From e89b5bfd2f5d1030658fa68e267abd19064d1344 Mon Sep 17 00:00:00 2001
From: DreamingOfClouds <Ellerbroek.Mason@gmail.com>
Date: Sun, 11 Oct 2026 09:15:41 -0500
Subject: [PATCH 1/2] =?UTF-8?q?Format=20obstructions=20(no=20new=20=CE=BA)?=
 =?UTF-8?q?:=20window=20width,=20copy=20cost=20and=20label=20dimension?=
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Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Adds research/format-obstructions: statements, proofs and exact offline checks
for Theorems A-C, the clique bound, Theorem P (fixed-role F2 windows),
Theorem S (mutable-basis simulation), h=9 E8 labelling uniqueness and k=2
plane UNSAT results, checked on gx.check1 programs (#327, #193) and #425's
word. Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via
Claude Code.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 .github/workflows/format-obstructions.yml     |  27 +
 research/format-obstructions/NOTICE           |  43 ++
 research/format-obstructions/PROOF.md         | 404 +++++++++++++
 research/format-obstructions/README.md        |  96 +++
 research/format-obstructions/SOURCE.json      |  12 +
 research/format-obstructions/checks/certs.py  | 157 +++++
 research/format-obstructions/checks/e8lib.py  |  50 ++
 research/format-obstructions/checks/f2.py     |  39 ++
 .../format-obstructions/checks/families.py    | 153 +++++
 .../format-obstructions/checks/mutable425.py  | 360 ++++++++++++
 research/format-obstructions/checks/pivots.py | 146 +++++
 .../format-obstructions/checks/pricing.py     |  97 ++++
 research/format-obstructions/checks/sat_e8.py | 476 +++++++++++++++
 research/format-obstructions/checks/toy.py    | 128 ++++
 research/format-obstructions/expected.json    | 545 ++++++++++++++++++
 research/format-obstructions/gx/gx.py         | 296 ++++++++++
 .../inputs/gcert1-p10-anneal-mw-flow.json.gz  | Bin 0 -> 324061 bytes
 .../inputs/gcert1-p11-pr193.json.gz           | Bin 0 -> 441023 bytes
 .../inputs/gcert1-p3-cone-R16.json            |   1 +
 .../inputs/pr425/FULL-WORD.json.gz            | Bin 0 -> 41088 bytes
 .../inputs/pricing/cost.py                    |  62 ++
 .../inputs/pricing/hist_p10b.json             |   1 +
 research/format-obstructions/requirements.txt |   4 +
 research/format-obstructions/verify.py        | 213 +++++++
 24 files changed, 3310 insertions(+)
 create mode 100644 .github/workflows/format-obstructions.yml
 create mode 100644 research/format-obstructions/NOTICE
 create mode 100644 research/format-obstructions/PROOF.md
 create mode 100644 research/format-obstructions/README.md
 create mode 100644 research/format-obstructions/SOURCE.json
 create mode 100644 research/format-obstructions/checks/certs.py
 create mode 100644 research/format-obstructions/checks/e8lib.py
 create mode 100644 research/format-obstructions/checks/f2.py
 create mode 100644 research/format-obstructions/checks/families.py
 create mode 100644 research/format-obstructions/checks/mutable425.py
 create mode 100644 research/format-obstructions/checks/pivots.py
 create mode 100644 research/format-obstructions/checks/pricing.py
 create mode 100644 research/format-obstructions/checks/sat_e8.py
 create mode 100644 research/format-obstructions/checks/toy.py
 create mode 100644 research/format-obstructions/expected.json
 create mode 100644 research/format-obstructions/gx/gx.py
 create mode 100644 research/format-obstructions/inputs/gcert1-p10-anneal-mw-flow.json.gz
 create mode 100644 research/format-obstructions/inputs/gcert1-p11-pr193.json.gz
 create mode 100644 research/format-obstructions/inputs/gcert1-p3-cone-R16.json
 create mode 100644 research/format-obstructions/inputs/pr425/FULL-WORD.json.gz
