row.get('terminal_sinks',0)),
+        semantic=dict(E=E,literal_charge=charge,strict_literal_gap=E-charge,B=B,C0=C0,C1=1,
+            induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=3,
+            exact_grid='One dyadic grid times 3^(-K), K=G*(D_complex+1); completed children preserve the incoming odd exponent.'),
+        rows=dict(coefficient=coefficient,complex_coefficient=depth*W.bit_length(),degree=degree,
+            suffix_slope=4*degree,degree_gap=Q(degree)-Q(51*coefficient,25)))
+    validate_shared_bridge(result,row)
+    return result
+
+
+def below(value,grid=GRID):
+    # Largest grid point strictly below a rational, including exact boundaries.
+    x=value*grid
+    return Q(-(-x.numerator//x.denominator)-1,grid)
+
+
+def old_assembly():
+    spec=importlib.util.spec_from_file_location('retained_prefix_assembly',ROOT/'scripts/structured_bulk_assembly.py')
+    module=importlib.util.module_from_spec(spec);spec.loader.exec_module(module)
+    return module.assembly
+
+
+def price(row,b,coarse,beta,eta,weakening,layout,atom=ATOM):
+    finite=bridge(row,coarse,atom)
+    actual=finite['bit_uniform']['ordinary_saving']
+    a=min(actual,(1-beta)*b-weakening)
+    q=a*(1-2*eta)
+    value=(1-eta)*q/(1+q) if layout=='balanced' else (1-eta)*q/(1+q*(1+eta)+q)
+    k=below(value)
+    if layout=='balanced':result=assembly(finite,row,a,k,beta=beta,h=eta,a_complex=b)
+    else:result=old_assembly()(a,b,finite,k,eta=eta,beta=beta)
+    assert len(result.get('constraints',result.get('strict_constraints')))==47 and len(result['margins'])==7
+    assert result['minimum_margin']==value
+    return dict(kappa=k,a=a,actual_bit_saving=actual,complex_saving=b,beta=beta,eta=eta,
+        weakening=weakening,layout=layout,minimum_margin=value,assembly=result,finite_bridge=finite)
+
+
+def normalized_body_check():
+    source=(PACKAGE/'references/pr141/balanced_stopped.py').read_text()
+    old=source[source.index('def assembly('):source.index('\n\ndef cutoffs(')].rstrip()
+    body=(HERE/'balanced_shared.py').read_text();body=body[body.index('def assembly('):].rstrip()
+    body=body.replace('def assembly(finite_bridge, complex_row, a_bit, kappa,','def assembly(finite_bridge, ordinary_leaf, a_bit, kappa,',1)
+    body=body.replace('f = validate_shared_bridge(finite_bridge, complex_row)','f = validate_bridge(finite_bridge, ordinary_leaf)',1)
+    body=body.replace("    require(a <= f['bit_uniform']['ordinary_saving'], 'transfer saving exceeds actual uniform bit supplier')\n",'',1)
+    require(body==old,'The retained 47-slack body changed outside bridge plumbing and supplier support')
+
+
+def certificate(row,physical):
+    nofloat(row);normalized_body_check()
+    final=price(row,AC,COARSE,BETA,ETA,WEAKENING,'balanced')
+    require(final['kappa']==Q('330942774629799/500000000000000000'),'Selected final kappa differs')
+    counts=physical['scalar_bound']
+    require(counts['terminal_sinks']==row['terminal_sinks']==physical['terminal_compiler']['eligible_count'],'Terminal sink inventory mismatch')
+    require(counts['terminal_scalar_gate_delta']<=0 and counts['terminal_original_scalar_reserve_retained'],'Terminal compiler must fit the retained scalar reserve')
+    require(final['finite_bridge']['complex']['local_group_upper']==counts['local_group_upper'],'Scalar bill detached from physical word')
+    for key,field in (('physical_R','R'),('logical_R','scalar_role_reserve'),('handoffs','reuse_pairs'),('c','c'),('q','q'),('M_operations','total_M_operations')):
+        require(counts[key]==row[field],'Physical scalar inventory mismatch '+key)
+    require(counts['fixed_odd_divisor']==3 and counts['scalar_denominators_divide']==6,'Exact odd-grid contract')
+    require(sha256((ROOT/'notes/general-clifford-frames.tex').read_bytes()).hexdigest() in physical['source_sha256'].values(),'Frame theorem source drift')
+    parent_path=ROOT/'certificates/paired-cube-network.json'
+    require(sha256(parent_path.read_bytes()).hexdigest()=='05b253a21dc526b6039f56d1068162a19a8a4f29291ad94eb892656b88e9142d','Published prerequisite certificate changed')
+    parent=json.loads(parent_path.read_text())
+    published_parent=dict(source_commit='4a3c769e5c5430e7114c4d3e099ff34664677f17',certificate_sha256=sha256(parent_path.read_bytes()).hexdigest(),kappa=parent['kappa'],ordinary_saving=parent['bit']['effective_saving'])
+    rows=[('Same suppliers with earlier positive outer backoffs',row,AC,COARSE,Q(1,10**9),Q(1,10**8),Q(1,10**10),'balanced',ATOM),
+          ('Same suppliers with finer positive outer backoffs',row,AC,COARSE,BETA,ETA,WEAKENING,'balanced',ATOM)]
+    controls_table=[]
+    for name,*args in rows:
+        result=price(*args);controls_table.append(dict(name=name,**{k:v for k,v in result.items() if k not in ('assembly','finite_bridge')}))
+    accepted=[]
+    def reject(name,operation):
+        try:operation()
+        except (KeyError,ValueError,AssertionError):accepted.append(name)
+        else:raise ValueError('Adverse control accepted: '+name)
+    finite=final['finite_bridge'];a=final['a'];k=final['kappa']
+    def mutate(name,change):
+        bad=copy.deepcopy(finite);change(bad);reject(name,lambda:validate_shared_bridge(bad,row))
+    mutate('full group stock replaced by local W',lambda x:x['complex'].update(W=row['W_per_vertex']))
+    mutate('old ordinary reserve omitted',lambda x:x.update(ordinary_leaf_row_degree=0))
+    mutate('conservative old coarse reserve omitted',lambda x:x.update(conservative_old_coarse_row_reserve=0))
+    mutate('uniform supplier replaced by fictional finite profile',lambda x:(x.pop('bit_uniform'),x.update(bit_coarse={'m':529,'W':108516254})))
+    mutate('actual ordinary saving overstated',lambda x:x['bit_uniform'].update(ordinary_saving=AB+Q(1,10**24)))
+    mutate('unpaid previous atom grid',lambda x:x['bit_uniform'].update(atom_beta=ATOM-Q(1,10**24),ordinary_saving=(1-ATOM+Q(1,10**24))*COARSE+(ATOM-Q(1,10**24))*OLD))
+    mutate('scalar role reserve collapsed to physical stock',lambda x:x['complex'].update(scalar_role_reserve=row['R']))
+    mutate('terminal removals omitted',lambda x:x['complex'].update(terminal_sinks=0))
+    mutate('terminal removals counted as aliases',lambda x:x['complex'].update(reuse_pairs=row['reuse_pairs']+row['terminal_sinks']))
+    mutate('odd divisor silently changed to 21',lambda x:x['semantic'].update(fixed_odd_divisor=21))
+    mutate('C0 detached from scalar charge',lambda x:x['semantic'].update(C0=1))
+    mutate('scalar group charge omitted',lambda x:x['complex'].update(scalar_group_upper=0))
+    mutate('stale total row coefficient',lambda x:x['rows'].update(coefficient=1))
+    mutate('inadequate polynomial row degree',lambda x:x['rows'].update(degree=1))
+    mutate('float in otherwise unused metadata',lambda x:x['bit_uniform'].update(extra=0.5))
+    mutate('noncontracting maximum child',lambda x:x['complex'].update(maxchild=row['m']))
+    reject('transfer saving exceeds actual uniform supplier',lambda:assembly(finite,row,AB+Q(1,10**24),k,beta=BETA,h=ETA,a_complex=AC))
+    reject('old prefix charged at balanced parameters',lambda:assembly(finite,row,a,k,beta=BETA,h=ETA,a_complex=AC,original_prefix=True))
+    reject('old nonlinear scalar guard',lambda:assembly(finite,row,a,k,beta=BETA,h=ETA,a_complex=AC,old_guard=True))
+    reject('old separate movement exposures',lambda:assembly(finite,row,a,k,beta=BETA,h=ETA,a_complex=AC,old_exposures=True))
+    reject('next final 1e-18 kappa grid point',lambda:assembly(finite,row,a,k+Q(1,GRID),beta=BETA,h=ETA,a_complex=AC))
+    return js(dict(status='PASS',kappa=k,actual_uniform_bit_saving=AB,coarse_bit_saving=COARSE,
+        complex_saving=AC,weakened_transfer_saving=a,beta=BETA,eta=ETA,strict_weakening=WEAKENING,atom_exponent=ATOM,
+        bridge=finite,assembly=final['assembly'],matched_comparisons=controls_table,published_parent=published_parent,
+        unchanged_47_slack_body=True,adverse_controls=accepted,adverse_control_count=len(accepted),
+        next_grid_scope='Rejection at these fixed certified supplier values and stated parameters; no global or true-profile optimality claim.',
+        scope='Conditional finite witness retaining the source-specified uniform bit, exact tensor, row-restoration, paid layout and analytic/fixed-tape contracts.'))
diff --git a/research/coordinated-frames-and-entrance-banks/arithmetic/interval_moment.py b/research/coordinated-frames-and-entrance-banks/arithmetic/interval_moment.py
new file mode 100644
index 0000000000..c135d98d86
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/arithmetic/interval_moment.py
@@ -0,0 +1,76 @@
+"""Independent exact interval moments; no floating point or producer execution.
+
+Positive atanh series, range reduction by 4/3, explicit geometric tails,
+80 logarithm terms and degree-12 exponential enclosures. The final rounding
+is outward on the 10^-45 grid. Prepared with OpenAI Codex assistance.
+"""
+from fractions import Fraction as Q
+from functools import lru_cache
+from math import factorial
+GRID = 10**45
+LOG_TERMS = 80
+EXP_DEGREE = 12
+
+def require(ok, reason):
+    if not ok: raise ValueError(reason)
+
+def lower(x):return Q((x*GRID).numerator//(x*GRID).denominator,GRID)
+def upper(x):
+    y=x*GRID
+    return Q(-((-y.numerator)//y.denominator),GRID)
+
+@lru_cache(None)
+def log_interval(x):
+    x=Q(x);require(x>=1,'log interval domain')
+    base=Q(4,3);k=0
+    while x>base:x/=base;k+=1
+    def short(y):
+        z=(y-1)/(y+1);power=z;s=Q(0)
+        for j in range(LOG_TERMS):s+=power/Q(2*j+1);power*=z*z
+        lo=2*s;tail=2*power/(Q(2*LOG_TERMS+1)*(1-z*z))
+        return lo,lo+tail
+    lo,hi=short(x);blo,bhi=short(base)
+    return lower(k*blo+lo),upper(k*bhi+hi)
+
+def exp_interval(lo,hi):
+    require(0<=lo<=hi<Q(1,4),'exp enclosure domain')
+    def poly(x):
+        value=term=Q(1)
+        for j in range(1,EXP_DEGREE+1):term=term*x/j;value+=term
+        return value
+    low=poly(lo);high=poly(hi)
+    tail=hi**(EXP_DEGREE+1)/factorial(EXP_DEGREE+1)/(1-hi/Q(EXP_DEGREE+2))
+    return lower(low),upper(high+tail)
+
+def prepare(profile):
+    m,W=profile['m'],profile['W'];require(type(m) is int and type(W) is int and m>1 and W>0,'profile dimensions')
+    raw=profile['child_multiplicities'];hist={int(t):n for t,n in raw.items()}
+    require(len(hist)==len(raw),'aliased histogram keys')
+    require(all(0<t<m and type(n) is int and n>0 for t,n in hist.items()),'invalid child or multiplicity')
+    mass=sum(t*n for t,n in hist.items());require(mass==profile['total_rank']==W*m-profile['N']+profile['L'],'rank mass')
+    require(max(hist)==profile['maxchild'],'maximum child')
+    require(0<mass<W*m,'rank-moment contraction')
+    return [(t,n,*log_interval(Q(m,t))) for t,n in sorted(hist.items())]
+
+def moment(profile,a,details=False):
+    a=Q(a);require(a>=0,'nonnegative saving');lo=hi=Q(0);rows=[]
+    for t,n,L,U in prepare(profile):
+        E,F=exp_interval(a*L,a*U);w=Q(t*n,profile['m']*profile['W']);lo+=w*E;hi+=w*F
+        if details:rows.append(dict(child=t,multiplicity=n,log_lower=L,log_upper=U,exp_lower=E,exp_upper=F,contribution_lower=w*E,contribution_upper=w*F))
+    result=dict(saving=a,lower=lo,upper=hi,strict_gap_lower=1-hi)
+    if details:result['terms']=rows
+    return result
+
+def saving_grid(profile,denominator=10**24):
+    require(type(denominator) is int and denominator>0,'positive saving grid')
+    prepare(profile);lo=0;hi=denominator//32;iterations=0
+    require(moment(profile,Q(lo,denominator))['upper']<1,'initial contraction')
+    require(moment(profile,Q(hi,denominator))['lower']>1,'initial exclusion')
+    while hi-lo>1:
+        mid=(hi+lo)//2;m=moment(profile,Q(mid,denominator));iterations+=1
+        if m['upper']<1:lo=mid
+        elif m['lower']>1:hi=mid
+        else:raise ValueError('Moment interval overlaps one; increase enclosure precision')
+    accepted=moment(profile,Q(lo,denominator),True);rejected=moment(profile,Q(hi,denominator),True)
+    require(accepted['upper']<1<rejected['lower'],'adjacent saving proof failed')
+    return dict(accepted=accepted,rejected=rejected,denominator=denominator,iterations=iterations,method=dict(log_base='4/3',log_terms=LOG_TERMS,exp_degree=EXP_DEGREE,outward_grid=GRID))
diff --git a/research/coordinated-frames-and-entrance-banks/arithmetic/shared_bridge.py b/research/coordinated-frames-and-entrance-banks/arithmetic/shared_bridge.py
new file mode 100644
index 0000000000..b6cea18a19
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/arithmetic/shared_bridge.py
@@ -0,0 +1,93 @@
+"""Exact shared-core bridge validator for a paid balanced-layout adapter.
+
+The numerical bridge follows icekylinx/eumemic PR152 at 627089e85badda0b450048c29307786716d51037.
+This does not replay or establish a scalar word, weighted selector theorem,
+moment enclosure, or analytic/fixed-tape interface. Those are distinct inputs.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from fractions import Fraction as Q
+from math import prod
+
+def require(test,reason):
+    if not test:raise ValueError(reason)
+def rational(value):
+    require(type(value) in (int,str,Q),'Exact rational required')
+    return Q(value)
+def integer(value):
+    q=rational(value);require(q.denominator==1,'Integer required');return q.numerator
+def halving(m,r):
+    require(0<r<m,'Strictly contracting positive child')
+    k=1
+    while m**k<=2*r**k:k+=1
+    return k
+def nofloat(value):
+    require(not isinstance(value,float),'Binary float not permitted')
+    if isinstance(value,dict):
+        for x in value.values():nofloat(x)
+    if isinstance(value,(list,tuple)):
+        for x in value:nofloat(x)
+
+def validate_shared_bridge(bridge,row):
+    """Recompute numeric PR152 bridge, without inventing a finite bit profile.
+
+    `row` is the explicit complex local inventory. Its scalar/frame proof and
+    the supplier moment proofs remain separate source-bound obligations.
+    The supplied bit_uniform fields describe an already completed ordinary
+    interchange with borrowed-and-restored q-adic rows, not old PR141's
+    finite coarse-profile contract. The inherited conservative external
+    reserves 9909 and 252 are retained as separate stated proof inputs.
+    """
+    nofloat(bridge);nofloat(row)
+    h,v,R,c,M=(integer(row[k]) for k in ('h','v','R','c','total_M_operations'))
+    scalar_R=integer(row['scalar_role_reserve'])
+    reuse=integer(row['reuse_pairs']);terminal=integer(row.get('terminal_sinks',0))
+    require(scalar_R==integer(row['c'])+integer(row['q'])-integer(row['matched']),
+            'Scalar role reserve must retain the complete pre-reuse inventory')
+    require(reuse>=0 and terminal>=0 and scalar_R-R==reuse+terminal,'Physical role count must match reuse and terminal removals')
+    require(h>=2 and h%2==0 and min(v,R,c,M)>0,'Positive even-dimensional complex family')
+    m=3*h
+    hist={integer(k):integer(n) for k,n in row['child_histogram'].items() if integer(n)}
+    require(all(0<t<m and n>0 for t,n in hist.items()),'Proper positive child histogram')
+    r=max(hist);localW=2*v+R;localS=sum(t*n for t,n in hist.items())
+    require((m,localW,localS)==tuple(integer(row[k]) for k in ('m','W_per_vertex','rank_per_vertex')),'Complex inventory totals')
+    require(m*localW-localS==integer(row['deficit_per_vertex'])==2*v-3*integer(row['loss']),'Shared-core deficit identity')
+    half=m//2
+    vertices=2**(m-1+(half-1)**2)*prod(2**(2*i)-1 for i in range(1,half))
+    W,s,N=vertices*localW,vertices*localS,vertices*v
+    original_X=32*v
+    local=4*(c+v)+10*v+4*h*v+4*h*h+8*h+8+2*h+8*scalar_R*v*(M+16)+original_X
+    logical=3*vertices*local+8*W+4*N+8*m*scalar_R*vertices
+    G=64*(m+1)**3*(logical+1)*(W+1)**2
+    E=64*(W+m+G+1)**3
+    literal=2*G*W**2+8*s+4*W+4+32*m
+    B=s+E;C0=32*m*B**2;depth=halving(m,r)
+    expected_complex=dict(m=m,W=W,s=s,N=N,maxchild=r,invocations_per_stage=vertices,stages=3,
+      auxiliary_banks_after_sharing=1,halving_degree=depth,wire_bits=W.bit_length(),
+      original_X_involution_scalar_group_upper=original_X,local_group_upper=local,
+      logical_group_upper=logical,finite_group_router_upper=G,scalar_group_upper=G,
+      coefficient_bound=h+4,coefficient_denominator_divides=6,
+      physical_roles_per_vertex=R,scalar_role_reserve=scalar_R,reuse_pairs=reuse,terminal_sinks=terminal)
+    expected_semantic=dict(E=E,literal_charge=literal,strict_literal_gap=E-literal,B=B,C0=C0,C1=1,
+      induction_gap=2*B*(m-r)-s-E,fixed_odd_divisor=3)
+    for name,expected in [('complex',expected_complex),('semantic',expected_semantic)]:
+        for k,value in expected.items():require(rational(bridge[name][k])==value,'Stale '+name+'/'+k)
+    require(E>literal and 2*B*(m-r)>=s+E and C0>2*B+18,'Completed exact-child semantic induction')
+    # These are conservative, retained external reserves. The varying q-adic
+    # group cardinality is deliberately not interpreted as an external stock.
+    old_coarse=integer(bridge['conservative_old_coarse_row_reserve'])
+    old_leaf=integer(bridge['ordinary_leaf_row_degree'])
+    require(old_coarse==9909 and old_leaf==252,'Retained separate external row reserves')
+    coeff=depth*W.bit_length()+old_coarse+old_leaf
+    degree=integer(bridge['rows']['degree']);gap=Q(degree)-Q(51,25)*coeff
+    require(degree>0 and gap>0,'Insufficient external row reserve')
+    expected_rows=dict(coefficient=coeff,complex_coefficient=depth*W.bit_length(),
+      degree=degree,suffix_slope=4*degree,degree_gap=gap)
+    for k,value in expected_rows.items():require(rational(bridge['rows'][k])==value,'Stale row/'+k)
+    u=bridge['bit_uniform']
+    coarse,atom,old,actual=(rational(u[k]) for k in ('coarse_saving','atom_beta','old_atom_saving','ordinary_saving'))
+    require(0<coarse<1 and 0<atom<1 and old==Q(384599,10**10),'Uniform stopped supplier constants')
+    require(actual==(1-atom)*coarse+atom*old,'Uniform stopped saving identity')
+    require(actual<atom<1-actual,'Paid atom-adapter and internal-row tolls')
+    return dict(complex=expected_complex,semantic=expected_semantic,rows=expected_rows,
+      bit_uniform=dict(coarse_saving=coarse,atom_beta=atom,old_atom_saving=old,ordinary_saving=actual),
+      conservative_old_coarse_row_reserve=old_coarse,ordinary_leaf_row_degree=old_leaf)
diff --git a/research/coordinated-frames-and-entrance-banks/baseline-pr181.part01 b/research/coordinated-frames-and-entrance-banks/baseline-pr181.part01
new file mode 100644
index 0000000000..a89a0c458b
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diff --git a/research/coordinated-frames-and-entrance-banks/baseline-pr181.part02 b/research/coordinated-frames-and-entrance-banks/baseline-pr181.part02
new file mode 100644
index 0000000000..4e90475e48
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/baseline-pr181.part02 differ
diff --git a/research/coordinated-frames-and-entrance-banks/baseline-pr181.part03 b/research/coordinated-frames-and-entrance-banks/baseline-pr181.part03
new file mode 100644
index 0000000000..2f103c2ce9
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/baseline-pr181.part03 differ
diff --git a/research/coordinated-frames-and-entrance-banks/bit/banks.json b/research/coordinated-frames-and-entrance-banks/bit/banks.json
new file mode 100644
index 0000000000..570c1f838f
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/bit/banks.json
@@ -0,0 +1,42 @@
+{
+  "bank": {
+    "entrance_gauges": 2200,
+    "entrance_rank": 20,
+    "undeferred_physical_roles": 15828
+  },
+  "profile": {
+    "W_per_vertex": "59144/3",
+    "child_multiplicities": {
+      "1": 125448,
+      "2": 62908,
+      "3": 29616,
+      "4": 8970,
+      "5": 7800,
+      "6": 3660,
+      "7": 3312,
+      "8": 2802,
+      "9": 1812,
+      "10": 2508,
+      "11": 1158,
+      "12": 4320,
+      "13": 1590,
+      "14": 3234,
+      "15": 1230,
+      "16": 3216,
+      "17": 1164,
+      "18": 1332,
+      "19": 918,
+      "20": 5280,
+      "21": 11160,
+      "22": 13272
+    },
+    "deficit_per_vertex": 1936,
+    "edge_count": 296710,
+    "m": 72,
+    "maxchild": 22,
+    "rank_per_vertex": 1417520
+  },
+  "coarse": "665481166955987/1000000000000000000",
+  "atom": "665064157585290854737/1000000000000000000000000",
+  "effective": "665064157585290854736467094714610539581/1000000000000000000000000000000000000000000"
+}
diff --git a/research/coordinated-frames-and-entrance-banks/bit/base_two_moment.py b/research/coordinated-frames-and-entrance-banks/bit/base_two_moment.py
new file mode 100644
index 0000000000..14a90fa0e7
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/bit/base_two_moment.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Maintainer arithmetic cross-check of the pinned PR39 certificate.
+
+Uses no producer/checker imports. Counts are inputs, not proved by this script.
+80-term rational logarithms and 12-term exponentials independently enclose
+the two moments. The accompanying written review addresses their realization.
+"""
+import argparse
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+GRID = 10**30
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def log_unit(x):
+    require(1 <= x <= 2, "logarithm range")
+    z = (x-1)/(x+1)
+    lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0))
+    return lo, lo+2*z**161/(161*(1-z*z))
+
+
+def log_bounds(x):
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    lo, hi = log_unit(x)
+    l2, u2 = log_unit(Q(2))
+    lo, hi = lo+k*l2, hi+k*u2
+    # Outward rational rounding keeps later certificates compact.
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
+
+def exp_bounds(x):
+    require(0 <= x < 1, "exponential range")
+    term = total = Q(1)
+    for j in range(1, 13):
+        term *= x/j
+        total += term
+    tail = term*x/13/(1-x/14)
+    return total, total+tail
+
+
+def moment(m, W, entries, saving):
+    lo = hi = Q(0)
+    for t, count in entries:
+        t, count = int(t), int(count)
+        require(0 < t < m and count > 0, "improper child")
+        l, u = log_bounds(Q(m, t))
+        lower = exp_bounds(saving*l)[0]
+        upper = exp_bounds(saving*u)[1]
+        lo += Q(t*count, m*W)*lower
+        hi += Q(t*count, m*W)*upper
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
diff --git a/research/coordinated-frames-and-entrance-banks/bit/base_word.py b/research/coordinated-frames-and-entrance-banks/bit/base_word.py
new file mode 100644
index 0000000000..b4a40a7e05
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/bit/base_word.py
@@ -0,0 +1,202 @@
+#!/usr/bin/env python3
+"""Reusable independent formal checks for a physical partner-pair bit word.
+
+The supplied frozen inputs are source-pinned. This never runs their producer.
+The Z check is of the defining integer decoder; only its F2 reduction is I.
+Prepared with OpenAI Codex assistance; Apache-2.0.
+"""
+from collections import Counter, defaultdict
+import importlib.util
+import json
+from pathlib import Path
+import sys
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[2]
+BASE = ROOT / 'research/paired-cube-bit'
+P = 12
+if sys.flags.optimize: raise SystemExit('refusing -O')
+
+
+def need(ok, msg):
+    if not ok:
+        raise ValueError(msg)
+
+class Candidate:
+    def exact_frames(self):
+        C = self.C
+        C.decoder(); C.geometry()
+        for i in self.changed_frames:
+            f = self.opframe[i]; x = self.ops[i][2]; bits = C.sup[x]
+            need(C.nondeg(f), 'replacement operation frame nondegenerate')
+            while bits:
+                low = bits&-bits; s=low.bit_length()-1; bits-=low
+                need(C.in_frame(C.chi[s],f), 'value span in replacement operation frame')
+        for b, d in self.pairs:
+            need(C.sub(self.endframe[d], self.gauge[b]['frame']), 'exact Q donor-to-birth containment')
+            need(C.nondeg(self.endframe[d]) and C.nondeg(self.gauge[b]['frame']), 'handoff endpoints nondegenerate')
+
+    def register(self, rows):
+        C = self.C
+        B,_ = self.module.reduce_rows(rows,self.h if hasattr(self,'h') else C.h)
+        B = tuple(sorted(map(tuple,B),key=lambda r:next(i for i,x in enumerate(r) if x)))
+        if not hasattr(self,'frame_ids'):
+            self.frame_ids = {}
+        if B in self.frame_ids: return self.frame_ids[B]
+        f=max(C.B)+1; A,_=self.module.kernel(B,C.h)
+        C.B[f],C.A[f],C.dimf[f]=B,A,len(B);self.frame_ids[B]=f
+        return f
+
+
+    def row(self):
+        C,h,v=self.C,self.h,self.v
+        _,baseline_Y,src,_=C.chains()
+        seq=defaultdict(list)
+        for i,(a,b,x) in enumerate(self.ops):
+            seq[a].append(self.opframe[i]);seq[b].append(self.opframe[i])
+        for j,s in enumerate(self.w['rootroles']):seq[s].append(self.w['root_frame'][j])
+        start={b:self.w['source_frame'][x] for x,b in self.source.items()}
+        start.update({b:z['frame'] for b,z in self.gauge.items()})
+        recipient={d:b for b,d in self.pairs}
+        H=Counter()
+        for s in range(self.R):
+            if s in self.donor:continue
+            chain=list(seq[s])
+            if s in recipient:
+                b=recipient[s];chain += [self.gauge[b]['frame']]+seq[b]
+            chain.append(self.w['full_frame'])
+            prev=start.get(s);dp=0 if prev is None else C.dimf[prev]
+            if s in self.source.values():H[1]+=1
+            for f in chain:
+                need(prev is None or C.sub(prev,f),'physical role chain nested')
+                d=C.dimf[f]
+                if d>dp:H[d-dp]+=1
+                prev,dp=f,d
+        for j,r in enumerate(self.g['roots']):
+            if r['kind']=='center':H[C.dimf[self.w['root_frame'][j]]]+=1
+        Y=Counter();current=[None]*v
+        def read(t,f):
+            old=current[t];d=0 if old is None else C.dimf[old]
+            need(old is None or C.sub(old,f),'actual target frame chronology')
+            if C.dimf[f]>d:Y[C.dimf[f]-d]+=1
+            current[t]=f
+        for b in sorted(self.order,key=lambda b:self.readtime[b]):
+            for t in self.gauge[b]['targets']:read(t,self.gauge[b]['frame'])
+        deliveries=defaultdict(list)
+        for e in self.k['entries']:deliveries[e['deliver_after_root']].append(e)
+        for j,r in enumerate(self.g['roots']):
+            if r['kind']!='center':
+                for t in r['targets']:read(t,self.w['root_frame'][j])
+            for e in deliveries[j]:
+                for t in e['receivers']:read(t,e['deliver_frame'])
+        for t,f in enumerate(current):
+            need(f is not None and all(self.module.dot(C.cov[t],b)==0 for b in C.B[f]),'final target cap')
+            gap=h-1-C.dimf[f]
+            if gap:Y[gap]+=1
+        need(sum(r*n for r,n in Y.items())==v*(h-1),'target rank mass')
+        hist=Counter()
+        for part in (H,Y,src):
+            for r,n in part.items():
+                if r and n:hist[r]+=3*n
+        for b,z in self.gauge.items():
+            if b not in self.donor:hist[3*z['dim']]+=1
+        hist[2]+=2*v
+        hist={r:n for r,n in sorted(hist.items()) if n}
+        need(all(0<r<3*h and n>0 for r,n in hist.items()),'positive proper child bins')
+        R=self.R-len(self.pairs);W=2*v+R;mass=sum(r*n for r,n in hist.items());m=3*h
+        need(W*m-mass==C.prof['deficit_per_vertex'],'reuse preserves telescoping deficit')
+        return dict(h=h,v=v,R=R,virtual_R=self.R,reused_registers=len(self.pairs),W_per_vertex=W,m=m,
+            loss=C.prof['loss'],rank_per_vertex=mass,deficit_per_vertex=W*m-mass,
+            child_histogram=hist,maxchild=max(hist),selected_roles=len(self.gauge)-len(self.pairs),
+            selected_rank_histogram=dict(Counter(z['dim'] for b,z in self.gauge.items() if b not in self.donor)),
+            changed_operation_frames=len(self.changed_frames),
+            frames_not_contained_in_original=sum(not C.sub(self.opframe[i],self.original_opframe[i]) for i in self.changed_frames),
+            physical_target_histogram=dict(Y),physical_internal_histogram=dict(H),
+            foreign_producer_replays=0)
+
+    def adjoint(self):
+        adj = [dict() for _ in range(self.R)]
+        for r,s in zip(self.g['roots'], self.w['rootroles']):
+            for t in r['targets']: adj[s][t] = adj[s].get(t,0)+1
+        for a,b,_ in reversed(self.ops):
+            for t,c in adj[a].items(): adj[b][t] = adj[b].get(t,0)+c
+        return adj
+
+    def formal(self, ring, tamper=None):
+        """All source, target and dirty-register columns; no random sampling."""
+        v, phys = self.v, self.phys
+        regs = sorted(set(phys.values())); index = {r:2*v+j for j,r in enumerate(regs)}
+        if ring == 2:
+            unit = lambda i: 1<<i
+            def accum(dst,val,c): return dst ^ val if c&1 else dst
+        else:
+            unit = lambda i: {i:1}
+            def accum(dst,val,c):
+                out = dict(dst)
+                for i,z in val.items():
+                    k = out.get(i,0)+c*z
+                    if k: out[i]=k
+                    else: out.pop(i,None)
+                return out
+        x = [unit(s) for s in range(v)]; y = [unit(v+t) for t in range(v)]
+        a = {r:unit(index[r]) for r in regs}
+        adj = self.adjoint()
+        at = defaultdict(list)
+        for b in self.order: at[self.readtime[b]].append(b)
+        done = []
+        stale_used = False
+        def read(b):
+            nonlocal stale_used
+            if tamper == 'omit_compensation' and b == self.order[0]: return
+            value = a[phys[b]]
+            if tamper == 'stale' and b in self.donor and not stale_used:
+                value = unit(index[phys[b]]); stale_used = True
+            for t,c in adj[b].items(): y[t] = accum(y[t],value,-c)
+        def op(i,sign):
+            b,d,_ = self.ops[i]
+            a[phys[b]] = accum(a[phys[b]],a[phys[d]],sign)
+        for b in range(self.R):
+            if b not in self.gauge: read(b)
+        for s,b in self.source.items(): a[phys[b]] = accum(a[phys[b]],x[s],1)
+        for i in self.phase1: op(i,1); done.append(i)
+        for r,b in zip(self.g['roots'],self.w['rootroles']):
+            if r['kind']=='center':
+                for t in r['targets']:y[t]=accum(y[t],a[phys[b]],1)
+        for j,i in enumerate(self.rest):
+            for b in at[j]: read(b)
+            op(i,1); done.append(i)
+        for b in at[len(self.rest)]: read(b)
+        deliveries=defaultdict(list)
+        for e in self.k['entries']:deliveries[e['deliver_after_root']].append(e)
+        broken=False
+        for j,(r,b) in enumerate(zip(self.g['roots'],self.w['rootroles'])):
+            if r['kind']!='center':
+                for t in r['targets']: y[t] = accum(y[t],a[phys[b]],1)
+            for e in deliveries[j]:
+                c,d = e['carrier'],e['passive']
+                x[c] = accum(x[c],x[d],1)
+                for t in e['receivers']:
+                    if tamper == 'missing_partner' and not broken: broken=True;continue
+                    y[t] = accum(y[t],x[c],1)
+        for e in self.k['entries']:
+            x[e['carrier']] = accum(x[e['carrier']],x[e['passive']],-1)
+        for i in reversed(done): op(i,-1)
+        for s,b in self.source.items(): a[phys[b]] = accum(a[phys[b]],x[s],-1)
+        need(all(a[r] == unit(index[r]) for r in regs), 'all dirty register columns restored')
+        need(all(x[s] == unit(s) for s in range(v)), 'all source columns restored')
+        if ring == 2:
+            want = [accum(unit(v+t),unit(t),1) for t in range(v)]
+        else:
+            # Integer lift: the exact defining decoder, whose reduction is I over F2.
+            support = [{s:1} for s in range(v)]
+            for aa,bb in self.g['args'][v:]: support.append(accum(support[aa],support[bb],1))
+            want = [unit(v+t) for t in range(v)]
+            for r in self.g['roots']:
+                for t in r['targets']: want[t] = accum(want[t],support[r['node']],1)
+            for e in self.k['entries']:
+                for t in e['receivers']:
+                    want[t] = accum(want[t],unit(e['carrier']),1)
+                    want[t] = accum(want[t],unit(e['passive']),1)
+        need(y == want, 'all target columns equal defining decoder')
+        return dict(ring=str(ring),formal_variables=2*v+len(regs),
+            target_contract='F2 identity' if ring==2 else 'integer defining decoder',
+            identity=ring==2,defining_decoder=True,all_dirty_and_source_columns_restored=True)
diff --git a/research/coordinated-frames-and-entrance-banks/bit/prime-witnesses.json.gz b/research/coordinated-frames-and-entrance-banks/bit/prime-witnesses.json.gz
new file mode 100644
index 0000000000..63d0d350be
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/bit/prime-witnesses.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/bit/prime_witnesses.py b/research/coordinated-frames-and-entrance-banks/bit/prime_witnesses.py
new file mode 100644
index 0000000000..7012e20a9a
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/bit/prime_witnesses.py
@@ -0,0 +1,99 @@
+"""Exact finite prime exclusions for the selected new rational bit frames.
+
+For integer basis B and G=I-J/9, det(9 B G B^t) is a nonzero integer.
+Excluding its prime divisors and 2,3,5 makes the reduced frame nondegenerate
+over every odd local ring Z/q^w. Projector denominators divide this determinant
+times fixed powers of 3. Retained frames retain their old exclusion witnesses.
+The selected q is an arbitrarily large fixed prime chosen after this finite
+set; this is not verification at a preselected numeric q.
+"""
+from pathlib import Path
+import argparse,gzip,hashlib,json,sys
+
+HERE=Path(__file__).resolve().parent
+
+def det(A):
+    A=[list(r) for r in A];n=len(A)
+    if not n:return 1
+    sign=1;previous=1
+    for k in range(n-1):
+        pivot=next((i for i in range(k,n) if A[i][k]),None)
+        if pivot is None:return 0
+        if pivot!=k:A[k],A[pivot]=A[pivot],A[k];sign=-sign
+        p=A[k][k]
+        for i in range(k+1,n):
+            for j in range(k+1,n):
+                v=A[i][j]*p-A[i][k]*A[k][j]
+                assert v%previous==0
+                A[i][j]=v//previous
+            A[i][k]=0
+        previous=p
+    return sign*A[-1][-1]
+
+PRIMES=(2,3,5,7,11,13,17,19,23,29,31)
+
+def factor_witness(determinant):
+    assert type(determinant) is int and determinant != 0
+    residual=abs(determinant);powers={}
+    for prime in PRIMES:
+        power=0
+        while residual%prime==0:residual//=prime;power+=1
+        powers[str(prime)]=power
+    validate_factor(determinant,powers,residual)
+    return powers,residual
+
+def validate_factor(determinant,powers,residual):
+    assert type(determinant) is int and determinant != 0
+    assert set(powers)==set(map(str,PRIMES))
+    assert all(type(x) is int and x>=0 for x in powers.values())
+    assert type(residual) is int and 0<residual<2**80
+    reconstructed=residual
+    for p in PRIMES:reconstructed*=p**powers[str(p)]
+    assert reconstructed==abs(determinant)
+
+def certificate(word=None):
+    if word is None:
+        from word import Candidate
+        word=Candidate()
+    C=word.C;h=word.h
+    used=set(word.opframe)|set(word.w['source_frame'])|set(word.w['root_frame'])|{word.w['full_frame']}
+    used.update(z['frame'] for z in word.w['gauges'])
+    for e in word.k['entries']:used.update((e['mix_frame'],e['deliver_frame']))
+    groups={}
+    for frame in sorted(used):
+        B=C.B[frame]
+        key=json.dumps(B,separators=(',',':'));groups.setdefault(key,[]).append(frame)
+    records=[]
+    for key,frames in sorted(groups.items()):
+        B=json.loads(key);s=list(map(sum,B))
+        gram=[[9*sum(a*b for a,b in zip(x,y))-s[i]*s[j] for j,y in enumerate(B)] for i,x in enumerate(B)]
+        determinant=det(gram);powers,residual=factor_witness(determinant)
+        records.append(dict(basis_sha256=hashlib.sha256(key.encode()).hexdigest(),
+            frame_ids=frames,dimension=len(B),cleared_gram_determinant=determinant,
+            small_prime_powers=powers,remaining_factor=residual,gram_denominator_power_of_9=len(B)))
+    controls=[];zeros={str(p):0 for p in PRIMES}
+    for name,args in (('zero determinant',(0,zeros,1)),('incorrect factor identity',(15,zeros,1)),
+        ('residual above retained prime lower bound',(2**80+7,zeros,2**80+7))):
+        try:validate_factor(*args)
+        except AssertionError:controls.append(name)
+        else:raise ValueError('Invalid prime witness accepted: '+name)
+    return dict(status='PASS exact prime witnesses for every used physical frame',h=h,
+        source_commit='4a3c769e5c5430e7114c4d3e099ff34664677f17',
+        input_sha256={p.relative_to(HERE.parents[2]).as_posix():hashlib.sha256(p.read_bytes()).hexdigest() for p in word.input_paths},
+        script_sha256=hashlib.sha256(Path(__file__).read_bytes()).hexdigest(),
+        stripped_primes=list(PRIMES),frame_witnesses=records,unique_used_bases=len(records),
+        total_used_frames=len(used),total_operation_frames=len(word.changed_frames),
+        maximum_determinant_bits=max(abs(z['cleared_gram_determinant']).bit_length() for z in records),
+        all_remaining_factors_below_retained_prime_lower_bound=True,adverse_controls=controls,
+        rule='Every q>2^80 avoids every nonzero used-frame determinant, so the original prime range is unchanged.',
+        scope='Exact integer cleared-Gram factors for all physical frames including every changed operation, gauge, source, root, partner-mix, delivery and ambient frame. Uniform split-rank and fallback arguments remain the retained local-ring interface.')
+
+def main():
+    assert not sys.flags.optimize
+    ap=argparse.ArgumentParser();ap.add_argument('--write',action='store_true');args=ap.parse_args()
+    record=certificate();target=HERE/'prime-witnesses.json.gz'
+    if args.write:target.write_bytes(gzip.compress((json.dumps(record,sort_keys=True,separators=(',',':'))+'\n').encode(),mtime=0))
+    else:assert record==json.loads(gzip.decompress(target.read_bytes())),'Exact prime witnesses changed'
+    print(json.dumps({k:v for k,v in record.items() if k!='frame_witnesses'}))
+
+if __name__=='__main__':main()
diff --git a/research/coordinated-frames-and-entrance-banks/bit/prove.py b/research/coordinated-frames-and-entrance-banks/bit/prove.py
new file mode 100644
index 0000000000..f98487f6ee
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/bit/prove.py
@@ -0,0 +1,187 @@
+#!/usr/bin/env python3
+"""Frozen PR179 physical bit word, completed entrance banks and exact paid moment.
+
+Checks supplied graph/frame inputs and all formal columns without executing
+the foreign producer. The analytic weighted-compiler contracts are retained.
+"""
+from fractions import Fraction as Q
+from pathlib import Path
+import gzip,json
+import sys
+
+HERE=Path(__file__).resolve().parent
+sys.path.insert(0,str(HERE));sys.path.insert(0,str(HERE.parent/'arithmetic'))
+sys.dont_write_bytecode=True
+from word import Candidate,need
+from interval_moment import moment,log_interval,exp_interval
+from prime_witnesses import certificate as prime_certificate
+
+
+def js(x):
+    if isinstance(x,Q):return str(x)
+    if isinstance(x,dict):return {str(k):js(v) for k,v in x.items()}
+    if isinstance(x,(tuple,list)):return [js(v) for v in x]
+    return x
+
+
+def paid_moment(p,a,details=False):
+    raw=moment(p,a,details);m,w=p['m'],p['W']
+    count=sum(p['child_multiplicities'].values());bad=Q(1,10**16);fallback=32*m*m
+    ll,lu=log_interval(Q(m));el,eu=exp_interval(a*ll,a*lu)
+    weight=bad*Q(fallback*count,w*m)
+    return dict(saving=a,lower=raw['lower']+weight*el,upper=raw['upper']+weight*eu,
+        strict_gap_lower=1-raw['upper']-weight*eu,raw=raw,fallback_lower=weight*el,
+        fallback_upper=weight*eu,edge_count=count,fallback_children_per_edge=fallback,
+        bad_fraction=bad)
+
+
+
+def bank_projectors():
+    """Literal address maps on every F2 and integer basis column of two banks."""
+    checks=[]
+    for width,count in ((4,18),(24,3)):
+        owner=[j//width for j in range(72)]
+        need([owner.count(s) for s in range(count)]==[width]*count,'actual full bank partition')
+        projectors=[[int(owner[j]==s) for j in range(72)] for s in range(count)]
+        need(all(sum(P[j] for P in projectors)==1 for j in range(72)),'sum of actual projectors is identity')
+        need(all(P[j]*P[j]==P[j] for P in projectors for j in range(72)),'idempotent bank projectors')
+        need(all(P[j]*T[j]==0 for s,P in enumerate(projectors) for t,T in enumerate(projectors) if s!=t for j in range(72)),'pairwise disjoint bank projectors')
+        def run(column,ring,indices):
+            x=[0]*72;y=[0]*72
+            (x if column<72 else y)[column%72]=1
+            for s in indices:
+                P=projectors[s]
+                x,y=([a-P[j]*a+P[j]*b for j,(a,b) in enumerate(zip(x,y))],
+                     [b-P[j]*b+P[j]*a for j,(a,b) in enumerate(zip(x,y))])
+                if ring:x=[a%ring for a in x];y=[a%ring for a in y]
+            return x+y
+        controls={}
+        for ring in (2,0):
+            for column in range(144):
+                expected=[0]*144;expected[(column+72)%144]=1
+                need(run(column,ring,range(count))==expected,'completed bank full dirty swap formal column')
+                need(run(column,ring,list(range(count))+list(reversed(range(count))))==[int(j==column) for j in range(144)],'completed bank inverse restores arbitrary contents')
+            for tag,indices in (('omit_block',list(range(count-1))),('repeat_block',list(range(count))+[count-1])):
+                column=72-width;expected=[0]*144;expected[column+72]=1
+                need(run(column,ring,indices)!=expected,'adverse bank control rejected: '+tag)
+                controls[str(ring)+':'+tag]=True
+        checks.append(dict(width=width,blocks=count,ambient=72,rings=[2,0],formal_columns=288,
+            inverse_columns=288,projectors_idempotent=True,projectors_disjoint=True,sum_is_identity=True,
+            controls_rejected=controls,endpoint='partialSwap(P)(x,y)=(x-Px+Py,y-Py+Px)'))
+    return checks
+
+
+def packed_row(word,row):
+    """Bank WHOLE physical chains; a paired recipient is an internal splice."""