 create mode 100644 research/format-obstructions/inputs/pricing/cost.py
 create mode 100644 research/format-obstructions/inputs/pricing/hist_p10b.json
 create mode 100644 research/format-obstructions/requirements.txt
 create mode 100644 research/format-obstructions/verify.py

diff --git a/.github/workflows/format-obstructions.yml b/.github/workflows/format-obstructions.yml
new file mode 100644
index 0000000000..99a6263d02
--- /dev/null
+++ b/.github/workflows/format-obstructions.yml
@@ -0,0 +1,27 @@
+name: format-obstructions
+
+on:
+  pull_request:
+    paths:
+      - 'research/format-obstructions/**'
+      - '.github/workflows/format-obstructions.yml'
+  push:
+    paths:
+      - 'research/format-obstructions/**'
+      - '.github/workflows/format-obstructions.yml'
+
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    timeout-minutes: 60
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: '3.12'
+      - name: Install the pinned packages (python-sat for the CaDiCaL checks)
+        run: python -m pip install -r research/format-obstructions/requirements.txt
+      - name: Replay the certificates, the toy, Theorem P, the #425 word and Theorem S, the pricing context and the E8 label SAT checks
+        run: python3 -B research/format-obstructions/verify.py --output "$RUNNER_TEMP/format-obstructions-out"
+      - name: Require an unchanged tree
+        run: git diff --exit-code
diff --git a/research/format-obstructions/NOTICE b/research/format-obstructions/NOTICE
new file mode 100644
index 0000000000..d77b237141
--- /dev/null
+++ b/research/format-obstructions/NOTICE
@@ -0,0 +1,43 @@
+Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code. Apache-2.0.
+Inherited files retain their original notices and assistance disclosures. This package claims no exclusive
+priority over concurrent work and reports no new kappa. The credits below do not imply review or endorsement.
+
+What this package contributes: PROOF.md and README.md (the statements and proofs of Theorems A, B, C, P and S, the
+clique bound, and their scope), the checks in checks/ (taint replay and Thm A/B/C data on gcert/1 programs, the p = 3
+toy and its exhaustive optimum, the fixed-role cross-Gram computation, the overlap measurements, the replay and the
+simulation of PR #425's word, the E8 label SAT instances, the clique numbers, the pricing context), verify.py,
+expected.json, SOURCE.json and the small p = 3 program inputs/gcert1-p3-cone-R16.json (local research, not published
+upstream). Everything else is vendored byte for byte or re-used as credited below.
+
+Vendored byte for byte (sha256 in SOURCE.json):
+
+- gx/gx.py: Jacob Sussman (Anthropic Claude assistance), gcert/1 and the reference checker gx.check1; Apache-2.0,
+  Copyright 2026 Jacob Sussman, jacobalansussman/wht-power-saving-lean at f010392c (as vendored in PR #327).
+- inputs/gcert1-p11-pr193.json.gz: Jacob Sussman's gcert/1 certificate of Avi Eisenberg (ikeboy)'s PR #193 complex
+  helper (as vendored in research/five-stage-gen4-banks/inputs/complex on main).
+- inputs/gcert1-p10-anneal-mw-flow.json.gz: PR #327 (DreamingOfClouds, head 40ed0456), the p = 10 complex program
+  compiled with PR #304's recipe. Its lineage: icekylinx (PR #144 paired cubes and signed decoder, PR #184 frame flow
+  and exact lift), Avi Eisenberg (ikeboy) (PR #191/#193/#194), chafreaky / Chafik Boukhalfa (PR #200 operation
+  frames and complex checker, PR #233 maximum-weight reuse pairs, PR #256 gcert emitter), eumemic (PR #168 modules and
+  physical frames), DaysSky (PR #162 carrier closure), Jacob Sussman (gcert/1, gx, the five-stage layout of
+  hcg890's PR #234, the unit convention of gxconv). Their notices are in PR #327 and PR #304.