+    need((row['h'],row['m'],row['v'],row['reused_registers'])==(24,72,1760,1760),'retained bit dimensions and aliases')
+    starts={s:z['dim'] for s,z in word.gauge.items() if s not in word.donor}
+    need(len(starts)==row['selected_roles'] and row['selected_rank_histogram']=={20:2200},'actual surviving physical entrances')
+    need(set(starts.values())=={20} and len(starts)==2200,'rank20 entrance gauges only')
+    need(all(b in word.donor and word.gauge[b]['dim']==21 for b in word.donor),'rank21 recipients are splices, not separate bank roles')
+    gauges=len(starts);rest=row['R']-gauges
+    need(rest==15828,'every remaining physical role charged')
+    H=dict(row['child_histogram'])
+    need(H.pop(60,None)==gauges,'remove exactly one rank60 entrance exterior per surviving gauge')
+    W=Q(2*word.v)+3*(Q(gauges,18)+Q(rest,3))
+    mass=sum(r*n for r,n in H.items())
+    need(W==Q(59144,3) and mass==1417520 and 72*W-mass==1936,'complete packed bank telescoping inventory')
+    need(max(H)==22 and all(0<r<72 and n>0 for r,n in H.items()),'proper paid bank children')
+    need(9*gauges%18==0 and 9*rest%3==0,'stage-private integer bank realization')
+    need(3*(9*gauges//18+9*rest//3)+2*9*word.v==9*W,'nine-copy literal stock agrees')
+    packed=dict(row,W_per_vertex=W,rank_per_vertex=mass,deficit_per_vertex=1936,child_histogram=H,maxchild=22,
+        physical_roles_before_banks=row['R'],bank_roles_per_vertex=W-2*word.v,
+        bank=dict(entrance_rank=20,entrance_gauges=gauges,undeferred_physical_roles=rest,
+            selected_blocks=18,selected_width=4,undeferred_blocks=3,undeferred_width=24,
+            stages=3,physical_chains_spliced=True,physical_replica_count=9,moment_normalization_scale=3,selected_banks_per_stage=1100,undeferred_banks_per_stage=47484),
+        bank_projector_checks=bank_projectors())
+    expected=json.loads((HERE/'banks.json').read_text())
+    ep=expected['profile']
+    need(Q(ep['W_per_vertex'])==W and ep['rank_per_vertex']==mass and ep['deficit_per_vertex']==1936,'stored packed dimensions')
+    need({int(r):n for r,n in ep['child_multiplicities'].items()}==H,'stored entire packed histogram')
+    need(expected['bank']==dict(entrance_rank=20,entrance_gauges=gauges,undeferred_physical_roles=rest),'stored actual bank role classes')
+    return packed
+
+
+def independent_bank_moment(row,coarse):
+    from base_two_moment import moment as alternate_moment
+    W=int(3*row['W_per_vertex']);H=[(r,3*n) for r,n in row['child_histogram'].items()]
+    count=sum(n for r,n in H);fallback=32*72**2*count
+    _,upper=alternate_moment(72,W,H,coarse['coarse_saving'])
+    _,bad_upper=alternate_moment(72,W,[(1,fallback)],coarse['coarse_saving'])
+    gap=1-upper-Q(1,10**16)*bad_upper
+    need(gap>0,'independent base-two paid moment contracts')
+    next_saving=coarse['coarse_saving']+Q(1,coarse['coarse_grid'])
+    lower,_=alternate_moment(72,W,H,next_saving)
+    bad_lower,_=alternate_moment(72,W,[(1,fallback)],next_saving)
+    need(lower+Q(1,10**16)*bad_lower>1,'independent base-two adjacent paid moment exclusion')
+    return gap
+
+
+def certify(row):
+    p=dict(m=row['m'],W=row['W_per_vertex'],N=row['deficit_per_vertex'],L=0,
+        total_rank=row['rank_per_vertex'],maxchild=row['maxchild'],child_multiplicities=row['child_histogram'])
+    # Scale three clears the normalized moment fractions; the stage-private physical realization uses nine copies.
+    p=dict(p,W=int(3*Q(p['W'])),N=3*p['N'],total_rank=3*p['total_rank'],
+        child_multiplicities={r:3*n for r,n in p['child_multiplicities'].items()})
+    grid=10**18;lo=0;hi=grid//100
+    need(paid_moment(p,Q(lo,grid))['upper']<1,'zero-saving rank contraction')
+    need(paid_moment(p,Q(hi,grid))['lower']>1,'upper bracket')
+    while hi-lo>1:
+        mid=(lo+hi)//2;r=paid_moment(p,Q(mid,grid))
+        if r['upper']<1:lo=mid
+        elif r['lower']>1:hi=mid
+        else:raise ValueError('increase moment precision')
+    coarse=Q(lo,grid);accepted=paid_moment(p,coarse,True);rejected=paid_moment(p,Q(hi,grid),True)
+    need(accepted['upper']<1<rejected['lower'],'adjacent grid paid-moment proof')
+    old=Q(384599,10**10);threshold=coarse/(1+coarse-old);atomgrid=10**24
+    floor=(threshold*atomgrid).numerator//(threshold*atomgrid).denominator
+    atom=Q(floor+1,atomgrid);actual=(1-atom)*coarse+atom*old
+    need(actual<atom<1-actual,'paid atom and row adapter toll')
+    need(Q(2*p['m']**3,2**80)<Q(1,10**16),'fixed prime rare-class bound')
+    need(Q(p['total_rank'])+accepted['bad_fraction']*accepted['fallback_children_per_edge']*accepted['edge_count']<p['W']*p['m'],'contaminated mass contracts')
+    previous=atom-Q(1,atomgrid)
+    need(previous<=(1-previous)*coarse+previous*old,'previous atom grid does not pay strict toll')
+    return dict(coarse_saving=coarse,accepted=accepted,rejected=rejected,coarse_grid=grid,
+        atom_beta=atom,old_atom_saving=old,ordinary_saving=actual,atom_threshold=threshold,
+        atom_grid=atomgrid,atom_lower_gap=atom-actual,atom_upper_gap=1-actual-atom,
+        controls=dict(next_coarse_grid_rejected=True,previous_atom_grid_rejected=True),
+        scope='Adjacent coarse exclusion for this fixed worst-case bad-class envelope and least atom on the stated grid; no true bad-fraction or global optimality claim.')
+
+
+def main():
+    need(not sys.flags.optimize,'assertions enabled')
+    word=Candidate();word.exact_frames();row=word.row()
+    need(row['changed_operation_frames']==2884 and row['reused_registers']==1760,'selected PR168-v4 replacement frames and physical aliases')
+    reference=json.loads((HERE.parent/'references/pr168/bit-physical-profile.json').read_text())
+    for k in ('h','v','m','W_per_vertex','rank_per_vertex','deficit_per_vertex','loss'):
+        need(row[k]==reference[k],'independent physical profile agrees: '+k)
+    need(row['R']==reference['physical_R'],'independent physical role count')
+    need(row['child_histogram']=={int(k):n for k,n in reference['child_histogram'].items()},'independent complete physical histogram')
+    formal=[word.formal(r) for r in (2,0)]
+    controls=[]
+    for tamper in ('omit_compensation','missing_partner','stale'):
+        try:word.formal(2,tamper)
+        except ValueError:controls.append(tamper)
+        else:raise ValueError('Adverse word control accepted: '+tamper)
+    i=word.changed_frames[0];old=word.opframe[i];word.opframe[i]=word.register([])
+    try:word.exact_frames()
+    except ValueError:controls.append('zero_operation_frame')
+    else:raise ValueError('Zero frame accepted')
+    word.opframe[i]=old
+    packed=packed_row(word,row)
+    coarse=certify(packed)
+    independent_gap=independent_bank_moment(packed,coarse)
+    expected=json.loads((HERE/'banks.json').read_text())
+    need(coarse['coarse_saving']==Q(expected['coarse']) and coarse['atom_beta']==Q(expected['atom']) and coarse['ordinary_saving']==Q(expected['effective']),'stored exact paid bank savings')
+    primes=prime_certificate(word)
+    need(primes==json.loads(gzip.decompress((HERE/'prime-witnesses.json.gz').read_bytes())),'source-bound all-frame prime witness reproduction')
+    need(primes['total_operation_frames']==row['changed_operation_frames'] and primes['h']==row['h'],'prime witnesses cover changed frames')
+    from hashlib import sha256
+    prime_summary={k:v for k,v in primes.items() if k!='frame_witnesses'}
+    prime_summary['witness_file']='bit/prime-witnesses.json.gz'
+    prime_summary['witness_sha256']=sha256((HERE/'prime-witnesses.json.gz').read_bytes()).hexdigest()
+    print(json.dumps(js(dict(status='PASS',profile=packed,inherited_profile=row,formal=formal,coarse=coarse,prime_witnesses=prime_summary,word_controls=controls,independent_moment_gap=independent_gap,
+        foreign_producer_replays=0,scope='Complete F2 identity and integer decoder on every source, target and dirty column; exact spliced physical frames; completed18x4/3x24 bank projectors; independent paid moments and atom toll. The inherited weighted compiler, bank routing, stage cover and all-size recovery contracts remain conditional.')),sort_keys=True))
+
+
+if __name__=='__main__':main()
diff --git a/research/coordinated-frames-and-entrance-banks/bit/word.py b/research/coordinated-frames-and-entrance-banks/bit/word.py
new file mode 100644
index 0000000000..53dc827200
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/bit/word.py
@@ -0,0 +1,104 @@
+"""Independent complete formal admission of the frozen PR168-v4 physical bit word.
+
+Retains eumemic's exact integer linear algebra and original decoder/geometry
+predicates. Physical chains, chronological formal columns and paid moments
+use our separate verifier. No producer or foreign aggregate checker runs.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+from hashlib import sha256
+import gzip,importlib.util,json,sys,time
+HERE=Path(__file__).resolve().parent
+SOURCE=HERE.parents[2]
+PKG=HERE.parent
+def module(name,path):
+    spec=importlib.util.spec_from_file_location(name,path);m=importlib.util.module_from_spec(spec);spec.loader.exec_module(m);return m
+def read(p):
+    raw=p.read_bytes();return json.loads(gzip.decompress(raw) if p.suffix=='.gz' else raw)
+old=module('own_frozen_word',PKG/'bit/base_word.py');need=old.need
+
+class Candidate(old.Candidate):
+    def __init__(self):
+        self.module=backend=module('exact_integer_backend',SOURCE/'research/paired-cube-bit/check_paired_cube_bit.py')
+        C=backend.Checker.__new__(backend.Checker);self.C=C;C.mut=None
+        B=SOURCE/'research/paired-cube-bit/out';F=SOURCE/'references/paired-cube/bit-physical'
+        self.input_paths=[B/'graph_p12.json',B/'profile_p12.json',F/'word_p12.json.gz',F/'frames_p12.json.gz',F/'kchron_p12.json']
+        C.g,C.prof,C.w,C.fr,C.k=map(read,self.input_paths)
+        C.h=C.g['h'];C.v=C.g['v'];C.frames()
+        self.h,self.v,self.R=C.h,C.v,C.prof['R'];self.w,self.g,self.k=C.w,C.g,C.k
+        self.ops=self.w['ops'];self.nf={int(n):f for n,f in self.w['node_frame'].items()}
+        self.source={int(n):s for n,s in self.w['sources'].items()};self.gauge={z['role']:z for z in self.w['gauges']}
+        need(len(self.gauge)==len(self.w['gauges']),'unique gauge roles')
+        self.original_opframe=tuple(self.nf[n] for a,b,n in self.ops);self.opframe=self.w['op_frame'][:]
+        need(len(self.opframe)==len(self.ops),'complete operation frame vector')
+        self.changed_frames=[i for i,(a,b) in enumerate(zip(self.original_opframe,self.opframe)) if a!=b]
+        self.phase1=sorted(self.w['phase1']);phase=set(self.phase1)
+        self.rest=[i for i in range(len(self.ops)) if i not in phase];order=self.phase1+self.rest
+        self.position={i:p for p,i in enumerate(order)};pos={i:p for p,i in enumerate(self.rest)}
+        self.order=[z['role'] for z in reversed(self.w['gauges'])]
+        self.role_ops=defaultdict(list)
+        for i in order:
+            for s in self.ops[i][:2]:self.role_ops[s].append(i)
+        need(all(xs==sorted(xs) for xs in self.role_ops.values()),'execution order preserves every role chronology')
+        self.first={s:pos.get(xs[0],-1) for s,xs in self.role_ops.items()}
+        self.last={s:pos.get(xs[-1],-1) for s,xs in self.role_ops.items()}
+        self.endframe={s:self.opframe[xs[-1]] for s,xs in self.role_ops.items()}
+        self.rootroles=set(self.w['rootroles']);self.rootkind={s:r['kind'] for s,r in zip(self.w['rootroles'],self.g['roots'])}
+        need(len(self.rootroles)==len(self.w['rootroles']),'unique root roles')
+        reads={int(s):p for s,p in self.w['reads'].items()}
+        need(set(reads)<=set(self.gauge),'only gauges have explicit reads')
+        self.readtime={s:reads.get(s,len(self.phase1))-len(self.phase1) for s in self.order}
+        need(all(0<=self.readtime[s]<=self.first[s] for s in self.order),'all gauge reads after centre phase and before first use')
+        self.pairs=[[b,a] for a,b in self.w['pairs']];self.donor=dict(self.pairs)
+        need(len(self.donor)==len(self.pairs)==len(set(self.donor.values())),'one-to-one aliases')
+        need(not set(self.donor)&set(self.donor.values()),'disjoint donor recipient sets')
+        for b,d in self.pairs:
+            need(d not in self.gauge and d not in self.rootroles and b in self.gauge,'ungauged non-root donor and gauged recipient')
+            need(self.last[d]<self.readtime[b],'donor dead before recipient read')
+        self.phys={s:self.donor.get(s,s) for s in range(self.R)}
+        need(len(set(self.phys.values()))==self.R-len(self.pairs),'physical slot count')
+        need(all(self.phys[a]!=self.phys[b] for a,b,n in self.ops),'distinct physical gate ports')
+        C.chains=self.base_chains
+
+    def base_chains(self):
+        """Reconstruct phase/gauge/source contracts; row() counts physical paths."""
+        C,w,h,v=self.C,self.w,self.h,self.v
+        order=self.phase1+self.rest
+        roleprev={};pred=[]
+        for i,(a,b,n) in enumerate(self.ops):
+            pred.append((roleprev.get(a,-1),roleprev.get(b,-1)));roleprev[a]=roleprev[b]=i
+        pending=[roleprev[s] for s in self.rootroles if self.rootkind[s]=='center' and s in roleprev];closure=set()
+        while pending:
+            i=pending.pop()
+            if i not in closure:closure.add(i);pending.extend(j for j in pred[i] if j>=0)
+        need(closure==set(self.phase1),'phase one is exactly centre closure')
+        content=[0]*self.R
+        for n,s in self.source.items():content[s]=1<<n
+        for i in order:
+            a,b,n=self.ops[i];args=self.g['args'][n]
+            if content[a]:need(args is not None and {content[a],content[b]}=={C.sup[t] for t in args},'actual addition operands')
+            else:need(content[b]==C.sup[n],'actual copied node value')
+            content[a]|=content[b]
+        co=[set() for _ in range(self.R)]
+        for r,s in zip(self.g['roots'],w['rootroles']):co[s].update(r['targets'])
+        for i in reversed(order):
+            a,b,n=self.ops[i];co[b].update(co[a])
+        touched={s for i in self.phase1 for s in self.ops[i][:2]}
+        for s,z in self.gauge.items():
+            need(s not in self.source.values() and s not in touched,'gauge untouched in phase one and source injection')
+            need(set(z['targets'])==co[s] and C.dimf[z['frame']]==z['dim']>0,'gauge dimension and complete response support')
+        src=Counter();members=Counter();labels=list(map(set,self.g['labels']))
+        for e in self.k['entries']:
+            c,d,f,cap=e['carrier'],e['passive'],e['mix_frame'],e['deliver_frame']
+            need(len(labels[c]&labels[d])==1,'orthogonal partner pair')
+            need(C.dimf[f]==2 and C.in_frame(C.chi[c],f) and C.in_frame(C.chi[d],f),'partner mix span')
+            need(C.sub(f,cap) and all(self.module.dot(C.cov[t],u)==0 for t in e['receivers'] for u in C.B[cap]),'partner mix delivery cap')
+            need(e['carrier_chain']==[w['source_frame'][c],f,cap,w['full_frame']] and e['passive_chain']==[w['source_frame'][d],f,w['full_frame']],'exact partner chains')
+            root=self.g['roots'][e['deliver_after_root']]
+            need(root['kind']=='side' and sorted(e['receivers'])==sorted(root['targets']),'partner delivery at own receiver root')
+            for chain in (e['carrier_chain'],e['passive_chain']):
+                for a,b in zip(chain,chain[1:]):
+                    need(C.sub(a,b),'nested source chronology');src[C.dimf[b]-C.dimf[a]]+=1
+            members[c]+=1;members[d]+=1
+        need(all(members[s]==1 for s in range(v)),'every source in one partner pair')
+        return Counter(),Counter(),src,self.R
diff --git a/research/coordinated-frames-and-entrance-banks/certificate.json b/research/coordinated-frames-and-entrance-banks/certificate.json
new file mode 100644
index 0000000000..cd1eecca63
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/certificate.json
@@ -0,0 +1,3510 @@
+{
+  "arithmetic": {
+    "actual_uniform_bit_saving": "665064157585290854736467094714610539581/1000000000000000000000000000000000000000000",
+    "adverse_control_count": 21,
+    "adverse_controls": [
+      "full group stock replaced by local W",
+      "old ordinary reserve omitted",
+      "conservative old coarse reserve omitted",
+      "uniform supplier replaced by fictional finite profile",
+      "actual ordinary saving overstated",
+      "unpaid previous atom grid",
+      "scalar role reserve collapsed to physical stock",
+      "terminal removals omitted",
+      "terminal removals counted as aliases",
+      "odd divisor silently changed to 21",
+      "C0 detached from scalar charge",
+      "scalar group charge omitted",
+      "stale total row coefficient",
+      "inadequate polynomial row degree",
+      "float in otherwise unused metadata",
+      "noncontracting maximum child",
+      "transfer saving exceeds actual uniform supplier",
+      "old prefix charged at balanced parameters",
+      "old nonlinear scalar guard",
+      "old separate movement exposures",
+      "next final 1e-18 kappa grid point"
+    ],
+    "assembly": {
+      "absorption_gap": "110023995087094205009646790079892270240761981863478586733621100949/500330661635287890212109786896227205924217575782424880899051000000000000000000000000",
+      "constraints": {
+        "K_dominates_log": "331161634787890712109411565440570761327238672637984671845075782424217575119100949/500330661635287890212109786896227205924217575782424880899051000000000000000000000000",
+        "K_geometry": "375330786635162890337109786896227205924217575782424880899051/500330661635287890212109786896227205924217575782424880899051000000000000000000000000",
+        "a_below_b": "663323931899051/1000000000000000000000000000000000000000000",
+        "a_positive": "662323931899050999999999336676068100949/1000000000000000000000000000000000000000000",
+        "alpha_below_one": "999999000000999999000000500330661635287890212109786896227205924217575782424880899051/1000661323270575780424219573792454411848435151564849761798102000000000000000000000000",
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+        "E": 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+        "exact_grid": "One dyadic grid times 3^(-K), K=G*(D_complex+1); completed children preserve the incoming odd exponent.",
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+            "last_write_chronology": 24693,
+            "pivot": 976,
+            "post_after_op": 24517,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              10,
+              4,
+              4
+            ],
+            "role": 10219,
+            "root_index": 1342,
+            "targets": [
+              976,
+              977,
+              978,
+              979,
+              980,
+              981,
+              982,
+              983
+            ],
+            "write_ranks": [
+              10,
+              10,
+              14,
+              18
+            ],
+            "writes": [
+              16243,
+              16464,
+              21921,
+              24517
+            ]
+          },
+          {
+            "first_any_old_read": 25559,
+            "first_pivot_old_read": 27716,
+            "last_write_chronology": 24892,
+            "pivot": 1016,
+            "post_after_op": 24716,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              16,
+              2
+            ],
+            "role": 10638,
+            "root_index": 1397,
+            "targets": [
+              1016,
+              1017,
+              1018,
+              1019,
+              1020,
+              1021,
+              1022,
+              1023
+            ],
+            "write_ranks": [
+              16,
+              18
+            ],
+            "writes": [
+              21994,
+              24716
+            ]
+          },
+          {
+            "first_any_old_read": 25770,
+            "first_pivot_old_read": 27752,
+            "last_write_chronology": 24804,
+            "pivot": 1024,
+            "post_after_op": 24628,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              16,
+              2
+            ],
+            "role": 10621,
+            "root_index": 1408,
+            "targets": [
+              1024,
+              1025,
+              1026,
+              1027,
+              1028,
+              1029,
+              1030,
+              1031
+            ],
+            "write_ranks": [
+              16,
+              18
+            ],
+            "writes": [
+              20861,
+              24628
+            ]
+          },
+          {
+            "first_any_old_read": 27108,
+            "first_pivot_old_read": 27502,
+            "last_write_chronology": 24734,
+            "pivot": 1088,
+            "post_after_op": 24558,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10623,
+            "root_index": 1496,
+            "targets": [
+              1088,
+              1089,
+              1090,
+              1091,
+              1092,
+              1093,
+              1094,
+              1095
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              20870,
+              24558
+            ]
+          },
+          {
+            "first_any_old_read": 25431,
+            "first_pivot_old_read": 27682,
+            "last_write_chronology": 24746,
+            "pivot": 1128,
+            "post_after_op": 24570,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10622,
+            "root_index": 1551,
+            "targets": [
+              1128,
+              1129,
+              1130,
+              1131,
+              1132,
+              1133,
+              1134,
+              1135
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              20867,
+              24570
+            ]
+          },
+          {
+            "first_any_old_read": 27182,
+            "first_pivot_old_read": 27718,
+            "last_write_chronology": 24533,
+            "pivot": 1136,
+            "post_after_op": 24357,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              12,
+              6
+            ],
+            "role": 10496,
+            "root_index": 1562,
+            "targets": [
+              1136,
+              1137,
+              1138,
+              1139,
+              1140,
+              1141,
+              1142,
+              1143
+            ],
+            "write_ranks": [
+              12,
+              18
+            ],
+            "writes": [
+              19241,
+              24357
+            ]
+          },
+          {
+            "first_any_old_read": 25737,
+            "first_pivot_old_read": 27564,
+            "last_write_chronology": 24687,
+            "pivot": 1184,
+            "post_after_op": 24511,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              4,
+              14
+            ],
+            "role": 9127,
+            "root_index": 1628,
+            "targets": [
+              1184,
+              1185,
+              1186,
+              1187,
+              1188,
+              1189,
+              1190,
+              1191
+            ],
+            "write_ranks": [
+              4,
+              18,
+              18
+            ],
+            "writes": [
+              12519,
+              24455,
+              24511
+            ]
+          },
+          {
+            "first_any_old_read": 25428,
+            "first_pivot_old_read": 27612,
+            "last_write_chronology": 24753,
+            "pivot": 1240,
+            "post_after_op": 24577,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10516,
+            "root_index": 1705,
+            "targets": [
+              1240,
+              1241,
+              1242,
+              1243,
+              1244,
+              1245,
+              1246,
+              1247
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              20090,
+              24577
+            ]
+          },
+          {
+            "first_any_old_read": 27652,
+            "first_pivot_old_read": 27794,
+            "last_write_chronology": 24816,
+            "pivot": 1304,
+            "post_after_op": 24640,
+            "pre_at_phase_cut_after_center_before_old_reads": true,
+            "prefix_children": [
+              15,
+              3
+            ],
+            "role": 10628,
+            "root_index": 1793,
+            "targets": [
+              1304,
+              1305,
+              1306,
+              1307,
+              1308,
+              1309,
+              1310,
+              1311
+            ],
+            "write_ranks": [
+              15,
+              18
+            ],
+            "writes": [
+              20892,
+              24640
+            ]
+          }
+        ],
+        "status": "AUTHOR_EXACT_FINITE_COMPILER_CHECK_PASS",
+        "target_histogram": {
+          "0": 6270,
+          "1": 3960,
+          "10": 9,
+          "11": 2,
+          "12": 6,
+          "13": 2,
+          "14": 4,
+          "15": 12,
+          "16": 8,
+          "18": 1274,
+          "2": 24,
+          "3": 16,
+          "4": 9,
+          "5": 2,
+          "6": 7,
+          "7": 3,
+          "8": 5,
+          "9": 1
+        },
+        "targets_disjoint": true,
+        "v": 1320
+      },
+      "terminal_formal": {
+        "absolute_residual_bound": 416406,
+        "columns": [
+          {
+            "columns": 13326,
+            "direction": 1,
+            "dirty": 10686,
+            "outputs": true,
+            "restored": true
+          },
+          {
+            "columns": 13326,
+            "direction": -1,
+            "dirty": 10686,
+            "outputs": true,
+            "restored": true
+          }
+        ],
+        "controls": {
+          "omit-ancestor-response": "REJECTED",
+          "omit-first-write": "REJECTED",
+          "omit-pre-shear": "REJECTED"
+        },
+        "digit_bits": 24,
+        "input_pins": {
+          "baseline.json.gz": "f4ad8cd1635f279d201225c174548cb03db13b63156cdf647ca8e834d4746524",
+          "frames.json.gz": "b516b59e7bd7e68c56c02e0041e7455b2eff445e81ac5fb69570d424cf0178a1",
+          "graph.json.gz": "9f73a6a7f1a4410e954246c6ff95d9cad0582d0057f707a693cc2fd170995ead",
+          "physical-frames.json.gz": "0e92261dbedacb1174ee58712f0a86dab00a4f2411e377faab7c08f54f8307f0",
+          "physical-pairs.json.gz": "869b852cbfd63ef691336ea398280ef57896dd876f6a6c78edac88a2a304fffb",
+          "physical.py": "496a5227d525e9291b05db7e6050ec866a7160e5bcd5da0a00bb7e5210efb996",
+          "profile-before.json.gz": "bd39675ce03fd1d4d456f736bcfc0b389532eb4f0a19c92877c2af5cb6f7310c",
+          "profile.json.gz": "cf80065ea589fcb0e9e83a2dfc76f694e3ca8bc2d8290f4175c45e2a100b8f84",
+          "sinks.json": "498f7ccc51db7d09d9dc191f0b0d44fd70f3f8bf374cf3f52548880bd42f04ec",
+          "word.json.gz": "0c4b720c57717c7e55e88f86a9e17ed75b3891f9d2e7ed29df0a188f8dfee67a"
+        },
+        "source_sha256": "d7fefe86e637584ad1bf53c8f7714b88d5abed92eaa29d110ef099536e51f516",
+        "status": "PASS full modified finite word"
+      }
+    },
+    "profile": {
+      "R": 10686,
+      "W_per_vertex": 13326,
+      "c": 21249,
+      "child_histogram": {
+        "1": 88140,
+        "10": 2580,
+        "11": 987,
+        "12": 3552,
+        "13": 642,
+        "14": 2142,
+        "15": 3258,
+        "16": 327,
+        "17": 132,
+        "18": 7782,
+        "19": 5334,
+        "2": 45426,
+        "20": 1551,
+        "3": 29823,
+        "4": 10488,
+        "5": 7248,
+        "6": 3987,
+        "7": 2784,
+        "8": 3603,
+        "9": 1434
+      },
+      "deficit_per_vertex": 1320,
+      "h": 22,
+      "loss": 440,
+      "m": 66,
+      "matched": 11364,
+      "maxchild": 20,
+      "q": 3157,
+      "rank_per_vertex": 878196,
+      "reuse_pairs": 2310,
+      "scalar_role_reserve": 13042,
+      "terminal_sinks": 46,
+      "total_M_operations": 32971,
+      "v": 1320
+    },
+    "status": "PASS"
+  },
+  "kappa": "330942774629799/500000000000000000",
+  "scope": "Finite word, paid moments, full finite bridge and 47 strict arithmetic constraints; source-specified all-size contracts remain assumptions.",
+  "status": "PASS conditional finite witness"
+}
diff --git a/research/coordinated-frames-and-entrance-banks/complex/check_sinks.py b/research/coordinated-frames-and-entrance-banks/complex/check_sinks.py
new file mode 100644
index 0000000000..d24d869c07
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/complex/check_sinks.py
@@ -0,0 +1,216 @@
+#!/usr/bin/env python3
+"""Author exact finite check for target-accumulating disjoint-root deletion.
+
+Portability-only adaptation by Chafik Boukhalfa with OpenAI Codex assistance.
+Original PR166 author James Chang. Uses frozen selected complex data; never imports upstream programs,
+regenerates a source graph, searches a new producer, or computes a new kappa.
+Python standard library. Copyright 2026; Apache-2.0.
+"""
+import argparse,gzip,hashlib,json,signal,sys,time
+try:
+ import resource
+except ImportError:
+ class resource:
+  RUSAGE_SELF=0
+  RLIMIT_CPU=0
+  @staticmethod
+  def setrlimit(*_): pass
+  @staticmethod
+  def getrusage(_):
+   class Usage: ru_maxrss=0
+   return Usage()
+from collections import Counter
+from fractions import Fraction
+from pathlib import Path
+
+
+def main():
+ if sys.flags.optimize:raise SystemExit('Run without -O')
+ ap=argparse.ArgumentParser(description=__doc__)
+ ap.add_argument('--candidate',type=Path,required=True)
+ ap.add_argument('--output',type=Path,required=True)
+ a=ap.parse_args();start_time=time.monotonic()
+ resource.setrlimit(resource.RLIMIT_CPU,(60,60));getattr(signal,"alarm",lambda _:None)(60)
+ pins=json.loads((a.candidate/'result.json').read_text())['pins'];data={}
+ for name in ['graph','word','frames','physical-frames','physical-pairs','profile-before','profile']:
+  name+='.json.gz';b=(a.candidate/name).read_bytes();assert hashlib.sha256(b).hexdigest()==pins[name]
+  data[name[:-8]]=json.loads(gzip.decompress(b))
+ g,w,fw,changes,pairs,row,profile=[data[n] for n in ['graph','word','frames','physical-frames','physical-pairs','profile-before','profile']]
+ H,V,R=g['h'],g['v'],row['R'];assert (H,V)==(22,1320)
+ def basis(rows):
+  b={}
+  for x in rows:
+   for p,y in sorted(b.items(),reverse=True):
+    if x>>p&1:x^=y
+   if x:
+    p=x.bit_length()-1
+    for k,y in list(b.items()):
+     if y>>p&1:b[k]=y^x
+    b[p]=x
+  return tuple(b[k] for k in sorted(b,reverse=True))
+ def perp(rows):
+  rows=basis(rows);p={r.bit_length()-1:r for r in rows};out=[]
+  for j in range(H):
+   if j in p:continue
+   x=1<<j
+   for k,r in p.items():
+    if r>>j&1:x|=1<<k
+   out.append(x)
+  return basis(out)
+ def within(A,B):return len(basis(tuple(A)+tuple(B)))==len(B)
+ def fmt(c):return {str(k):v for k,v in sorted(c.items()) if v}
+ FULL=basis(1<<i for i in range(H));sources={int(k):v for k,v in w['sources'].items()}
+ aliases={b:a for a,b,t in pairs};donors={a:b for a,b,t in pairs};deadline={b:t for a,b,t in pairs};alias=lambda s:aliases.get(s,s)
+ live=set(range(R))-set(aliases);assert len(live)==profile['physical_R']
+ phase=set(w['phase1']);chron=w['phase1']+[i for i in range(len(w['ops'])) if i not in phase];when={i:j for j,i in enumerate(chron)}
+ ofs=[perp(fw['annihilators'][n]) for aa,bb,n in w['ops']]
+ for i,F in changes:assert basis(F)==tuple(F);ofs[i]=tuple(F)
+ roleops=[[] for _ in range(R)];roots_of=[[] for _ in range(R)]
+ for i,(aa,bb,n) in enumerate(w['ops']):
+  assert alias(aa)!=alias(bb);roleops[aa].append(i);roleops[bb].append(i)
+ for j,s in enumerate(w['rootroles']):roots_of[s].append(j)
+ spans=[()]*len(g['args'])
+ for n,args in enumerate(g['args']):spans[n]=(g['inputs'][n],) if args is None else basis(spans[args[0]]+spans[args[1]])
+ rootframes=[spans[r['node']] if r['kind']=='center' else perp(g['inputs'][t] for t in r['targets']) for r in g['roots']]
+ selected={z['role']:z for z in w['selected']};first_read=[len(chron)+1]*V
+ for z in selected.values():
+  t=deadline.get(z['role']);position=len(phase) if t is None else when[t]
+  for y in z['targets']:first_read[y]=min(first_read[y],position)
+ # Recount the complete old physical row paths and compare every positive bin.
+ starts=[()]*R
+ for n,s in sources.items():starts[s]=(g['inputs'][n-1],)
+ for s,z in selected.items():starts[s]=perp(z['annihilator'])
+ rframe={s:rootframes[j] for j,s in enumerate(w['rootroles'])};rkind={s:g['roots'][j]['kind'] for j,s in enumerate(w['rootroles'])}
+ def chain(s):return [starts[s]]+[ofs[i] for i in roleops[s]]+([rframe[s]] if s in rframe else [])+[FULL]
+ local=Counter()
+ for s in live:
+  seq=chain(s)
+  if s in donors:seq=seq[:-1]+chain(donors[s])
+  assert all(within(A,B) for A,B in zip(seq,seq[1:]))
+  local.update(len(B)-len(A) for A,B in zip(seq,seq[1:]) if len(B)>len(A))
+  if s in sources.values():local[1]+=1
+  for t in (s,donors.get(s)):
+   if t is not None and rkind.get(t)=='center':local[len(rframe[t])]+=1
+ assert fmt(local)==profile['local_histogram']
+ source=Counter({int(k):v for k,v in profile['source_data_histogram'].items()});target=Counter({int(k):v for k,v in profile['target_data_histogram'].items()})
+ def children(L,T):
+  c=Counter({r:3*n for r,n in L.items() if r});c.update({r:3*n for r,n in T.items() if r});c.update({r:3*n for r,n in source.items() if r});c[2]+=2*V
+  for s,z in selected.items():
+   if s not in aliases:c[3*z['rank']]+=1
+  return c
+ oldchildren=children(local,target);assert fmt(oldchildren)==profile['child_histogram']
+ rejected=Counter();eligible=[]
+ for j,r in enumerate(g['roots']):
+  if r.get('channel')!='disjoint':continue
+  s=w['rootroles'][j];writes=[i for i in roleops[s] if w['ops'][i][0]==s];uses=[i for i in roleops[s] if w['ops'][i][1]==s];why=[];U=rootframes[j]
+  if s in aliases or s in donors:why.append('alias_or_donor')
+  if s in sources.values():why.append('source_injection')
+  if s in selected:why.append('selected_gauge')
+  if roots_of[s]!=[j]:why.append('multiple_roots')
+  if uses:why.append('retained_producer_consumer')
+  if not writes:why.append('no_writes')
+  if any(w['opcoeff'][i][0]!=1 for i in writes):why.append('nonadditive_destination')
+  if any(not within(ofs[i],U) for i in writes):why.append('outside_read_frame')
+  if any(i in phase for i in writes):why.append('center_phase_touch')
+  available_pivots=[t for t in r['targets'] if writes and first_read[t]>max(when[i] for i in writes)]
+  if not available_pivots:why.append('no_uncorrected_pivot')
+  if why:rejected.update(why);continue
+  assert len(r['targets'])==8 and len(U)==18 and r['coefficients']==['1/2']*8
+  # The sink's whole original trajectory is copied by only one target pivot.
+  # Every other target stays at0 until the common rank18 post-read.
+  path=[()]+[ofs[i] for i in writes]+[U];assert all(within(A,B) for A,B in zip(path,path[1:]))
+  prefix=[len(B)-len(A) for A,B in zip(path,path[1:]) if len(B)>len(A)]
+  assert sum(prefix)==18
+  eligible.append(dict(root_index=j,role=s,targets=r['targets'],pivot=available_pivots[0],writes=writes,write_ranks=[len(ofs[i]) for i in writes],prefix_children=prefix,pre_at_phase_cut_after_center_before_old_reads=True,post_after_op=writes[-1],last_write_chronology=max(when[i] for i in writes),first_pivot_old_read=first_read[available_pivots[0]],first_any_old_read=min(first_read[t] for t in r['targets'])))
+ assert len(set(t for x in eligible for t in x['targets']))==8*len(eligible)
+ removed={x['role'] for x in eligible};removed_ops={i for x in eligible for i in x['writes']}
+ assert all(aa not in removed and bb not in removed for i,(aa,bb,n) in enumerate(w['ops']) if i not in removed_ops)
+ # All formal source and dirty coefficients of every selected sink are checked
+ # by exact adjoint pullback of its current final value through actual aliases.
+ # Remaining source/dirty identity follows from literal inverse cancellation
+ # of the unchanged retained word; no sampler or packed-digit hypothesis.
+ def final_form(s,stop):
+  c={s:1}
+  for i in reversed(chron[:stop+1]):
+   aa,bb,n=w['ops'][i];aa,bb=alias(aa),alias(bb);ca,cb=w['opcoeff'][i];v=c.get(aa,0)
+   if v:
+    c[aa]=ca*v;c[bb]=c.get(bb,0)+cb*v
+    if not c[bb]:del c[bb]
+  fresh={n-1:c[alias(s)] for n,s in sources.items() if c.get(alias(s),0)}
+  return c,fresh
+ dirtyterms=[]
+ for item in eligible:
+  s=item['role'];c,fresh=final_form(s,item['last_write_chronology']);assert c[s]==1
+  assert not (set(c)&(removed-{s}))
+  cube={x//2 for x in g['labels'][item['targets'][0]]}
+  expected={n:1 for n,t in enumerate(g['labels']) if cube.isdisjoint(x//2 for x in t)}
+  assert fresh==expected and len(fresh)==448;dirtyterms.append(len(c)-1)
+  # Exact independent-target/sink/accumulation-symbol sandwich, both signs.
+  # Algebra includes arbitrary previous target corrections in y[t].
+  for direction in (1,-1):
+   N=16+len(item['writes']);ys=[[Fraction(int(i==t)) for i in range(N)] for t in range(8)]
+   for t in range(1,8):ys[t]=[a-b for a,b in zip(ys[t],ys[0])]
+   # Arbitrary intervening non-pivot corrections are explicit independent
+   # columns. The chosen pivot has none until after its post-read.
+   for t in range(1,8):ys[t][8+len(item['writes'])+t]+=1
+   for k,i in enumerate(item['writes']):ys[0][8+k]+=direction*Fraction(w['opcoeff'][i][1],2)
+   for t in range(1,8):ys[t]=[a+b for a,b in zip(ys[t],ys[0])]
+   for t in range(8):
+    expected=[Fraction(int(i==t)) if i<8 else direction*Fraction(w['opcoeff'][item['writes'][i-8]][1],2) if i<8+len(item['writes']) else Fraction(int(t>0 and i==8+len(item['writes'])+t)) for i in range(N)]
+    assert ys[t]==expected
+ # Whole simultaneous local and shared three-core ledger.
+ L=local.copy();T=target.copy()
+ for item in eligible:
+  L.subtract(Counter(item['prefix_children']+[4]));T[18]-=1;T.update(item['prefix_children'])
+ assert all(v>=0 for v in L.values()) and all(v>=0 for v in T.values())
+ # Replay every target event in the actual deadline order, interleaving the
+ # new pivot writes and post-reads. Pre-shears are at the common zero frame
+ # just after the center phase and before its first compensation read.
+ by_write={i:item for item in eligible for i in item['writes']}
+ after={item['writes'][-1]:item for item in eligible}
+ read_at={}
+ for z in reversed(w['selected']):
+  d=deadline.get(z['role']);position=len(phase) if d is None else when[d]
+  read_at.setdefault(position,[]).append(z)
+ current=[() for _ in range(V)];actual_target=Counter();target_events=[]
+ def move_target(t,U,kind,index):
+  U=tuple(U);assert within(current[t],U),(kind,index,t,current[t],U)
+  actual_target[len(U)-len(current[t])]+=1
+  target_events.append([kind,index,t,list(current[t]),list(U)])
+  current[t]=U
+ for position,i in enumerate(chron):
+  for z in read_at.get(position,[]):
+   U=perp(z['annihilator'])
+   for t in z['targets']:move_target(t,U,'old-read',z['role'])
+  if i in by_write:
+   item=by_write[i];move_target(item['pivot'],ofs[i],'pivot-write',i)
+  if i in after:
+   item=after[i];U=rootframes[item['root_index']]
+   for t in item['targets']:move_target(t,U,'post-read',i)
+   assert all(current[t]==current[item['pivot']]==U for t in item['targets'])
+ for j,r in enumerate(g['roots']):
+  if r['kind']=='side' and w['rootroles'][j] not in removed:
+   for t in r['targets']:move_target(t,rootframes[j],'retained-root',j)
+ for t,q in enumerate(g['inputs']):move_target(t,perp((q,)),'target-finish',t)
+ assert fmt(Counter({r:n for r,n in actual_target.items() if r}))==fmt(Counter({r:n for r,n in T.items() if r}))
+ # Independent reverse of each scalar sandwich, including arbitrary inserted
+ # non-pivot corrections, yields the inverse rather than assuming a sign rule.
+ for item in eligible:
+  N=16+len(item['writes']);initial=[[Fraction(int(i==t)) for i in range(N)] for t in range(8)]
+  state=[r[:] for r in initial];events=[]
+  for t in range(1,8):events.append(('shear',t,0,Fraction(-1)))
+  for k,i in enumerate(item['writes']):events.append(('basis',0,8+k,Fraction(w['opcoeff'][i][1],2)))
+  for t in range(1,8):events.append(('basis',t,8+len(item['writes'])+t,Fraction(1)))
+  for t in range(1,8):events.append(('shear',t,0,Fraction(1)))
+  def apply(e,sign):
+   kind,t,s,c=e;c*=sign
+   if kind=='basis':state[t][s]+=c
+   else:state[t]=[a+c*b for a,b in zip(state[t],state[s])]
+  for e in events:apply(e,1)
+  for e in reversed(events):apply(e,-1)
+  assert state==initial
+ C=children(L,T);n=len(eligible);delta={str(k):C[k]-oldchildren[k] for k in C.keys()|oldchildren.keys() if C[k]!=oldchildren[k]}
+ assert delta==({'4':-3*n,'18':-3*n} if n else {});W=profile['W_per_vertex']-n;rank=sum(k*v for k,v in C.items());assert rank==profile['rank_per_vertex']-66*n and 66*W-rank==1320
+ result=dict(status='AUTHOR_EXACT_FINITE_COMPILER_CHECK_PASS',scope='Only all-eight disjoint roots; retained full proof contracts unchanged',input_pins={k:pins[k] for k in pins if k in [n+'.json.gz' for n in data]},h=H,v=V,old_physical_R=len(live),new_physical_R=len(live)-n,old_W=profile['W_per_vertex'],new_W=W,old_rank=profile['rank_per_vertex'],new_rank=rank,deficit=1320,eligible_count=n,rejected_conditions=dict(rejected),targets_disjoint=True,simultaneous_compatible=True,all_formal_source_and_dirty_coefficients=True,formal_original_source_dirty_target_columns=2*V+len(live),remaining_formal_columns=2*V+len(live)-n,chronological_inverse_by_literal_reverse=True,full_old_physical_histogram_verified=True,all_new_target_events_replayed=True,scalar_sandwich_literal_reverse_checked=True,shared_child_delta=delta,scalar_gate_delta=-sum(len(x['writes'])+2 for x in eligible),new_odd_denominator=False,new_kappa_claimed=False,local_histogram=fmt(L),target_histogram=fmt(T),child_histogram=fmt(C),sinks=eligible,dirty_coefficient_nonzero_range=[min(dirtyterms),max(dirtyterms)],seconds=time.monotonic()-start_time,rss_kib=resource.getrusage(resource.RUSAGE_SELF).ru_maxrss)
+ a.output.write_text(json.dumps(result,indent=2)+'\n');print(json.dumps({k:v for k,v in result.items() if k not in ['sinks','local_histogram','target_histogram','child_histogram','input_pins']},indent=2))
+if __name__=='__main__':main()
diff --git a/research/coordinated-frames-and-entrance-banks/complex/formal_sinks.py b/research/coordinated-frames-and-entrance-banks/complex/formal_sinks.py
new file mode 100644
index 0000000000..703d57f386
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/complex/formal_sinks.py
@@ -0,0 +1,137 @@
+"""Independent full execution of PR166 sink substitution on our fused168.
+
+James Chang's PR166 compiler; fused168 source lineage in its SOURCE.json.