+- inputs/pr425/FULL-WORD.json.gz: PR #425 (jacklightChen / Zhihao Chen, with OpenAI Codex assistance; head e11ccea1),
+  the mutable-basis Hamming complex word, regenerated with its own hamming_packets.py and hamming_full.py under
+  Python 3.12 (sha256 e55a0d4a... equals the one in its EXPECTED.json). It builds on Dylan Crow's PR #370 table of
+  E8 root lines in Construction-A coordinates (with Anthropic Claude assistance) and on Jacob Sussman's E8 complex
+  unit as published in DaysSky's PR #352. Original notices: PR #425.
+- inputs/pricing/cost.py: PR #346 (Louis Harrison), the cost model of PR #310 (Dylan Crow / utcorvusvolat-dotcom, w3
+  supplier work); only solve_moment is used.
+- inputs/pricing/hist_p10b.json: the five-stage normalised histogram of PR #325 (DreamingOfClouds) as copied from the
+  math.json that its verification writes (a run output at head 0eca9340, not a committed file); PR #325 builds on #315/#285/#276 (DreamingOfClouds), eumemic's PR #168 generator and source527
+  package, chafreaky's PR #200 physical rules, hcg890 and Jacob Sussman's five-stage architecture, and Evan
+  McKinney's PR #197 completed banks.
+
+Used as references and not vendored: PR #386 (jacklightChen) for the width m = 64 of its compressed nine-stage windows;
+PR #433 (icekylinx, split centres) and PR #442 (DaysSky, idle-axis hosting) for where recent gains came from;
+PR #370 (Dylan Crow) for the E8 bit word and its copy loss 72; SovereignSteak's PR #250 and hcg890's PR #234 for the
+five-stage ledger D = 4kv - 5 cst. checks/e8lib.py uses the E8 line coordinates of PR #370; checks/sat_e8.py and
+checks/families.py are adapted from local research scripts of DreamingOfClouds (Claude assistance).
+
+The five-stage layout, the frame lemma (Lean) and gx.check1 are Jacob Sussman's. OpenAI: the original complex phase
+network of "Integer multiplication below n log n".
diff --git a/research/format-obstructions/PROOF.md b/research/format-obstructions/PROOF.md
new file mode 100644
index 0000000000..d2e0ffc663
--- /dev/null
+++ b/research/format-obstructions/PROOF.md
@@ -0,0 +1,404 @@
+# Format obstructions: statements, proofs and the checks behind them
+
+Labels: **[P]** proved here by hand; **[C]** checked by computer, exactly (F₂, integer or `Fraction` arithmetic, or
+SAT); **[M]** a pricer or model output, not a theorem; **[U]** a hypothesis read from upstream and not re-derived
+here. `verify.py` runs every [C] and [M] item; §11 maps each statement to its check.
+
+## 0. Conventions
+
+- A label space L of dimension m carries a symmetric bilinear form ⟨·,·⟩ over a field F (F₂ with the dot product for
+  the complex words; ℚ with the E8 form or x ↦ xᵀ(I − J/9)y for the bit words). For a subspace V, V^⊥ is its
+  orthogonal, rk V = dim V − dim(V ∩ V^⊥) is the rank of the form restricted to V, and V is *nondegenerate* if
+  V ∩ V^⊥ = 0. If V is nondegenerate then L = V ⊕ V^⊥ and V^⊥ is nondegenerate (L is nondegenerate).
+- A pair (port) a has a *label* U_a ⊆ L, nondegenerate of dimension k_a. In every program checked here U_a = ⟨q_a⟩
+  with ⟨q_a, q_a⟩ ≠ 0 (k = 1).
+- `h` is the window dimension of one invocation, `v` the number of ports, `R` the number of helper registers,
+  `cst` the copy cost, `m` the dimension of the whole label space of a multi-stage word.
+
+## 1. The model
+
+**Frames and steps.** Every register stands at a *frame*, a subspace of L, and changes frame by *steps*. A step
+f → g is *nested* if f ⊆ g and a *drop* otherwise. It costs d(f, g) = dim f + dim g − 2 dim(f ∩ g), the rank of the
+block (upstream's general frame lemma, Jacob Sussman's `frame_lemma` in the Lean repository; used as a black box and
+only for prices, never in Theorems A and B).
+
+**Gates.** A gate is a free linear addition between registers that all stand at one common frame; each register
+first reaches that frame by a step.
+
+**Data wires.** The source wire x_a starts at U_a. The target wire y_a starts at 0 and every frame it ever occupies
+lies inside U_a^⊥. Helper registers ("slots") start at 0. These are the rules E0–E3 of `gx.check1` (see `gx/gx.py`):
+`E1` labels (nested steps, equal frame at a gate, x: ⟨q⟩ → full, y: 0 → q^⊥, slots 0 → full), `E2` the scatter cut
+(every y at frame 0 when the retained totals are copied), `E3` gate kinds, `E6` the block ledger.
+
+**Drop-free paths.** A contribution of x_T reaches a register w *drop-free* if there are registers
+x_T = r₀, r₁, …, r_k = w and times 0 = t₀ < t₁ < … < t_k such that a gate at time t_i adds r_{i−1} into r_i, and
+r_{i−1} takes only nested steps between t_{i−1} and t_i. (A drop of the *receiving* register before the gate does
+not matter.)
+
+**Two semantics in which the statements are checked.**
+1. `gx.check1` (gcert/1): the rules above, with the star scatter, plus the exact scalar identity E5. Checked on
+   three programs: the p = 3 toy (v = 8), the p = 10 complex program of PR #327 (v = 960) and Sussman's p = 11
+   program of #193 (v = 1,320). All three are complex-side programs.
+2. PR #425's mutable-basis semantics (jacklightChen): gates (adds and dyadic scalings) between registers at the
+   same current label, nested climbs except the eight centre copies (full → 0), source registers and target
+   registers may be modified and read, no target feeds a source or helper register, a frame-0 preconditioner
+   y ← V⁻¹(y − K s) and a cleanup of all source and helper registers at the full frame. The model above contains
+   this semantics (it never assumed immutable sources); §9 checks it on #425's own word.
+
+Not covered by any check here: the bit words (their checker is the community's paired-cube checker, not gcert/1), and
+the Lean layer.
+
+## 2. Theorem A (content-frame invariant) [P][C]
+
+**Statement.** If a register w receives at time t a contribution of x_T along a drop-free path, then
+frame(w, t′) ⊇ U_T for every t′ ≥ t until w's next drop.
+
+**Proof.** Induction along the path x_T = r₀, …, r_k = w of the definition. x_T starts at U_T and takes only nested
+steps until t₁, so frame(x_T, t₁) ⊇ U_T. The gate at t₁ puts both registers at one frame F₁ = frame(x_T, t₁) ⊇ U_T, so
+r₁ stands at F₁ ⊇ U_T after it. Between t_i and t_{i+1} the register r_i takes only nested steps, so its frame at
+t_{i+1} still contains U_T, and the gate at t_{i+1} puts r_{i+1} at that frame. ∎
+
+**Corollary A1.** If a drop-free path leads from x_T to y_S, then U_T ⊆ frame(y_S) ⊆ U_S^⊥, so U_T ⊥ U_S. This holds
+for any number of hops, any gate kinds (x→x, y→y, s→s, x→y, s→y), any write order, any layout, and for recursive
+invocations (a nested invocation is a sub-network; flatten it).
+
+**Corollary A2.** U_S is nondegenerate and nonzero, so U_S ⊄ U_S^⊥: no drop-free path leads from x_S to y_S. Every
+route that delivers x_S to y_S passes a drop.
+
+**Check [C]** (`checks/certs.py`). Replay the gates of each program and carry, for every register, the set of
+sources that reached it drop-free (the scatter, a drop, passes none). At every gate every newly received source port
+must lie in the gate's frame (exact F₂ reduction). Result: 110 / 1,081,204 / 2,063,837 gate checks (p = 3 / 10 / 11),
+0 violations. The final set of y_S is orthogonal to q_S, never contains S, and has size 4 / 620 / 884. Controls:
+cyclically shifting the labels makes 12 of the lossless pairs of the p = 3 program non-orthogonal; corrupting one port
+(odd weight kept) is rejected by `gx.check1` at E1 and gives 14 violations in the replay.