+New full packed-column executor by Chafik Boukhalfa with OpenAI Codex.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import defaultdict
+from hashlib import sha256
+import argparse,gzip,json,time,importlib.util,sys
+
+BASE=Path(__file__).resolve().parent
+ap=argparse.ArgumentParser(description=__doc__)
+ap.add_argument('--candidate',type=Path,default=BASE.parents[1]/'r13-search/fused168')
+ap.add_argument('--selection',type=Path,default=BASE/'fused168-screen.json')
+ap.add_argument('--formal-helper',type=Path)
+ap.add_argument('--output',type=Path,default=BASE/'formal-sinks.json')
+args=ap.parse_args();INPUT=args.candidate;HELPER=args.formal_helper or INPUT/'prove.py'
+if sys.flags.optimize:raise SystemExit('Run without -O')
+spec=importlib.util.spec_from_file_location('original_formal',HELPER)
+old=importlib.util.module_from_spec(spec);spec.loader.exec_module(old)
+def load(name):return json.loads(gzip.decompress((INPUT/(name+'.json.gz')).read_bytes()))
+g,w,row=[load(n) for n in ['graph','word','profile-before']]
+pairs=load('physical-pairs');selection=json.loads(args.selection.read_text())
+for name,digest in selection['input_pins'].items():assert sha256((INPUT/name).read_bytes()).hexdigest()==digest,name
+sinks=selection['sinks'];removed={s['role'] for s in sinks};sinkroots={s['root_index'] for s in sinks}
+alias={b:a for a,b,t in pairs};R=row['R'];v=g['v']
+live=[s for s in range(R) if s not in alias and s not in removed]
+slot={s:i for i,s in enumerate(live)}
+phys=lambda s:slot[alias.get(s,s)]
+D,adj=old.adjoint(g,w,R)
+ops=w['ops'];coef=w['opcoeff'];phase=w['phase1'];rest=[i for i in range(len(ops)) if i not in set(phase)]
+deadlines=dict((b,t) for a,b,t in pairs);at=defaultdict(list)
+for s,t in deadlines.items():
+ if t is not None:at[t].append(s)
+selected=[z['role'] for z in w['selected']];deferred=set(selected)
+sources={int(x):s for x,s in w['sources'].items()}
+bywrite={i:s for s in sinks for i in s['writes']};after={s['writes'][-1]:s for s in sinks}
+cnum=[[int(6*Q(c)) for c in r['coefficients']] for r in g['roots']]
+assert len(live)==selection['new_physical_R']
+
+def execute(direction,bits=None,mutation=None):
+ # With bits=None this is an absolute coefficient L1 majorant. With bits
+ # supplied it is the actual signed word on an injectively packed basis.
+ major=bits is None; top=6; count=2*v+len(live)
+ def basis(i):return 6 if major else 6<<(bits*i)
+ def add(x,c,y):return x+abs(c)*y if major else x+c*y
+ def track(xs):
+  nonlocal top
+  if major:top=max(top,max(xs,default=0))
+ X=[basis(i) for i in range(v)];Y=[basis(v+i) for i in range(v)];Z=[basis(2*v+i) for i in range(len(live))]
+ initial=Y[:];z=Z[:]
+ # Retain the ORIGINAL response of every retained dirty coordinate,
+ # including responses propagated through the deleted sinks.
+ virtual=[0 if s in deferred or s in alias or s in removed else Z[phys(s)] for s in range(R)]
+ def gate(values,i):
+  a,b,_=ops[i];ca,cb=coef[i]
+  values[a]=abs(ca)*values[a]+abs(cb)*values[b] if major else ca*values[a]+cb*values[b]
+ def decode(values,kind=None,skip=False):
+  out=[0]*v
+  for j,(root,s,cs) in enumerate(zip(g['roots'],w['rootroles'],cnum)):
+   if kind is not None and root['kind']!=kind:continue
+   if skip and j in sinkroots:continue
+   for t,c in zip(root['targets'],cs):out[t]=add(out[t],c,values[s])
+  assert all(x%6==0 for x in out)
+  return [x//6 for x in out]
+ for i in range(len(ops)):gate(virtual,i)
+ response=decode(virtual,skip=mutation=='omit-ancestor-response')
+ Y=[add(y,-direction,r) for y,r in zip(Y,response)];track(Y);del virtual,response
+ for n,s in sources.items():z[phys(s)]=add(z[phys(s)],1,X[n-1])
+ done=[]
+ def actualgate(i):
+  a,b,_=ops[i];ca,cb=coef[i];a,b=phys(a),phys(b)
+  z[a]=abs(ca)*z[a]+abs(cb)*z[b] if major else ca*z[a]+cb*z[b]
+  if major:
+   nonlocal top
+   top=max(top,z[a])
+  done.append(i)
+ def read(s):
+  for t,c in D[s].items():
+   assert z[phys(s)]%6==0
+   Y[t]=add(Y[t],-direction*c,z[phys(s)]//6)
+  track(Y)
+ for i in phase:actualgate(i)
+ virtual=[0 if s in removed else z[phys(s)] for s in range(R)]
+ center=decode(virtual,'center');Y=[add(y,direction,c) for y,c in zip(Y,center)];track(Y);del virtual,center
+ for sink in sinks:
+  pivot=sink['pivot']
+  for t in sink['targets']:
+   if t!=pivot and mutation!='omit-pre-shear':Y[t]=add(Y[t],-1,Y[pivot])
+ track(Y)
+ for s in reversed(selected):
+  if deadlines.get(s) is None:read(s)
+ for i in rest:
+  for s in at.get(i,[]):read(s)
+  if i in bywrite:
+   sink=bywrite[i];pivot=sink['pivot'];b=phys(ops[i][1]);assert z[b]%2==0
+   if mutation!='omit-first-write' or i!=sinks[0]['writes'][0]:Y[pivot]=add(Y[pivot],direction*coef[i][1],z[b]//2)
+   track(Y)
+  else:actualgate(i)
+  if i in after:
+   sink=after[i];pivot=sink['pivot']
+   for t in sink['targets']:
+    if t!=pivot:Y[t]=add(Y[t],1,Y[pivot])
+   track(Y)
+ virtual=[0 if s in removed else z[phys(s)] for s in range(R)]
+ side=decode(virtual,'side',True);Y=[add(y,direction,c) for y,c in zip(Y,side)];del virtual,side
+ for a in range(0,v,8):
+  for j,q in enumerate(g['inputs'][a:a+8]):
+   value=0
+   for r,x in zip(g['inputs'][a:a+8],X[a:a+8]):
+    c=1 if (q^r).bit_count()==6 else -1 if (q^r).bit_count()==2 else 0
+    value=add(value,c,x)
+   assert value%2==0;Y[a+j]=add(Y[a+j],direction,value//2)
+ track(Y)
+ if not major:assert Y==[y+direction*x for y,x in zip(initial,X)],'all formal output columns'
+ for i in reversed(done):
+  a,b,_=ops[i];ca,cb=coef[i];a,b=phys(a),phys(b)
+  z[a]=z[a]+abs(cb)*z[b] if major else ca*(z[a]-cb*z[b])
+  if major:top=max(top,z[a])
+ for n,s in sources.items():z[phys(s)]=add(z[phys(s)],-1,X[n-1])
+ track(z)
+ if not major:assert z==Z,'retained arbitrary dirty columns restored'
+ # Include the expected map in a safe final residual bound.
+ return top+12 if major else dict(direction=direction,columns=count,dirty=len(live),outputs=True,restored=True)
+
+if __name__=='__main__':
+ t=time.monotonic();bound=execute(1);bits=8*((bound.bit_length()+2+7)//8);assert 2**bits>2*bound
+ print('majorant',bound,'digit bits',bits,flush=True)
+ columns=[execute(d,bits) for d in [1,-1]]
+ controls={}
+ for mutation in ['omit-first-write','omit-ancestor-response','omit-pre-shear']:
+  try:execute(1,bits,mutation)
+  except AssertionError:controls[mutation]='REJECTED'
+  else:raise AssertionError('accepted '+mutation)
+ result=dict(status='PASS full modified finite word',absolute_residual_bound=bound,digit_bits=bits,columns=columns,controls=controls,source_sha256=sha256(Path(__file__).read_bytes()).hexdigest(),input_pins={p.name:sha256(p.read_bytes()).hexdigest() for p in list(INPUT.glob('*.json.gz'))+[args.selection,HELPER]},seconds=time.monotonic()-t)
+ args.output.write_text(json.dumps(result,indent=2)+'\n');print(json.dumps(result))
diff --git a/research/coordinated-frames-and-entrance-banks/complex/physical.py b/research/coordinated-frames-and-entrance-banks/complex/physical.py
new file mode 100644
index 0000000000..8e55fa721d
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/complex/physical.py
@@ -0,0 +1,177 @@
+#!/usr/bin/env python3
+"""Exact audit of the extended-link fused PR161/162 complex physical word, including every input.
+
+Chafik Boukhalfa with OpenAI Codex assistance. Apache-2.0.
+The scalar signed graph, binary frames and original compiler are credited to
+icekylinx (PR144); searched modules and physical reuse baseline to eumemic
+(PR157 and PR161), extended carrier closure by DaysSky (PR162), with the lineage retained there. New checks independently execute
+all physical aliases, deadline reads and inverse cleanup on an injective
+packed formal basis, using a separately derived absolute coefficient bound.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import defaultdict,Counter
+from hashlib import sha256
+import argparse,gzip,json,sys,time,copy,gc
+try:
+ import resource
+except ImportError:
+ class resource:
+  RUSAGE_SELF=0
+  @staticmethod
+  def getrusage(_):
+   class Usage: ru_maxrss=0
+   return Usage()
+
+
+def adjoint(g,word,R):
+    D=[{} for _ in range(R)]
+    for root,s in zip(g['roots'],word['rootroles']):
+        for t,c in zip(root['targets'],root['coefficients']):
+            c=Q(c)*6;assert c.denominator==1;D[s][t]=D[s].get(t,0)+int(c)
+    for (a,b,_),(ca,cb) in zip(reversed(word['ops']),reversed(word['opcoeff'])):
+        for t,c in D[a].items():D[b][t]=D[b].get(t,0)+cb*c
+        if ca<0:D[a]={t:-c for t,c in D[a].items()}
+    D=[{t:c for t,c in row.items() if c} for row in D]
+    for z in word['selected']:assert set(D[z['role']])<=set(z['targets'])
+    receipt=dict(nonzeros=sum(map(len,D)),max_numerator_over_6=max(abs(c) for row in D for c in row.values()),L1=sum(abs(c) for row in D for c in row.values()),gauge_supports_contained=True)
+    selected={z['role']:D[z['role']] for z in word['selected']}
+    return selected,receipt
+
+class Formal:
+    def __init__(self,g,word,row,pairs,D):
+        self.g,self.word,self.row,self.D=g,word,row,D;self.v=g['v'];self.R=row['R']
+        self.merge={b:a for a,b,t in pairs};assert len(self.merge)==len(pairs)==len(set(self.merge.values()))
+        assert not set(self.merge)&set(self.merge.values())
+        self.live=[s for s in range(self.R) if s not in self.merge];self.slot={s:i for i,s in enumerate(self.live)}
+        self.phys=[self.slot[self.merge.get(s,s)] for s in range(self.R)]
+        self.sources={int(x):s for x,s in word['sources'].items()}
+        self.selected=[z['role'] for z in word['selected']];self.deferred=set(self.selected)
+        self.phase=sorted(word['phase1']);pset=set(self.phase);self.rest=[i for i in range(len(word['ops'])) if i not in pset]
+        self.deadline={b:t for a,b,t in pairs};self.at=defaultdict(list)
+        for b,t in self.deadline.items():
+            if t is not None:self.at[t].append(b)
+        self.ops=[tuple(x) for x in word['ops']];self.coef=[tuple(x) for x in word['opcoeff']]
+        assert all(a in (-1,1) and b in (-1,1) for a,b in self.coef)
+        assert all(self.phys[a]!=self.phys[b] for a,b,x in self.ops)
+        self.cnum=[[int(6*Q(c)) for c in r['coefficients']] for r in g['roots']]
+        self.bound=self.coefficient_bound()
+        self.width=8*((self.bound.bit_length()+2+7)//8);self.base=1<<self.width
+        assert self.base>2*self.bound
+    def coefficient_bound(self):
+        # Unscaled formal variables: a physical slot has coefficients bounded
+        # by its absolute row sum b. Actual packed slots use 6 times the basis.
+        g,word,v,R=self.g,self.word,self.v,self.R
+        old=[int(s not in self.deferred and s not in self.merge) for s in range(R)]
+        for (a,b,_),(ca,cb) in zip(self.ops,self.coef):old[a]=abs(ca)*old[a]+abs(cb)*old[b]
+        y=[6]*v
+        for root,s,cs in zip(g['roots'],word['rootroles'],self.cnum):
+            for t,c in zip(root['targets'],cs):y[t]+=abs(c)*old[s]
+        b=[1]*len(self.live);top=max([6*x for x in old]+y)
+        for node,s in self.sources.items():b[self.phys[s]]+=1
+        done=[]
+        def gate(i):
+            nonlocal top
+            (a,c,_),(ca,cb)=self.ops[i],self.coef[i];a,c=self.phys[a],self.phys[c]
+            b[a]=abs(ca)*b[a]+abs(cb)*b[c];top=max(top,6*b[a]);done.append(i)
+        def read(s):
+            for t,c in self.D[s].items():y[t]+=abs(c)*b[self.phys[s]]
+        for i in self.phase:gate(i)
+        # All root/center positive readouts may be safely bounded by their
+        # value after the complete M: centers are closed in phase one.
+        for s in reversed(self.selected):
+            if self.deadline.get(s) is None:read(s)
+        for i in self.rest:
+            for s in self.at.get(i,[]):read(s)
+            gate(i)
+        for root,s,cs in zip(g['roots'],word['rootroles'],self.cnum):
+            for t,c in zip(root['targets'],cs):y[t]+=abs(c)*b[self.phys[s]]
+        # K has four +/-1/2 source contributions in every output row.
+        y=[x+12+6 for x in y];top=max(top,max(y))
+        for i in reversed(done):
+            (a,c,_),(ca,cb)=self.ops[i],self.coef[i];a,c=self.phys[a],self.phys[c]
+            b[a]=abs(ca)*b[a]+abs(ca*cb)*b[c];top=max(top,6*b[a])
+        for node,s in self.sources.items():b[self.phys[s]]+=1;top=max(top,6*b[self.phys[s]])
+        return top
+    def decode(self,z,kind=None):
+        out=[0]*self.v
+        for root,s,cs in zip(self.g['roots'],self.word['rootroles'],self.cnum):
+            if kind is not None and root['kind']!=kind:continue
+            for t,c in zip(root['targets'],cs):out[t]+=c*z[s]
+        assert all(x%6==0 for x in out);return [x//6 for x in out]
+    def run(self,direction,mutation=None):
+        v,R,w=self.v,self.R,self.width
+        X=[6<<(w*i) for i in range(v)];Y0=[6<<(w*(v+i)) for i in range(v)];Z=[6<<(w*(2*v+i)) for i in range(len(self.live))]
+        # Compute precisely the initial reads D[s]z_s without expanding
+        # nongauged D: exact transposition identity D=JM gives this cheaper
+        # independent algebraic evaluation. All late physical reads below
+        # use the explicitly computed integer adjoint rows.
+        old=[Z[self.phys[s]] if s not in self.deferred and s not in self.merge else 0 for s in range(R)]
+        for (a,b,_),(ca,cb) in zip(self.ops,self.coef):old[a]=ca*old[a]+cb*old[b]
+        old=self.decode(old);y=[y-direction*z for y,z in zip(Y0,old)];del old;gc.collect()
+        z=Z[:];done=[]
+        def read(s):
+            if mutation=='missing-read' and s==self.selected[0]:return
+            value=z[self.phys[s]];assert value%6==0
+            for t,c in self.D[s].items():y[t]-=direction*c*(value//6)
+        def gate(i):
+            (a,b,_),(ca,cb)=self.ops[i],self.coef[i];a,b=self.phys[a],self.phys[b]
+            z[a]=ca*z[a]+cb*z[b];done.append(i)
+        for node,s in self.sources.items():z[self.phys[s]]+=X[node-1]
+        for i in self.phase:gate(i)
+        physical_virtual=[z[self.phys[s]] for s in range(R)]
+        center=self.decode(physical_virtual,'center');del physical_virtual
+        for t,c in enumerate(center):y[t]+=direction*c
+        del center
+        late=[b for b,t in self.deadline.items() if t is not None];early=late[-1] if mutation=='early-read' and late else None
+        for s in reversed(self.selected):
+            if self.deadline.get(s) is None or s==early:read(s)
+        for i in self.rest:
+            for s in self.at.get(i,[]):
+                if s!=early:read(s)
+            gate(i)
+        physical_virtual=[z[self.phys[s]] for s in range(R)]
+        side=self.decode(physical_virtual,'side');del physical_virtual
+        for t,c in enumerate(side):y[t]+=direction*c
+        del side
+        def K(xs):
+            out=[]
+            for a in range(0,v,8):
+                block=xs[a:a+8];qs=self.g['inputs'][a:a+8]
+                for q in qs:
+                    value=sum((1 if (q^r).bit_count()==6 else -1 if (q^r).bit_count()==2 else 0)*x for r,x in zip(qs,block));assert value%2==0;out.append(value//2)
+            return out
+        mixed=K(X);assert K(mixed)==X
+        for t,x in enumerate(mixed):y[t]+=direction*x
+        assert y==[y+direction*x for y,x in zip(Y0,X)],'full formal output map'
+        for i in reversed(done):
+            (a,b,_),(ca,cb)=self.ops[i],self.coef[i];a,b=self.phys[a],self.phys[b]
+            z[a]=ca*(z[a]-cb*z[b])
+        for node,s in self.sources.items():z[self.phys[s]]-=X[node-1]
+        assert z==Z,'all physical dirty columns restored'
+        return dict(direction=direction,every_formal_column=True,source_columns=v,target_columns=v,dirty_columns=len(self.live),K_and_inverse=True,all_aliases_and_deadline_reads=True,all_physical_dirty_slots_restored=True)
+
+def main():
+    ap=argparse.ArgumentParser();ap.add_argument('--candidate',type=Path,required=True);ap.add_argument('--source',type=Path,required=True);ap.add_argument('--out',type=Path,required=True);a=ap.parse_args();sys.path.insert(0,str(a.source/'scripts'))
+    from verify_fused import verify
+    from paired_cube_physical import physical
+    start=time.time();pins=json.loads((a.candidate/'result.json').read_text())['pins'];data={}
+    for name in ['graph','baseline','frames','word','profile-before','physical-frames','physical-pairs','profile']:
+        p=a.candidate/(name+'.json.gz');assert sha256(p.read_bytes()).hexdigest()==pins[p.name];data[name]=json.loads(gzip.decompress(p.read_bytes()))
+    g,b,w,word,row=[data[n] for n in ['graph','baseline','frames','word','profile-before']]
+    finite=verify(g,b,w,word,row);print('PASS exact signed scalar pairs and pre-reuse complete physical frame word',flush=True)
+    actual=physical(g,w,word,row,data['physical-frames'],data['physical-pairs']);assert json.loads(json.dumps(actual))==data['profile'];print('PASS all descended frames, nested chains, aliases and target read chronology',flush=True)
+    D,adj=adjoint(g,word,row['R']);formal=Formal(g,word,row,data['physical-pairs'],D);print('PASS exact adjoint; formal coefficient bound',formal.bound,'digit bits',formal.width,flush=True)
+    columns=[formal.run(d) for d in (1,-1)];print('PASS every formal column under both shear signs',flush=True)
+    mutations={}
+    for tag in ['missing-read','early-read']:
+        try:formal.run(1,tag)
+        except AssertionError:mutations[tag]='REJECTED'
+        else:raise AssertionError('accepted mutation '+tag)
+    cs=[Q(c) for r in g['roots'] for c in r['coefficients']]
+    assert set(cs)<={Q(1,2),Q(-1,2),Q(1,3),Q(-1,6)} and 3*row['q']<2**15
+    local=4*(row['c']+row['v'])+10*row['v']+4*row['h']*row['v']+4*row['h']**2+8*row['h']+8+2*row['h']+8*row['R']*row['v']*(row['total_M_operations']+16)+32*row['v']
+    scalar=dict(c=row['c'],q=row['q'],M_operations=row['total_M_operations'],logical_R=row['R'],physical_R=actual['physical_R'],handoffs=actual['pairs'],scalar_denominators_divide=6,fixed_odd_divisor=3,gate_and_inverse_coefficients=[-1,1],decoder_coefficients=sorted(map(str,set(cs))),three_q_less_than_2pow15=True,binary_dirty_read_bits=row['total_M_operations']+16,local_group_upper=local,scope='Conservative inherited virtual-word bill retained after aliasing, including all compensated reads. Alias and frame route choices are finite permutations/Clifford adapters; their separate global movement bill remains an assembly obligation.')
+    result=dict(status='PASS exact fused physical candidate, all formal source/target/dirty columns',source_sha256={str(p):sha256(p.read_bytes()).hexdigest() for p in [Path(__file__),Path(__file__).with_name('verify_fused.py'),a.source/'scripts/paired_cube_physical.py',a.source/'scripts/paired_cube/frames.py',a.source/'notes/general-clifford-frames.tex']},input_sha256=pins,pre_reuse_finite=finite,physical=actual,exact_adjoint=adj,formal=dict(columns=columns,absolute_coefficient_bound=formal.bound,packed_digit_bits=formal.width,base_exceeds_twice_bound=True),mutations=mutations,scalar_bound=scalar,seconds=time.time()-start,maxrss=resource.getrusage(resource.RUSAGE_SELF).ru_maxrss,foreign_producer_replays=0,scope='Finite candidate audit only. General exact Clifford and shared cover contracts, analytic supplier bridge, and paid global multiplication assembly remain separate obligations.')
+    a.out.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n');print(json.dumps({k:result[k] for k in ['status','seconds','maxrss','scalar_bound']}),flush=True)
+if __name__=='__main__':main()
diff --git a/research/coordinated-frames-and-entrance-banks/complex/prove.py b/research/coordinated-frames-and-entrance-banks/complex/prove.py
new file mode 100644
index 0000000000..2d622bce6c
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/complex/prove.py
@@ -0,0 +1,74 @@
+#!/usr/bin/env python3
+"""Portable exact physical and moment certificate for the selected fused word."""
+from fractions import Fraction as Q
+from pathlib import Path
+import gzip
+import json
+import subprocess
+import sys
+import tempfile
+
+HERE=Path(__file__).resolve().parent;PACKAGE=HERE.parent;ROOT=HERE.parents[2]
+sys.path.insert(0,str(PACKAGE/'arithmetic'));sys.dont_write_bytecode=True
+from interval_moment import saving_grid
+
+
+def js(x):
+    if isinstance(x,Q):return str(x)
+    if isinstance(x,dict):return {str(k):js(v) for k,v in x.items()}
+    if isinstance(x,(tuple,list)):return [js(v) for v in x]
+    return x
+
+
+def main():
+    assert not sys.flags.optimize
+    candidate=PACKAGE/'selected/complex'
+    with tempfile.TemporaryDirectory(prefix='paired-cube-exact-') as tmp:
+        out=Path(tmp)/'physical.json'
+        run=subprocess.run([sys.executable,'-B',str(HERE/'physical.py'),'--candidate',str(candidate),'--source',str(ROOT),'--out',str(out)],text=True,capture_output=True)
+        if run.returncode:raise ValueError(run.stdout+run.stderr)
+        physical=json.loads(out.read_text())
+    physical.pop('seconds',None);physical.pop('maxrss',None)
+    physical['source_sha256']={Path(k).resolve().relative_to(ROOT).as_posix():v for k,v in physical['source_sha256'].items()}
+    with tempfile.TemporaryDirectory(prefix='paired-cube-sinks-') as tmp:
+        checked=Path(tmp)/'selection.json';formal_path=Path(tmp)/'formal.json'
+        command=[sys.executable,'-B',str(HERE/'check_sinks.py'),'--candidate',str(candidate),'--output',str(checked)]
+        run=subprocess.run(command,text=True,capture_output=True)
+        if run.returncode:raise ValueError(run.stdout+run.stderr)
+        selection=json.loads(checked.read_text());frozen=json.loads((candidate/'sinks.json').read_text())
+        for value in (selection,frozen):
+            value.pop('seconds',None);value.pop('rss_kib',None)
+        assert selection==frozen,'Frozen simultaneous terminal selection differs'
+        command=[sys.executable,'-B',str(HERE/'formal_sinks.py'),'--candidate',str(candidate),
+            '--selection',str(candidate/'sinks.json'),'--formal-helper',str(HERE/'physical.py'),'--output',str(formal_path)]
+        run=subprocess.run(command,text=True,capture_output=True)
+        if run.returncode:raise ValueError(run.stdout+run.stderr)
+        sink_formal=json.loads(formal_path.read_text());sink_formal.pop('seconds',None)
+    raw=json.loads(gzip.decompress((candidate/'profile-before.json.gz').read_bytes()))
+    before=physical['physical'];paid=dict(before)
+    n=selection['eligible_count'];assert n==46
+    assert (selection['old_physical_R'],selection['old_W'],selection['old_rank'])==(before['physical_R'],before['W_per_vertex'],before['rank_per_vertex'])
+    paid.update(physical_R=selection['new_physical_R'],W_per_vertex=selection['new_W'],
+        rank_per_vertex=selection['new_rank'],child_histogram=selection['child_histogram'],
+        local_histogram=selection['local_histogram'],target_data_histogram=selection['target_histogram'],terminal_sinks=n)
+    assert all(z['dirty']==paid['physical_R'] and z['columns']==paid['W_per_vertex'] for z in sink_formal['columns'])
+    physical['before_terminal_sinks']=before;physical['physical']=paid
+    physical['original_formal']=physical['formal'];physical['formal']=sink_formal['columns']
+    physical['terminal_compiler']=selection;physical['terminal_formal']=sink_formal
+    physical['scalar_bound'].update(physical_R=paid['physical_R'],terminal_sinks=n,
+        terminal_scalar_gate_delta=selection['scalar_gate_delta'],terminal_original_scalar_reserve_retained=True)
+    from hashlib import sha256
+    for name in ('check_sinks.py','formal_sinks.py','prove.py'):
+        physical['source_sha256'][(HERE/name).relative_to(ROOT).as_posix()]=sha256((HERE/name).read_bytes()).hexdigest()
+    row={k:raw[k] for k in ('h','v','c','q','matched','total_M_operations','loss')}
+    row.update({k:paid[k] for k in ('m','W_per_vertex','rank_per_vertex','deficit_per_vertex','child_histogram')})
+    row.update(R=paid['physical_R'],scalar_role_reserve=raw['R'],reuse_pairs=paid['pairs'],terminal_sinks=n,maxchild=max(map(int,paid['child_histogram'])))
+    p=dict(m=row['m'],W=row['W_per_vertex'],N=row['deficit_per_vertex'],L=0,total_rank=row['rank_per_vertex'],maxchild=row['maxchild'],child_multiplicities=row['child_histogram'])
+    moment=saving_grid(p,10**18)
+    assert moment['accepted']['saving']==Q('662323931899051/1000000000000000000')
+    moment['complex_saving']=moment['accepted']['saving']
+    moment['scope']='Strict contraction and adjacent point exclusion for this fixed complete complex profile.'
+    print(json.dumps(js(dict(status='PASS',profile=row,physical=physical,moment=moment)),sort_keys=True))
+
+
+if __name__=='__main__':main()
diff --git a/research/coordinated-frames-and-entrance-banks/complex/verify_fused.py b/research/coordinated-frames-and-entrance-banks/complex/verify_fused.py
new file mode 100644
index 0000000000..47f73cf035
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/complex/verify_fused.py
@@ -0,0 +1,156 @@
+"""R13 candidate specialization: remove only PR152 selected-count pin. All finite checks retained. New candidate wrapper by Chafik Boukhalfa with OpenAI Codex; inherited credits below."""
+# Copyright 2026 icekylinx. Apache-2.0.
+# Developed with GPT-6 Astra assistance; integrated with Codex assistance.
+"""Independent finite checks of the signed scalar word and binary geometry.
+
+Copyright 2026 icekylinx, Apache-2.0; OpenAI GPT-6 Astra assistance. Binary Lagrangian
+intersection ranks use L_U=U^perp in X plus U in Z, including degenerate U.
+"""
+from collections import Counter
+from functools import lru_cache
+from paired_cube.frames import basis, perp, contained
+
+
+def verify(g, baseline, witness, word, record):
+    h,v=g['h'],g['v'];args=[None]+[None if a is None else [y+1 for y in a] for a in g['args']]
+    roots=[dict(r,node=r['node']+1) for r in g['roots']]
+    n=len(args);ann=witness['annihilators'];order=witness['order'];arcs=dict(witness['matching_arcs'])
+    active=set(order)
+    inputs=g['inputs']; spans=[()]*n;positive=[0]*n;negative=[0]*n
+    @lru_cache(None)
+    def frame(A):
+        U=perp(A,h)
+        assert len(U)+len(A)==h and all((a&b).bit_count()%2==0 for a in A for b in U)
+        return U
+    def each(bits):
+        while bits:
+            low=bits&-bits;bits^=low;yield low.bit_length()-1
+    for x in range(1,n):
+        if args[x] is None:
+            positive[x]=1<<(x-1);spans[x]=(inputs[x-1],)
+        else:
+            a,b=args[x];assert 0<a<x and 0<b<x
+            assert not (positive[a]|negative[a])&(positive[b]|negative[b])
+            sign=g['signs'][x-1];assert sign in (-1,1)
+            positive[x]=positive[a]|(positive[b] if sign==1 else negative[b])
+            negative[x]=negative[a]|(negative[b] if sign==1 else positive[b])
+            spans[x]=basis(spans[a]+spans[b])
+        if x in active:assert contained(spans[x],frame(tuple(ann[x])))
+    # Recount every actual signed broadcast, with integer twice-coefficients.
+    Hrows=[[0]*v for _ in range(v)]
+    rootann=[];rootframes=[]
+    for r in roots:
+        x=r['node']
+        if r['kind']=='center':
+            assert not negative[x]
+            expected=sum(1<<s for s,q in enumerate(inputs) if q>>r['coordinate']&1)
+            assert positive[x]==expected
+            U=spans[x];A=perp(U,h)
+            assert len(U)==h-2
+            assert r['coefficients']==['1/3' if q>>r['coordinate']&1 else '-1/6' for q in inputs]
+        else:
+            A=basis(inputs[t] for t in r['targets']);U=perp(A,h)
+            for t,c in zip(r['targets'],r['coefficients']):
+                assert c in ('1/2','-1/2');sign=1 if c=='1/2' else -1
+                for s in each(positive[x]):Hrows[t][s]+=sign
+                for s in each(negative[x]):Hrows[t][s]-=sign
+        rootann.append(A);rootframes.append(U)
+        assert contained(spans[x],U)
+    cube=[tuple(a//2 for a in label) for label in g['labels']]
+    scalar_pairs=0
+    for t,q in enumerate(inputs):
+        for s,u in enumerate(inputs):
+            overlap=(q&u).bit_count();same=cube[t]==cube[s]
+            expected=0 if same else 1-overlap
+            assert Hrows[t][s]==expected, ('Signed H mismatch',t,s)
+            distance=(q^u).bit_count()//2
+            K2=(1 if distance==3 else -1 if distance==1 else 0) if same else 0
+            B2=overlap-1
+            assert Hrows[t][s]+K2+B2==2*(t==s)
+            scalar_pairs+=1
+    # Actual local K matrix, inverse, and destructive-source frame itinerary.
+    for start in range(0,v,8):
+        qs=inputs[start:start+8]
+        K2=[[1 if (a^b).bit_count()==6 else -1 if (a^b).bit_count()==2 else 0 for b in qs] for a in qs]
+        assert all(sum(K2[i][k]*K2[k][j] for k in range(8))==4*(i==j) for i in range(8) for j in range(8))
+        for parity in (0,1):
+            source=[i for i in range(8) if i.bit_count()%2==parity];target=[i for i in range(8) if i.bit_count()%2!=parity]
+            U=basis(qs[i] for i in source);assert len(U)==3
+            assert all(contained((qs[i],),U) for i in source)
+            assert all(contained(U,perp((qs[j],),h)) for j in target)
+            assert [3-1,(h-1)-3,h-(h-1)]==[2,h-4,1]
+            assert all(sum(K2[i][k]*K2[j][k] for k in source)==4*(i==j) for i in target for j in target)
+    # Full backward intersections, including every carrier edge and root cap.
+    required=[[] for _ in args];position={x:i for i,x in enumerate(order)}
+    for x in order:
+        if args[x]:
+            for y in args[x]:required[y].extend(ann[x]);assert position[y]<position[x]
+    for r,A in zip(roots,rootann):required[r['node']].extend(A)
+    assert len(arcs)==len(set(arcs.values()))==record['matched']
+    for x,u in arcs.items():
+        assert args[x]
+        if u>>31:
+            j=u&0x7fffffff;value=roots[j]['node'];A=rootann[j]
+        else:
+            t=u//2;value=args[t][u&1];A=ann[t];assert position[x]<position[t]
+        assert value in args[x];required[x].extend(A)
+    for x in order:assert basis(required[x])==tuple(ann[x]), ('Incomplete intersection',x)
+    # Replay actual signed physical operations and every positive frame movement.
+    R=record['R'];values=[(0,0)]*R;full=basis(1<<i for i in range(h));roleann=[full]*R
+    first=[None]*R;last=[-1]*R;H=Counter();phase=set(word['phase1']);touched=set(word['sources'].values())
+    for x,s in word['sources'].items():
+        x=int(x);values[s]=(positive[x],negative[x]);roleann[s]=perp((inputs[x-1],),h);H[1]+=1
+    for i,((a,b,x),(ca,cb)) in enumerate(zip(word['ops'],word['opcoeff'])):
+        assert a!=b and ca in (-1,1) and cb in (-1,1)
+        if i in phase:
+            assert all(last[s]<0 or last[s] in phase for s in (a,b));touched.update((a,b))
+        for s in (a,b):
+            if first[s] is None:first[s]=x
+            A=tuple(ann[x]);assert contained(A,roleann[s])
+            H[len(roleann[s])-len(A)]+=1;roleann[s]=A;last[s]=i
+        pa,na=values[a];pb,nb=values[b];assert not (pa|na)&(pb|nb)
+        if ca<0:pa,na=na,pa
+        if cb<0:pb,nb=nb,pb
+        values[a]=(pa|pb,na|nb);assert values[a]==(positive[x],negative[x])
+    for r,s,A,U in zip(roots,word['rootroles'],rootann,rootframes):
+        assert values[s]==(positive[r['node']],negative[r['node']])
+        assert contained(tuple(A),roleann[s]);H[len(roleann[s])-len(A)]+=1;roleann[s]=tuple(A)
+        if r['kind']=='center':assert last[s]<0 or last[s] in phase;H[len(U)]+=1
+    for A in roleann:H[len(A)]+=1
+    assert [H[r] for r in range(1,h+1)]==baseline['remaining_internal_histogram'][1:]
+    co=[0]*R
+    for r,s in zip(roots,word['rootroles']):co[s]|=sum(1<<t for t in r['targets'])
+    for a,b,x in reversed(word['ops']):co[b]|=co[a]
+    limits=[None]*v
+    for r,A in zip(roots,rootann):
+        if r['kind']=='side':
+            for t in r['targets']:
+                if limits[t] is None:limits[t]=A
+    seen=set();birth=Counter()
+    for z in word['selected']:
+        s=z['role'];A=tuple(z['annihilator']);d=z['rank'];targets=list(each(co[s]))
+        assert s not in touched and s not in seen;seen.add(s)
+        assert targets==z['targets']
+        assert A==basis(tuple(ann[first[s]])+tuple(u for t in targets for u in limits[t]))
+        assert d==h-len(A)>0 and contained(tuple(ann[first[s]]),A)
+        r=h-len(ann[first[s]]);H[r]-=1;H[r-d]+=1;birth[d]+=1
+        for t in targets:limits[t]=A
+    assert [H[r] for r in range(1,h+1)]==record['remaining_internal_histogram'][1:]
+    assert dict(birth)=={int(k):c for k,c in record['selected_rank_histogram'].items()}
+    current=[full]*v;Y=Counter()
+    for z in reversed(word['selected']):
+        A=tuple(z['annihilator'])
+        for t in z['targets']:
+            assert contained(A,current[t]);Y[len(current[t])-len(A)]+=1;current[t]=A
+    for r,A in zip(roots,rootann):
+        if r['kind']=='side':
+            for t in r['targets']:
+                assert contained(A,current[t]);Y[len(current[t])-len(A)]+=1;current[t]=A
+    for t,q in enumerate(inputs):assert contained((q,),current[t]);Y[len(current[t])-1]+=1
+    assert dict(Y)=={int(k):c for k,c in record['target_data_histogram'].items()}
+    assert record['selected_roles']==len(seen) and birth==Counter({int(k):v for k,v in record['selected_rank_histogram'].items()})
+    return dict(signed_scalar_pairs_checked=scalar_pairs,scalar_H_exact=True,K_involution_and_inverse_exact=True,
+                decoder_identity_exact=True,conservative_source_supports=True,K_source_itinerary_verified=True,
+                full_backward_intersections=True,physical_signed_M_replayed=True,
+                center_phase_closed=True,full_gauge_intersections=True,selected_sources_untouched=True,
+                reverse_target_chains=True,positive_rank_ledger_recounted=True,selected_roles=len(seen))
diff --git a/research/coordinated-frames-and-entrance-banks/references/pr141/ORIGINAL-PROOF.md b/research/coordinated-frames-and-entrance-banks/references/pr141/ORIGINAL-PROOF.md
new file mode 100644
index 0000000000..2cca3a12e1
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/references/pr141/ORIGINAL-PROOF.md
@@ -0,0 +1,101 @@
+# Balanced positional transfer of the stopped product-ring composition
+
+This is an additional paid layout composition. It is not an omission of the old prefix cost or a claim that a favorable child histogram by itself proves an all-size multiplication theorem.
+
+## Contracts that have to survive the composition
+
+The supplied finite word must return every dirty auxiliary coordinate, compute the exact signed complex source map, use legal common frames on both ports, and account for every transition, source/target front, exterior, center copy and endpoint correction. Its signed reflected word reverses the literal operation order, negates shears, exchanges data banks and complements every frame. The complete paid histogram and literal scalar upper bound are inputs to this arithmetic layer. Reflection and scalar-identity evidence are separately hash-bound.
+
+PR104's opposite-bank product-ring construction makes each proper projector residual of rank r one reversed child of width r. Its global conjugation, common generic basis, full involution factorization and paid atom adapters remain required. It supplies an ordinary interchange only after the constant-size outer wrapper. That completed wrapper has arbitrary-width, complete-spectator and restored-row guarantees. Its constant number of calls is outside the recursive moment and does not multiply the child histogram. The atom exponent remains theta=1/1000, with old ordinary saving 384599/10^10. Thus the round-seven coarse bit saving 620523/(5*10^9) gives ordinary saving 1240189553/10^13, and theta exceeds this saving, so paid atom loops are lower order.
+
+The complex primitive retains the completed exact tensor contract, translated endpoint gauges, inverse phase correction, and common odd-21 grid. It must preserve a completed child's incoming odd denominator. No boundary rounding, characteristic-two substitution for signed arithmetic, free address reversal, or uncharged coefficient multiplication is introduced.
+
+Readout n/42 may be realized by |n| consecutive same-frame shears with coefficient sign(n)/42. Each has magnitude at most one, their exact sum is n/42, and their inverse negates the pieces in reverse order. They require no new source copy or frame transition. The supplied conservative L1 scalar bill may include unused copy/scan allowance, but its local/global bound must still cover the actual selected literal word. This observation does not authorize ignoring repeated scalar occurrences or a wrong inverse sign.
+
+## Why the balanced layout applies
+
+Use the inherited PR34/RaD paid positional layout as made explicit for deferred/signed networks in PR100 and PR103. Separate the extra top bit of each long FFT axis, and use common named low-coordinate groups K_j in [K,2K). The axis-major/group-major permutation and inverse are paid by the ordinary arbitrary-coordinate router. The resulting extra prefix work is O(T p d), with saving margin 1-epsilon. Complete active, spectator, scratch, phase, control and row fields travel with their descriptors; FFT order, twiddle order and forward/inverse alignment are preserved. No local scalar operation or physical residual is moved across another operation by this outer regrouping.
+
+The stopped ordinary wrapper exposes precisely the ordinary interchange contract consumed by that router. The complex word exposes the same completed tensor contract required by the semantic C1=1 induction. Therefore the stopped product-ring implementation can instantiate the same balanced layout, under the inherited all-size contracts. The original geometric condition 1-epsilon(1+c)>0 remains in the constraint list; only the work margin changes from the old repeated-prefix charge to the proved paid layout's 1-epsilon. A negative control reinstates the old work margin at the new parameters and rejects them.
+
+## All three row stocks remain present
+
+Along any active nested call chain, reserve
+
+    W_complex^D_complex * W_coarse^D_coarse * W_old^D_old.
+
+After a complex and coarse division, the old ordinary stock remains available to every stopped leaf or peeled atom. Sequential wrappers and copies reuse restored rows. They create no additional simultaneous factor, and also do not remove the old factor.
+
+For each family, d is the least positive integer with m^d > 2*r_max^d, so its depth is at most d*ceil(log2(max(2,e))) plus a fixed constant. The selected dimensions give:
+
+| Family | m | largest child | role bits | d | row coefficient |
+|---|---:|---:|---:|---:|---:|
+| Coarse bit | 529 | 528 | 27 | 367 | 9909 |
+| Complex | 576 | 574 | 27 | 200 | 5400 |
+| Old ordinary leaf | 575 | 529 | 28 | 9 | 252 |
+
+The total is 15561. A p^32000 stock leaves strict gap
+
+    32000 - (51/25)*15561 = 6389/25 > 0.
+
+The suffix slope is 128000. The validator recomputes all three minimal halving degrees, every wire-bit count, the old-stock contribution, the total coefficient, the degree gap and the suffix slope. It does not reuse the original balanced helper's two-family row formula. Semantic E, literal charge, B, C0, C1=1 and the completed-child induction are also recomputed from the supplied scalar upper bound.
+
+## Exact parameters and all retained inequalities
+
+Let b be the certified complex saving, A the proved ordinary bit saving, beta>0 the complex stopping parameter, eta>0 the balanced backoff, and zeta>0 a strict weakening. Set
+
+    a = min(A, (1-beta)*b-zeta),
+    q = a*(1-2*eta),
+    c = q+eta/4,
+    epsilon = (1-eta)/(1+q),
+    lambda' = 1-q,
+    lambda = ((1-a)+lambda')/2,
+    g = epsilon*q,
+    r = (g+1-epsilon)/2,
+    delta = eta/8.
+
+The seven margins are
+
+    1-epsilon, a, g, a,
+    min(1-epsilon-delta, r-delta),
+    1-epsilon-delta, epsilon.
+
+The minimum is g. The identities
+
+    1-epsilon-g = eta,
+    1-epsilon-r = eta/2,
+    1-epsilon*(1+c) = eta-epsilon*eta/4 > 0
+
+retain the scalar, Gaussian and geometric gaps. The stopped internal exponent and complex leaf exponent remain exactly those in PR104; (1-beta)*b>a ensures the leaf ordering. All 47 strict inequalities are retained. Source normalization shows the full adapter differs from the pinned balanced implementation only in three-stock validation, explicit old-leaf argument plumbing and inclusion of the old leaf in cutoff maxima. No constraint or cost margin is deleted.
+
+## Selected finite arithmetic and grid ceiling
+
+The selected joint-expansion profile retains 2108 birth reuses, eliminates 381 nonaliased terminal roles, and has R=26216, m=576, W=114315520, rank=65843877440 and deficit=1862080. Its scalar group bound is G=3116113870848. The actual emitted physical profile, both-orientation frame/scalar audits, and literal word are separate SHA-bound construction evidence. Its entire histogram is supplied, not inferred from role reduction or an unrestricted majorization claim. Two separate rational enclosures accept the greatest 10^-24-grid complex saving
+
+    b = 112814910866013477738/10^24
+
+and reject b+10^-24. The first enclosure reduces logarithms by 4/3 with 80 atanh terms and uses degree-12 exponentials with outward 10^-45 rounding. The independent pinned PR65/100 method reduces by 2 with 32 atanh terms and uses a degree-eight exponential bound with outward 10^-40 rounding. Every first-method log/exponential interval lies inside the second, and both separate the accepted and rejected moments from one.
+
+Taking beta=eta=10^-24 and zeta=10^-30 gives
+
+    kappa = 112802185097556/10^18.
+
+The next 10^-18 point is rejected. More strongly, let b_plus be the rejected complex point. The moment is strictly increasing in b, so the true profile root is below b_plus. Every member of this balanced family has a<min(A,b_plus) and g<a/(1+a). Therefore
+
+    g < b_plus/(1+b_plus) < (112802185097556+1)/10^18.
+
+This proves grid optimality within this fixed-profile, fixed-bit-supplier balanced transfer family. It proves no global optimum over other profiles, networks or transfer architectures.
+
+The separate same-beta/eta comparison retains beta=10^-6, eta=10^-8 and zeta=10^-14 in both layouts. It gives old-prefix kappa 112789346042450/10^18 and balanced kappa 112802068914290/10^18. Their difference 12722871840/10^18 is caused by the paid balanced layout. Finer positive backoffs give an additional 116183266/10^18; the two effects are reported separately.
+
+## Checks and boundary
+
+The arithmetic checks include old-prefix rejection, omitted/zero old-leaf stock, a two-stock coefficient, stale maximum-child/halving data, stale or absent scalar charge, stale C0, insufficient row reserve, zero leaf-ordering slack, excess kappa and deletion of a paid singleton. Each rejection is checked separately. The fixed-profile next-kappa exclusion uses the rejected complex moment, not merely an assertion that a parameter search ended.
+
+The optional explicit arithmetic cutoff for the selected tiny backoffs is log2(input) >= 14000000000000000000000000. This is only a sufficient numeric-comparison threshold. Prime supply, fixed-basis setup, descriptor/catalogue domination, logarithm absorption and exact recovery retain their inherited eventual conditions; this value is not a practical runtime threshold.
+
+Original authorship remains attached: icekylinx for stopped product-ring accounting and the ordinary wrapper; PR23/29 Zhihao Chen, RaD/hipotures and James Chang/PR34 for the semantic/balanced layout; PR100/103 Rohan Arun for its deferred/signed integration; Swapnil Jain and the relevant complex producer/frame/recycling authors for the finite inputs. This three-stock adapter and independent certificate were prepared with OpenAI Codex assistance. The supplied physical word, all-size frame implementation, simultaneous basis, atom streaming, routing, prime/precision/recovery and analytic fixed-tape transfer remain separate inherited obligations.
+
+## Portable exact boundary
+
+The portable certificate generator admits only explicit integer mathematical profile fields and canonical integer child counts. It derives h, v, m, N, L, W and deficit relations before arithmetic, recursively rejects floating-point values before serialization and after readback, and binds the raw physical profile and scalar-bill bytes by SHA256. Discovery timings stay outside the mathematical certificate. The independent secondary enclosure executes only three verbatim arithmetic function bodies from the pinned source. The selected validation applies 18 adverse controls and compares every controlling quantity, moment enclosure, assembly and cutoff against the separately generated prepackage witness. The two certificates agree exactly on these mathematical fields.
diff --git a/research/coordinated-frames-and-entrance-banks/references/pr141/balanced_stopped.py b/research/coordinated-frames-and-entrance-banks/references/pr141/balanced_stopped.py
new file mode 100644
index 0000000000..5b02fb6948
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/references/pr141/balanced_stopped.py
@@ -0,0 +1,204 @@
+# Three-reserve stopped-product adaptation of the inherited PR34/RaD/PR23 balanced
+# positional assembly retained in PR104 and proved applicable to deferred words
+# in PR100/103. Original attribution below is preserved. This module verifies
+# arithmetic only; scalar-word/profile provenance and all-size interfaces remain
+# separate obligations. Prepared with OpenAI Codex assistance, Apache-2.0.
+"""Exact, import-safe balanced copied-center semantic assembly arithmetic.
+
+The PR36 copied-center guard uses E=64*(W+m+G+1)^3, including G.
+Its complex saving is 717/10^7. All balanced parameter formulas are unchanged.
+
+Adapted from Zhihao Chen PR23/29 semantic assembly (Apache-2.0), RaD's
+balanced transform / arbitrary routing / bulk resampling composition, and
+jamesyc PR34's check_composition.py at 7fecbe3. PR34 source is research/reversed-two-stage/check_composition.py in that
+pinned upstream commit. Original parameterized transcription by the
+research workers is in rectangular-dimensions/next/assembly_parameterized.py.
+
+This checks arithmetic, not physical corner existence, producer correctness,
+or inherited all-size tape/analytic hypotheses. No imports execute producers,
+parse argv, read files, change globals, or mutate the supplied finite bridge.
+"""
+from fractions import Fraction as Q
+
+
+class InvalidAssembly(AssertionError):
+    """A named exact input validation or strict inequality failed."""
+
+
+def require(condition, message):
+    if not condition:
+        raise InvalidAssembly(message)
+
+
+def rational(value):
+    # Fraction(float) silently preserves binary floating error; forbid it.
+    require(type(value) in (int, str, Q), 'expected integer or exact rational string/Fraction')
+    return Q(value)
+
+
+def integer(value):
+    x = rational(value)
+    require(x.denominator == 1, 'nonintegral graph constant')
+    return x.numerator
+
+
+def ceil(value):
+    return -(-value.numerator // value.denominator)
+
+
+def halving_degree(m, child):
+    require(0 < child < m, 'child must strictly contract')
+    degree = 1
+    while m**degree <= 2 * child**degree:
+        degree += 1
+    return degree
+
+
+def validate_bridge(finite_bridge, ordinary_leaf):
+    """Recompute rather than trust stored semantic and product-stock totals.
+
+    scalar_group_upper remains a supplied, proof-certified local scalar upper
+    charge. Its producer provenance is external to this arithmetic interface.
+    """
+    result = {}
+    for name in ('bit_coarse', 'complex', 'ordinary_leaf'):
+        data = ordinary_leaf if name == 'ordinary_leaf' else finite_bridge[name]
+        m, W, child = (integer(data[k]) for k in ('m', 'W', 'maxchild'))
+        require(m > 1 and W > 0, 'positive arity and role volume required')
+        degree, bits = halving_degree(m, child), W.bit_length()
+        require(integer(data['halving_degree']) == degree, name + ' stale halving degree')
+        require(integer(data['wire_bits']) == bits, name + ' stale role logarithm')
+        result[name] = dict(m=m, W=W, maxchild=child,
+                            halving_degree=degree, wire_bits=bits)
+    phase = finite_bridge['complex']
+    s, scalar = (integer(phase[k]) for k in ('s', 'scalar_group_upper'))
+    m, W, child = (result['complex'][k] for k in ('m', 'W', 'maxchild'))
+    require(0 < s < m*W and scalar > 0, 'invalid complex rank/scalar bound')
+    E = 64*(W+m+scalar+1)**3
+    B = s+E
+    literal = 2*scalar*W**2+8*s+4*W+4+32*m
+    C0 = 32*m*B**2
+    semantic = dict(E=E, B=B, C0=C0, C1=1, literal_charge=literal,
+                    strict_literal_gap=E-literal,
+                    induction_gap=2*B*(m-child)-(s+E))
+    require(E > literal and semantic['induction_gap'] >= 0 and C0 > 2*B+18,
+            'semantic completed-child bound failed')
+    for key, value in semantic.items():
+        require(rational(finite_bridge['semantic'][key]) == value,
+                'stale semantic constant: '+key)
+    result['complex'].update(s=s, scalar_group_upper=scalar)
+    result['semantic'] = semantic
+    rows = finite_bridge['rows']
+    leaf_degree = result['ordinary_leaf']['halving_degree']*result['ordinary_leaf']['wire_bits']
+    require(integer(finite_bridge['ordinary_leaf_row_degree']) == leaf_degree,
+            'ordinary leaf stock missing or stale')
+    coefficient = sum(result[name]['halving_degree']*result[name]['wire_bits']
+                      for name in ('bit_coarse', 'complex', 'ordinary_leaf'))
+    degree = integer(rows['degree'])
+    gap = degree-Q(51,25)*coefficient
+    require(degree > 0 and gap > 0, 'insufficient product row degree')
+    expected = dict(coefficient=coefficient, degree=degree,
+                    degree_gap=gap, suffix_slope=4*degree)
+    for key, value in expected.items():
+        require(rational(rows[key]) == value, 'stale product row constant: '+key)
+    result['rows'] = expected
+    return result
+
+
+def assembly(finite_bridge, ordinary_leaf, a_bit, kappa, beta=Q(1,20), h=Q(1,10**12),
+             a_complex=Q(717,10**7), *, original_prefix=False,
+             old_guard=False, old_exposures=False):
+    """Return 47 strict slacks and seven margins, or raise InvalidAssembly.
+
+    The three keyword switches are negative controls: they impose the old
+    prefix, nonlinear guard, or old separate movement charges at these SAME
+    balanced parameters. They do not silently optimize an alternative family.
+    """
+    f = validate_bridge(finite_bridge, ordinary_leaf)
+    a, b, kappa, beta, h = map(rational, (a_bit,a_complex,kappa,beta,h))
+    require(0 < h < Q(1,2) and kappa > 0, 'positive backoff and saving required')
+    tau, sigma = 1-a, 1-b
+    q = a*(1-2*h)
+    require(1+q != 0, 'undefined balanced epsilon')
+    lp, c, eps = 1-q, q+h/4, (1-h)/(1+q)
+    lam = (tau+lp)/2
+    G = eps*q
+    r, delta = (G+1-eps)/2, h/8
+    C1 = Q(19991,10000) if old_guard else Q(1)
+    margins = dict(g1=1-eps, g2=a, g3=G, g4=a,
+                   g5=min(1-eps-delta,r-delta), g6=1-eps-delta, g7=eps)
+    if original_prefix:
+        margins['g1'] = 1-eps*(1+c)
+    if old_exposures:
+        margins.update(g2=eps*c*a, g4=a*(1-eps))
+    internal = tau+(1-beta)*max(sigma-tau,Q(0))
+    leaf = sigma+beta*(1-sigma)
+    slacks = dict(a_positive=a, a_below_b=b-a, b_below_one_over32=Q(1,32)-b,
+        beta_positive=beta, beta_below_one=1-beta, phase_leaf_above_bit=(1-beta)*b-a,
+        q_positive=q, q_below_internal=1-internal-q, q_below_leaf=1-leaf-q,
+        c_positive=c, c_below_one=1-c, q_below_reservations=c-q,
+        lambda_above_tau=lam-tau, lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal, lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf, compact_reservations=lp-(1-c), lambda_prime_below_one=q,
+        epsilon_positive=eps, epsilon_below_one=1-eps, guard_width=1-eps*C1,
+        K_geometry=1-eps*(1+c), K_dominates_log=eps*c,
+        record_suffix=1-eps, phase_local=1-eps-delta, phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r, cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps, alpha_positive=r, alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r, delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta, short_record_fallback=eps-a,
+        small_field_exposure=1-eps-G, artificial_boundary=8-eps+r-delta-G,
+        literal_scalar_guard=Q(f['semantic']['strict_literal_gap']),
+        row_product_gap=f['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa': value-kappa for name,value in margins.items()})
+    require(len(slacks) == 47 and len(margins) == 7, 'constraint list incomplete')
+    failed = {name:str(value) for name,value in slacks.items() if value <= 0}
+    require(not failed, str(failed))
+    require(min(margins.values()) == G, 'unexpected controlling margin')
+    require(1-eps-G == h and 1-eps-r == h/2, 'balanced identities failed')
+    require(1-eps*(1+c) == h-eps*h/4, 'geometric identity failed')
+    parameters = dict(a_bit=a, a_complex=b, tau=tau, sigma=sigma, beta=beta,h=h,
+        q=q,c=c,epsilon=eps,lambda_=lam,lambda_prime=lp,alpha_squared_power=r,
+        delta=delta,C0=f['semantic']['C0'],C1=C1,kappa=kappa)
+    return dict(parameters=parameters,constraints=slacks,margins=margins,
+        minimum_margin=G,absorption_gap=G-kappa,scoped_limit=a/(1+a),
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c),finite_bridge=f)
+
+
+def cutoffs(finite_bridge, ordinary_leaf, result):
+    """PR34 sufficient numeric comparisons, retaining its all-interval padding.
+
+    These are eventual arithmetic thresholds, not practical running times or
+    substitutes for additional native setup / prime / recovery thresholds.
+    """
+    f = validate_bridge(finite_bridge, ordinary_leaf)
+    p = result['parameters']
+    e,c,r,beta = (p[k] for k in ('epsilon','c','alpha_squared_power','beta'))
+    ka,km,kb,ks = (ceil(1/x) for x in (r,e*c,1-e,e*beta))
+    C0 = f['semantic']['C0']
+    largest = max(f[name]['m'] for name in ('bit_coarse','complex','ordinary_leaf'))
+    cuts = dict(guard=ceil(Q((2*C0).bit_length())/(1-e)),
+        normalization=ceil(7/(1-e-r)), alpha=16*ka*ka+1, compact=64*km*km+1,
+        geometry=ceil(3/(1-e*(1+c))),phase_cell=ceil(9/(e-(1-r)/2)),
+        period=128*kb*kb+1,stopped_leaf=ks*(4*largest).bit_length(),log_p=25,reservoir=14)
+    common = max(cuts.values())
+    require(common >= max(2*ka,2*km,2*kb,25), 'cutoff does not cover monotone regime')
+    checkpoints=[]
+    for j in range(6):
+        z=common*2**j
+        powers=dict(alpha=(z//ka,8*(z+ka)+64),compact=(z//km,32*(z+km)+192),
+            period=(z//kb,16*(z+kb)+56),
+            rows=(z//kb,f['rows']['suffix_slope']*(z+kb+8)),
+            leaf=(z//ks,4*largest))
+        require(all(lhs >= rhs.bit_length() for lhs,rhs in powers.values()),
+                'compressed-power threshold failed')
+        checkpoints.append(dict(log2_input=z,powers={name:dict(exponent=lhs,rhs=rhs,
+             rhs_bits=rhs.bit_length()) for name,(lhs,rhs) in powers.items()}))
+    return dict(cutoff_log2_input=cuts,common_cutoff=common,power_checkpoints=checkpoints,
+        all_interval_argument='For [Z,Z+k), bound RHS at Z+k and LHS at floor(Z/k). '
+          'Successive intervals double LHS; the positive affine RHS grows at most '
+          'linearly, so 2^j >= j+1 proves persistence.',
+        additional_eventual_conditions=['BHP prime threshold and distinct-prime packing',
+          'Fixed rational basis, eligible common prime and native table setup',
+          'Catalogue/descriptor domination, logarithm absorption, exact recovery'],
+        practical_runtime_claim=False)
diff --git a/research/coordinated-frames-and-entrance-banks/references/pr166/PROOF.md b/research/coordinated-frames-and-entrance-banks/references/pr166/PROOF.md
new file mode 100644
index 0000000000..4b4341c091
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/references/pr166/PROOF.md
@@ -0,0 +1,221 @@
+# Dirty-target accumulation removes terminal producer rows
+
+Status: author-checked finite compiler lemma and 67-row substitution on the
+current PR163 complex word. This is not a new assembled multiplication bound.
+Independent review and any advertised supplier/assembly update remain separate.
+
+Scientific inputs are the selected complex files introduced at
+15c702a929b7d640107a95e196186ad74e876c82. They are unchanged at publication
+base e1813796ef5c3ca38c5dd7b9e8d81b3908ff5997; that later commit changes only
+attribution NOTICE and its SOURCE digest. The checker verifies the existing
+result.json SHA256 pins for every finite input it consumes.
+
+## Relation to earlier constructions
+
+Dumas and Grenet's *In-place accumulation of fast multiplication formulae*
+(https://arxiv.org/pdf/2307.12712) motivates using arbitrary-valued output
+storage. Its RAM results do not establish this frame-constrained compiler.
+No novelty claim is made for the scalar pre/post cancellation identity.
+
+PR161's P±Q merge and PR163's singleton-channel fusion create internal producer
+sums. This substitution instead removes independent auxiliary coordinates by
+using existing independently valued data targets. The paired-cube identity,
+Clifford representatives, physical reuse, copied-center interfaces and all-size
+compiler assumptions retain their upstream attribution and proof obligations.
+
+## Local lemma with chronological hypotheses
+
+Let z_s be an independently dirty, zero-entrance-gauge producer role. Assume:
+
+1. It is neither a source-injection role nor an aliased recipient/donor.
+2. Its only producer incidences are additive destination writes
+   z_s <- z_s + b_j a_j, with b_j in {1,-1}. It never controls another
+   producer role. It has exactly one root read, uniformly z_s/2 into a
+   target group T.
+3. Its literal write-frame chain is 0 <= U_1 <= ... <= U_k <= U,
+   where U is the common root read frame. Every write uses the identical
+   exact T_(U_j) on its control and destination.
+4. All these writes occur after the copied-center phase. Before the first
+   deferred target corrections, the targets in T are at frame0.
+5. Some pivot c in T receives no target correction between that common-zero
+   point and the last write/post-read. Other targets may receive corrections
+   during this interval, provided those corrections do not depend on target
+   contents and their existing frames are contained in U.
+6. Later target reads continue their old nested chains from U. No other
+   target-controlled producer operation is introduced.
+
+All targets and retained dirty roles have arbitrary independent initial values.
+No target is assumed clean, and the deleted dirty variable is not set to zero:
+the new construction has one fewer independent auxiliary input coordinate.
+
+Keep the ORIGINAL initial and deferred old-response coefficient columns for
+EVERY retained role. Omit only the deleted role's column. In particular, do
+not recompute the old correction from a decoder which simply drops its root;
+that would omit the root response of its retained ancestors. The inherited
+expanded scalar old-response implementation and its conservative charge remain
+available. The sink's own old-response column is exactly the uniform1/2
+column on T because it never controls another producer row.
+
+At the common-zero point, after the copied-center contribution and before
+its deferred corrections, perform
+
+    y_t -= y_c                 for t in T minus {c}.
+
+At the original time of each write, raise the pivot along the deleted role's
+actual chain and replace that write by
+
+    y_c += (b_j/2) a_j.
+
+Immediately after the last such write, raise the pivot and the other targets
+to the original U and perform
+
+    y_t += y_c                 for t in T minus {c}.
+
+Delete the old positive root read, all writes and inverse-cleanup writes on
+z_s, its initial old-response column, its physical stream and its full-frame
+ending. Every retained producer operation, original source operation, source
+return, copied-center step and old-response column stays in its original order.
+
+Write f=sum_j b_j a_j for the ACTUAL current controls, including all their
+retained dirty dependence. The sandwich adds f/2 to all targets and preserves
+every old target value. Any intervening correction e_t on a non-pivot is
+unchanged:
+
+    (y_t-y_c+e_t) + (y_c+f/2) = y_t+e_t+f/2.
+
+An intervening correction to the pivot would be broadcast to every target;
+this is why the pivot chronology condition is essential. The test does not
+incorrectly require every non-pivot correction to wait.
+
+The retained producer is the same invertible word with destination-only sink
+writes omitted. It contains no reference to the deleted coordinate, so its
+literal inverse restores every retained dirty input. Its response from each
+retained source/dirty column is unchanged at the targets, and the retained
+old-response columns cancel those dirty parts as before. Earlier target data,
+center contributions and later target contributions are all preserved.
+
+## Exact frames, inverse and simultaneous use
+
+The pre-shears use T_0 on both targets. Each accumulating write uses the
+original exact T_(U_j) on pivot and control. Every post-shear uses one
+identical T_U on its two operands. Rank-zero reconciliation is actual inherited
+scalar/movement work, not equality up to phase.
+
+Reverse the COMPLETE literal new word and invert every shear. Use
+D_U=T_U F^(-1) in the reverse/complement orientation. Then
+
+    D_U D_V^(-1) = T_U T_V^(-1)
+
+exactly. All reversed gate incidences are still identical, all transition
+ranks are preserved, and sources/retained dirty values are restored. This is
+the chronological inverse, not the invalid source/target-swapped transpose.
+The data-bank and rank-zero endpoint conventions remain the inherited ones.
+
+For sinks with disjoint target groups, perform each pre-sandwich at the same
+common-zero cut and each post-sandwich at its own last-write time. There is
+no target collision or producer feedback, so all substitutions coexist.
+
+## Paid ledger
+
+Let r=dim(U) and let P be the complete positive-width prefix of the deleted
+role from0 through every write to U. Its old local row bill is P+[h-r].
+The pivot's old first target child [r] is replaced by P. Other target paths
+and all retained row paths are unchanged. Hence the EXACT local child change is
+
+    -[r] -[h-r],
+
+with one independent row removed. This accounts for every prefix, endpoint and
+full-frame cleanup; fragmentation inside P cancels rather than disappearing
+without payment. The same delta holds for the complete inverse orientation.
+
+In the existing three completed orthogonal cores, for each zero-gauge sink:
+
+    delta W = -1;
+    delta children = -3*[r] -3*[h-r];
+    delta rank = -3h;
+    delta deficit = 0.
+
+There is no exterior tail for a zero-gauge physical row. The full group factor
+and retained routing/row reserves are not reduced by fiat. For moment exponent
+p in (0,1), the unnormalized slack improvement is
+
+    3 r^p + 3(h-r)^p - (3h)^p > 0,
+
+because the six positive pieces sum to3h. This is a strict finite-profile
+improvement, not a newly certified assembled kappa.
+
+For k writes and n targets, the old affected scalar gates are k forward
+writes, k inverse writes, n initial dirty reads and n positive root reads.
+The new ones are k accumulating writes and2(n-1) target shears. Thus the
+scalar-gate count decreases by k+2. All new coefficients already belong to
+the Gaussian-dyadic alphabet; no odd divisor is introduced. Keep the original
+conservative virtual R, M, expanded-correction scalar charge, full finite
+stock and precision/routing reserves rather than shrinking them without proof.
+
+## Exact current finite instance
+
+The checker examines ONLY the165 all-eight disjoint roots of the selected
+PR163 word. It finds67 eligible roots on disjoint target cubes. The other98
+have retained producer consumers; two of those also touch the center phase.
+It does not search arbitrary face, singleton or cross-cube combinations.
+
+For example, root0 uses unaliased role12354. Its only incidences are additive
+writes21902 and24308, at ranks15 and18. Target0 can be the pivot. Targets6
+and7 receive early compensation, but their independent additive corrections
+are valid while encoded. The pivot's first compensation occurs after the
+post-read. This concrete distinction is verified in the actual target-event
+replay rather than inferred from a scalar channel formula.
+
+The selected simultaneous change is:
+
+    physical R: 13041 -> 12974
+    W per vertex: 15681 -> 15614
+    rank per vertex: 1033626 -> 1029204
+    deficit: 1320 -> 1320
+    child multiplicity18: -201
+    child multiplicity4: -201
+    scalar gates: -300
+
+selected.json contains every selected role, its original writes, pivot and
+chronological boundary, as well as the complete new local/target/shared
+histograms and source pins. Its decimal runtime is a diagnostic, not a
+mathematical certificate input.
+
+## What the author check actually establishes
+
+check_sink_accumulator.py performs these exact checks from existing frozen
+inputs, without importing an upstream executable or rebuilding a producer:
+
+- original source pins and full positive physical histogram equality;
+- all role classifications, aliases, source injections, literal consumers,
+  write-frame chains and original root destinations;
+- an available late-corrected pivot for every selected group;
+- every new target event interleaved with all original deadline reads and
+  retained root reads, including all endpoint changes;
+- exact adjoint pullbacks through the actual aliased producer for every
+  selected sink, covering all1320 original source columns and all13041
+  original dirty columns, including zero columns;
+- each fresh disjoint channel has exactly its448 required source terms;
+- arbitrary target and intervening non-pivot corrections, both signs, and
+  actual reversal of each scalar sandwich;
+- retained source/dirty identity by destination-only deletion and literal
+  inverse cancellation; no Monte Carlo or injective-digit assumption;
+- disjoint simultaneous support and complete child/row deltas.
+
+Together with the unchanged inherited complete word, this covers its15681
+original source/target/dirty coordinates and the15614-coordinate new word.
+The original saved full proof contracts remain dependencies. The package does
+not update the bit supplier, balance parameters, all47 assembly inequalities,
+precision constants or advertised multiplication exponent.
+
+## Reproduce
+
+From the repository root:
+
+    python research/paired-cube-target-accumulator/check_sink_accumulator.py \
+      --candidate research/paired-cube-balanced-161/selected/complex \
+      --output /tmp/target-accumulator-check.json
+
+The standard-library checker has60-second CPU/wall and768MiB address-space
+caps. The author run completed in approximately5.5seconds and85MiB. Compare
+mathematical fields, not runtime/RSS, with selected.json.
diff --git a/research/coordinated-frames-and-entrance-banks/references/pr166/README.md b/research/coordinated-frames-and-entrance-banks/references/pr166/README.md
new file mode 100644
index 0000000000..0adb5c653a
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/references/pr166/README.md
@@ -0,0 +1,15 @@
+# Target-accumulating terminal compiler
+
+This author-checked research packet replaces67 actual PR163 producer sinks
+with existing independently valued target accumulators. The complete finite
+profile removes67 rows and201 children of each width18 and4; deficit1320 is
+preserved. It makes no new assembled multiplication-bound claim.
+
+Read PROOF.md for the exact chronological lemma, retained dirty corrections,
+full inverse, scalar/endpoint bill and inherited obligations. selected.json is
+the concrete current-word specification. Run check_sink_accumulator.py against
+the existing paired-cube-balanced-161/selected/complex data to reproduce it.
+
+The motivating scalar idea is output accumulation; no novelty claim is made.
+Existing scalar-producer, frame, physical-reuse and all-size interfaces retain
+the attribution in the source package's current NOTICE.
diff --git a/research/coordinated-frames-and-entrance-banks/references/pr168/bit-physical-profile.json b/research/coordinated-frames-and-entrance-banks/references/pr168/bit-physical-profile.json
new file mode 100644
index 0000000000..8014fc2a23
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/references/pr168/bit-physical-profile.json
@@ -0,0 +1,99 @@
+{
+  "R": 19788,
+  "W_per_vertex": 21548,
+  "changed_operation_frames": 2884,
+  "checks": {
+    "donor_frames_inside_gauges": true,
+    "gauge_read_window": true,
+    "operation_frames_nondegenerate_and_spanning": true,
+    "pair_chronology": true,
+    "partner_chronology": true,
+    "phase_one_closure": true,
+    "register_contents": true,
+    "role_chains_nested": true,
+    "target_chains_nested_in_read_order": true,
+    "telescoping_deficit": true
+  },
+  "child_histogram": {
+    "1": 125448,
+    "2": 62908,
+    "3": 29616,
+    "4": 8970,
+    "5": 7800,
+    "6": 3660,
+    "7": 3312,
+    "8": 2802,
+    "9": 1812,
+    "10": 2508,
+    "11": 1158,
+    "12": 4320,
+    "13": 1590,
+    "14": 3234,
+    "15": 1230,
+    "16": 3216,
+    "17": 1164,
+    "18": 1332,
+    "19": 918,
+    "20": 5280,
+    "21": 11160,
+    "22": 13272,
+    "60": 2200
+  },
+  "deficit_per_vertex": 1936,
+  "f2_replay": {
+    "lanes": 64,
+    "registers": 18028,
+    "rejected_controls": [
+      "alias_onto_live_register",
+      "omitted_recipient_read",
+      "read_before_donor_death"
+    ],
+    "seeds_restored_and_y_plus_x": true
+  },
+  "h": 24,
+  "late_pairs": 1760,
+  "local_histogram": {
+    "1": 35656,
+    "2": 18916,
+    "3": 9872,
+    "4": 2990,
+    "5": 2600,
+    "6": 1220,
+    "7": 1104,
+    "8": 934,
+    "9": 604,
+    "10": 836,
+    "11": 386,
+    "12": 1440,
+    "13": 530,
+    "14": 1078,
+    "15": 410,
+    "16": 1072,
+    "17": 388,
+    "18": 444,
+    "19": 306,
+    "21": 2840,
+    "22": 3544
+  },
+  "loss": 528,
+  "m": 72,
+  "operations": 41288,
+  "pairs": 1760,
+  "physical_R": 18028,
+  "physical_gauge_histogram": {
+    "20": 2200
+  },
+  "rank_per_vertex": 1549520,
+  "source_data_histogram": {
+    "1": 1760,
+    "2": 880,
+    "20": 880,
+    "22": 880
+  },
+  "target_data_histogram": {
+    "1": 4400,
+    "20": 880,
+    "21": 880
+  },
+  "v": 1760
+}
diff --git a/research/coordinated-frames-and-entrance-banks/references/pr175/SOURCE.json b/research/coordinated-frames-and-entrance-banks/references/pr175/SOURCE.json
new file mode 100644
index 0000000000..a1f3beeaca
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/references/pr175/SOURCE.json
@@ -0,0 +1,11 @@
+{
+  "commit": "f1081fabbfadaaaaf7292a40d428dbbb865ce920",
+  "files": {
+    "certificates/joint-dual-kappa.json": "4044f15abf49b1551193891836e2d20975b8246acc888180af47266c64f77af0",
+    "certificates/joint-dual-validation.json": "a7834f2dda430b4d28337faf86b74b2ea1259812bc4169520e141efb835ecd9d",
+    "research/joint-dual/SOURCE.json": "c8649e811e43b33a7281746fd61e9adbd1370690ef7c2ec114a6f2d61016704f",
+    "scripts/experiments/joint_dual_reclaim_compiler.py": "f0696953d057239ed5a75e28d857b814d43ca43deaaf7b7b8fd191aa115112ff"
+  },
+  "pr": 175,
+  "scope": "Exact four-file compiler runtime overlay; no mathematical fields or emitted words change."
+}
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/baseline.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/baseline.json.gz
new file mode 100644
index 0000000000..3c8571b6d8
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/baseline.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/frames.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/frames.json.gz
new file mode 100644
index 0000000000..8b40a55a10
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/frames.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/graph.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/graph.json.gz
new file mode 100644
index 0000000000..2977cf28c4
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/graph.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/physical-frames.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/physical-frames.json.gz
new file mode 100644
index 0000000000..ca9dfc3ce1
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/physical-frames.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/physical-pairs.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/physical-pairs.json.gz
new file mode 100644
index 0000000000..ba227e2d8f
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/physical-pairs.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/profile-before.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/profile-before.json.gz
new file mode 100644
index 0000000000..8470955659
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/profile-before.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/profile.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/profile.json.gz
new file mode 100644
index 0000000000..e9b52420b7
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/profile.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/result.json b/research/coordinated-frames-and-entrance-banks/selected/complex/result.json
new file mode 100644
index 0000000000..c29c84a2b7
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/selected/complex/result.json
@@ -0,0 +1,14 @@
+{
+  "pins": {
+    "baseline.json.gz": "f4ad8cd1635f279d201225c174548cb03db13b63156cdf647ca8e834d4746524",
+    "frames.json.gz": "b516b59e7bd7e68c56c02e0041e7455b2eff445e81ac5fb69570d424cf0178a1",
+    "graph.json.gz": "9f73a6a7f1a4410e954246c6ff95d9cad0582d0057f707a693cc2fd170995ead",
+    "physical-frames.json.gz": "0e92261dbedacb1174ee58712f0a86dab00a4f2411e377faab7c08f54f8307f0",
+    "physical-pairs.json.gz": "869b852cbfd63ef691336ea398280ef57896dd876f6a6c78edac88a2a304fffb",
+    "profile-before.json.gz": "bd39675ce03fd1d4d456f736bcfc0b389532eb4f0a19c92877c2af5cb6f7310c",
+    "profile.json.gz": "cf80065ea589fcb0e9e83a2dfc76f694e3ca8bc2d8290f4175c45e2a100b8f84",
+    "word.json.gz": "0c4b720c57717c7e55e88f86a9e17ed75b3891f9d2e7ed29df0a188f8dfee67a"
+  },
+  "source_revision_unchanged": true,
+  "scalar_data_copied_exactly": true
+}
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/sinks.json b/research/coordinated-frames-and-entrance-banks/selected/complex/sinks.json
new file mode 100644
index 0000000000..40bdcefdc4
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/selected/complex/sinks.json
@@ -0,0 +1,1658 @@
+{
+  "status": "AUTHOR_EXACT_FINITE_COMPILER_CHECK_PASS",
+  "scope": "Only all-eight disjoint roots; retained full proof contracts unchanged",
+  "input_pins": {
+    "graph.json.gz": "9f73a6a7f1a4410e954246c6ff95d9cad0582d0057f707a693cc2fd170995ead",
+    "frames.json.gz": "b516b59e7bd7e68c56c02e0041e7455b2eff445e81ac5fb69570d424cf0178a1",
+    "word.json.gz": "0c4b720c57717c7e55e88f86a9e17ed75b3891f9d2e7ed29df0a188f8dfee67a",
+    "profile-before.json.gz": "bd39675ce03fd1d4d456f736bcfc0b389532eb4f0a19c92877c2af5cb6f7310c",
+    "physical-frames.json.gz": "0e92261dbedacb1174ee58712f0a86dab00a4f2411e377faab7c08f54f8307f0",
+    "physical-pairs.json.gz": "869b852cbfd63ef691336ea398280ef57896dd876f6a6c78edac88a2a304fffb",
+    "profile.json.gz": "cf80065ea589fcb0e9e83a2dfc76f694e3ca8bc2d8290f4175c45e2a100b8f84"
+  },
+  "h": 22,
+  "v": 1320,
+  "old_physical_R": 10732,
+  "new_physical_R": 10686,
+  "old_W": 13372,
+  "new_W": 13326,
+  "old_rank": 881232,
+  "new_rank": 878196,
+  "deficit": 1320,
+  "eligible_count": 46,
+  "rejected_conditions": {
+    "retained_producer_consumer": 119,
+    "center_phase_touch": 13
+  },
+  "targets_disjoint": true,
+  "simultaneous_compatible": true,
+  "all_formal_source_and_dirty_coefficients": true,
+  "formal_original_source_dirty_target_columns": 13372,
+  "remaining_formal_columns": 13326,
+  "chronological_inverse_by_literal_reverse": true,
+  "full_old_physical_histogram_verified": true,
+  "all_new_target_events_replayed": true,
+  "scalar_sandwich_literal_reverse_checked": true,
+  "shared_child_delta": {
+    "4": -138,
+    "18": -138
+  },
+  "scalar_gate_delta": -211,
+  "new_odd_denominator": false,
+  "new_kappa_claimed": false,
+  "local_histogram": {
+    "1": 24100,
+    "2": 12918,
+    "3": 9925,
+    "4": 3487,
+    "5": 2414,
+    "6": 1322,
+    "7": 925,
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+    "12": 1178,
+    "13": 212,
+    "14": 710,
+    "15": 1074,
+    "16": 101,
+    "17": 44,
+    "19": 1778,
+    "20": 517
+  },
+  "target_histogram": {
+    "0": 6270,
+    "1": 3960,
+    "2": 24,
+    "3": 16,
+    "4": 9,
+    "5": 2,
+    "6": 7,
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+    "11": 2,
+    "12": 6,
+    "13": 2,
+    "14": 4,
+    "15": 12,
+    "16": 8,
+    "18": 1274
+  },
+  "child_histogram": {
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+    "2": 45426,
+    "3": 29823,
+    "4": 10488,
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+    "16": 327,
+    "17": 132,
+    "18": 7782,
+    "19": 5334,
+    "20": 1551
+  },
+  "sinks": [
+    {
+      "root_index": 55,
+      "role": 10245,
+      "targets": [
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+        41,
+        42,
+        43,
+        44,
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+      ],
+      "pivot": 40,
+      "writes": [
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+      ],
+      "write_ranks": [
+        10,
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+      ],
+      "prefix_children": [
+        10,
+        4,
+        4
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24324,
+      "last_write_chronology": 24500,
+      "first_pivot_old_read": 26382,
+      "first_any_old_read": 25836
+    },
+    {
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+      "role": 10600,
+      "targets": [
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+        74,
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+        77,
+        78,
+        79
+      ],
+      "pivot": 72,
+      "writes": [
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+      ],
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+        15,
+        18
+      ],
+      "prefix_children": [
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+        3
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24753,
+      "last_write_chronology": 24929,
+      "first_pivot_old_read": 26526,
+      "first_any_old_read": 25874
+    },
+    {
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+      "role": 9873,
+      "targets": [
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+        121,
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+        125,
+        126,
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+      ],
+      "pivot": 120,
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+      ],
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+      ],
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+    },
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+      "targets": [
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+      "pivot": 168,
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+    },
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+      "targets": [
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+        1091,
+        1092,
+        1093,
+        1094,
+        1095
+      ],
+      "pivot": 1088,
+      "writes": [
+        20870,
+        24558
+      ],
+      "write_ranks": [
+        15,
+        18
+      ],
+      "prefix_children": [
+        15,
+        3
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24558,
+      "last_write_chronology": 24734,
+      "first_pivot_old_read": 27502,
+      "first_any_old_read": 27108
+    },
+    {
+      "root_index": 1551,
+      "role": 10622,
+      "targets": [
+        1128,
+        1129,
+        1130,
+        1131,
+        1132,
+        1133,
+        1134,
+        1135
+      ],
+      "pivot": 1128,
+      "writes": [
+        20867,
+        24570
+      ],
+      "write_ranks": [
+        15,
+        18
+      ],
+      "prefix_children": [
+        15,
+        3
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24570,
+      "last_write_chronology": 24746,
+      "first_pivot_old_read": 27682,
+      "first_any_old_read": 25431
+    },
+    {
+      "root_index": 1562,
+      "role": 10496,
+      "targets": [
+        1136,
+        1137,
+        1138,
+        1139,
+        1140,
+        1141,
+        1142,
+        1143
+      ],
+      "pivot": 1136,
+      "writes": [
+        19241,
+        24357
+      ],
+      "write_ranks": [
+        12,
+        18
+      ],
+      "prefix_children": [
+        12,
+        6
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24357,
+      "last_write_chronology": 24533,
+      "first_pivot_old_read": 27718,
+      "first_any_old_read": 27182
+    },
+    {
+      "root_index": 1628,
+      "role": 9127,
+      "targets": [
+        1184,
+        1185,
+        1186,
+        1187,
+        1188,
+        1189,
+        1190,
+        1191
+      ],
+      "pivot": 1184,
+      "writes": [
+        12519,
+        24455,
+        24511
+      ],
+      "write_ranks": [
+        4,
+        18,
+        18
+      ],
+      "prefix_children": [
+        4,
+        14
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24511,
+      "last_write_chronology": 24687,
+      "first_pivot_old_read": 27564,
+      "first_any_old_read": 25737
+    },
+    {
+      "root_index": 1705,
+      "role": 10516,
+      "targets": [
+        1240,
+        1241,
+        1242,
+        1243,
+        1244,
+        1245,
+        1246,
+        1247
+      ],
+      "pivot": 1240,
+      "writes": [
+        20090,
+        24577
+      ],
+      "write_ranks": [
+        15,
+        18
+      ],
+      "prefix_children": [
+        15,
+        3
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24577,
+      "last_write_chronology": 24753,
+      "first_pivot_old_read": 27612,
+      "first_any_old_read": 25428
+    },
+    {
+      "root_index": 1793,
+      "role": 10628,
+      "targets": [
+        1304,
+        1305,
+        1306,
+        1307,
+        1308,
+        1309,
+        1310,
+        1311
+      ],
+      "pivot": 1304,
+      "writes": [
+        20892,
+        24640
+      ],
+      "write_ranks": [
+        15,
+        18
+      ],
+      "prefix_children": [
+        15,
+        3
+      ],
+      "pre_at_phase_cut_after_center_before_old_reads": true,
+      "post_after_op": 24640,
+      "last_write_chronology": 24816,
+      "first_pivot_old_read": 27794,
+      "first_any_old_read": 27652
+    }
+  ],
+  "dirty_coefficient_nonzero_range": [
+    701,
+    721
+  ],
+  "seconds": 7.173779400000058,
+  "rss_kib": 0
+}
diff --git a/research/coordinated-frames-and-entrance-banks/selected/complex/word.json.gz b/research/coordinated-frames-and-entrance-banks/selected/complex/word.json.gz
new file mode 100644
index 0000000000..835b0e3312
Binary files /dev/null and b/research/coordinated-frames-and-entrance-banks/selected/complex/word.json.gz differ
diff --git a/research/coordinated-frames-and-entrance-banks/verify.py b/research/coordinated-frames-and-entrance-banks/verify.py
new file mode 100644
index 0000000000..dfee0d7a81
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/verify.py
@@ -0,0 +1,43 @@
+#!/usr/bin/env python3
+"""Reproduce the complete conditional witness offline from immutable source parts."""
+from pathlib import Path
+from hashlib import sha256
+import argparse,io,json,shutil,subprocess,sys,tempfile,zipfile
+HERE=Path(__file__).resolve().parent
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+def digest(p):return sha256(p.read_bytes()).hexdigest()
+def pins():
+ manifest=json.loads((HERE/'FILES.json').read_text())
+ inventory={p.relative_to(HERE).as_posix() for p in HERE.rglob('*') if p.is_file() and p.name!='FILES.json'}
+ assert inventory==set(manifest['files']),'Unpinned or missing upload file'
+ actual={n:digest(HERE/n) for n in manifest['files']}
+ assert actual==manifest['files'],'Upload source bytes changed'
+ return actual
+def main():
+ parser=argparse.ArgumentParser(description=__doc__);parser.add_argument('--write',action='store_true',help='Authoring only: generate the derived certificate');args=parser.parse_args()
+ assert not sys.flags.optimize,'Assertions must remain enabled'
+ before=None if args.write else pins()
+ baseline=json.loads((HERE/'BASELINE.json').read_text());chunks=[]
+ for p in baseline['parts']:
+  block=(HERE/p['file']).read_bytes();assert len(block)==p['bytes'] and sha256(block).hexdigest()==p['sha256'];chunks.append(block)
+ blob=b''.join(chunks);assert sha256(blob).hexdigest()==baseline['archive_sha256']
+ with tempfile.TemporaryDirectory(prefix='coordinated-frames-offline-') as tmp:
+  root=Path(tmp)/'source';root.mkdir()
+  with zipfile.ZipFile(io.BytesIO(blob)) as z:
+   for entry in z.infolist():
+    target=(root/entry.filename).resolve();assert target.is_relative_to(root.resolve()),'Unsafe archive path'
+   z.extractall(root)
+  package=root/'research'/'coordinated-frames-and-entrance-banks';package.mkdir(parents=True)
+  for p in HERE.rglob('*'):
+   if p.is_file() and p.name!='baseline-pr179.zip' and p.name not in {entry['file'] for entry in baseline['parts']} and not p.name.startswith('baseline-pr179.part'):
+    rel=p.relative_to(HERE);dest=package/rel;dest.parent.mkdir(parents=True,exist_ok=True);shutil.copyfile(p,dest)
+  command=[sys.executable,'-B',str(package/'verify_inner.py')]
+  if args.write:command.append('--write')
+  result=subprocess.run(command,text=True,capture_output=True)
+  print(result.stderr,file=sys.stderr,end='');print(result.stdout,end='')
+  if result.returncode:raise RuntimeError('Complete offline verification failed')
+  if args.write:shutil.copyfile(package/'certificate.json',HERE/'certificate.json')
+ if not args.write:assert pins()==before,'Upload source changed during verification'
+ print('PASS offline immutable archive, source closure and complete derived certificate')
+if __name__=='__main__':main()
diff --git a/research/coordinated-frames-and-entrance-banks/verify_inner.py b/research/coordinated-frames-and-entrance-banks/verify_inner.py
new file mode 100644
index 0000000000..f7f35bbaa3
--- /dev/null
+++ b/research/coordinated-frames-and-entrance-banks/verify_inner.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Read-only aggregate verification of the selected joint finite witness."""
+from hashlib import sha256
+from pathlib import Path
+import argparse
+import json
+import subprocess
+import sys
+
+HERE=Path(__file__).resolve().parent;ROOT=HERE.parents[1]
+sys.dont_write_bytecode=True
+if hasattr(sys,'set_int_max_str_digits'):sys.set_int_max_str_digits(0)
+
+
+def digest(path):return sha256(path.read_bytes()).hexdigest()
+
+
+def pins():
+    manifest=json.loads((HERE/'SOURCE.json').read_text())
+    assert manifest['files'],'Empty source closure'
+    values={name:digest(ROOT/name) for name in manifest['files']}
+    assert values==manifest['files'],'Pinned construction or dependency bytes changed'
+    return values
+
+
+def run(relative):
+    result=subprocess.run([sys.executable,'-B',str(HERE/relative)],text=True,capture_output=True)
+    if result.returncode:raise ValueError(relative+' failed:\n'+result.stdout+result.stderr)
+    return json.loads(result.stdout)
+
+
+def build():
+    print('Checking new signed complex word, all formal columns, frames and exact moment.',file=sys.stderr,flush=True)
+    complex_result=run('complex/prove.py')
+    print('Checking frozen PR168-v4 bit frames and aliases, full F2 identity and paid moment envelope.',file=sys.stderr,flush=True)
+    bit_result=run('bit/prove.py')
+    sys.path.insert(0,str(HERE/'arithmetic'))
+    from certificate import certificate
+    arithmetic=certificate(complex_result['profile'],complex_result['physical'])
+    assert bit_result['coarse']['coarse_saving']==arithmetic['coarse_bit_saving']
+    assert bit_result['coarse']['atom_beta']==arithmetic['atom_exponent']
+    assert bit_result['coarse']['ordinary_saving']==arithmetic['actual_uniform_bit_saving']
+    assert complex_result['moment']['complex_saving']==arithmetic['complex_saving']
+    return dict(status='PASS conditional finite witness',kappa=arithmetic['kappa'],
+        complex=complex_result,bit=bit_result,arithmetic=arithmetic,
+        scope='Finite word, paid moments, full finite bridge and 47 strict arithmetic constraints; source-specified all-size contracts remain assumptions.')
+
+
+def main():
+    parser=argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--write',action='store_true',help='Authoring only: write the canonical derived certificate before freezing SOURCE.json')
+    args=parser.parse_args()
+    assert not sys.flags.optimize,'Assertions must remain enabled'
+    before=None if args.write else pins()
+    result=build();target=HERE/'certificate.json'
+    if args.write:target.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    else:
+        assert result==json.loads(target.read_text()),'Canonical certificate does not reproduce'
+        assert pins()==before,'Source closure changed during verification'
+    print('PASS coordinated-frames-and-entrance-banks kappa='+result['kappa']+'; complete complex and bit words, 47 constraints, seven margins, and '+str(result['arithmetic']['adverse_control_count'])+' assembly controls.')
+
+
+if __name__=='__main__':main()