+
+## 3. Theorem B (stage additivity) [P] under hypotheses [U]
+
+**Rank lemma.** If V₀, …, V_s ⊆ L are pairwise orthogonal then Σ rk V_i ≤ m.
+*Proof.* Let B be the list of the union of bases of the V_i (the V_i may overlap in isotropic vectors, so the list may
+be dependent). Its Gram matrix is block diagonal with blocks Gram(basis of V_i), so its rank is Σ rk V_i. The Gram
+matrix of a list is BᵀQB with B the m × |B| coordinate matrix, so its rank is at most m. ∎
+
+**Setting.** Fix a pair a and the stages j = 1, …, s that run an invocation containing it, in time order. Let 𝒯_j be
+the targets of stage j (all ports) and M_j = span{U_b : b ∈ 𝒯_j}. Let C_j be the span of the labels U_b of the
+ports b that reach Y_a drop-free during stage j (the *correctors* of a; by Cor. A1 C_j ⊆ U_a^⊥).
+
+- (B1) *Target family.* There is a nondecreasing family of subspaces Ψ(t) with frame(Y_a, t) ⊆ Ψ(t) ⊆ U_a^⊥. (In a
+  layout with twin gates, Ψ is the frame of the wires tied to Y_a.)
+- (B2) *Common cut.* There are times τ₁ < … < τ_s ≤ τ_{s+1} (the end) such that at τ_j every target b ∈ 𝒯_j stands at
+  the frame Ψ(τ_j), and all gates of stage j precede τ_{j+1}.
+  In `gx.check1` programs this is E2 (all y at frame 0 at the scatter).
+
+**Statement.** Put N_j = Ψ(τ_{j+1}) ∩ M_j. Then U_a, N₁, …, N_s are pairwise orthogonal, and C_j ⊆ N_j. Hence
+k_a + Σ_j rk C_j ≤ m.
+
+**Proof.** (i) N_j ⊆ Ψ(τ_{j+1}) ⊆ U_a^⊥. (ii) For j < j′: N_j ⊆ Ψ(τ_{j+1}) ⊆ Ψ(τ_{j′}) by monotonicity, and by (B2) Ψ(τ_{j′})
+is the frame of every target b of stage j′, hence ⊆ U_b^⊥; so N_j ⊥ M_{j′} ⊇ N_{j′}. (iii) C_j ⊆ N_j: a corrector b of a in
+stage j has U_b ⊆ frame(Y_a at its arrival) by Theorem A, which lies in Ψ(τ_{j+1}); and U_b ⊆ M_j. The rank of a
+subspace is monotone (a principal submatrix has rank at most that of the matrix), so rk C_j ≤ rk N_j. The rank lemma
+gives k_a + Σ rk N_j ≤ m. ∎
+
+**Corollary B1.** Suppose in every stage the correctors of a span all of U_a^⊥ ∩ (window), of dimension h − 1; this
+subspace is nondegenerate (complement of a nondegenerate line), so rk C_j = h − 1. Then
+m ≥ 1 + s(h − 1). For the five-stage layout s = 5, for K-stage layouts with twin gates s = 2K + 1:
+m ≥ 1 + (2K + 1)(h − 1). If the saving of a data wire is at most k, i.e. D ≤ N_data·k (the ledger of the five-stage
+layout, D = 4kv − 5·cst [U]; read from upstream and not derived here), the saving fraction obeys
+δ = D/(mW) ≤ (N_data/W)·k/(k + Σ_j rk C_j).
+
+**Measured neighbourhood ranks [C]** (`checks/certs.py`, `checks/families.py`, `checks/sat_e8.py`,
+`checks/mutable425.py`): the span of the drop-free correctors of every target is exactly q^⊥, of dimension h − 1:
+
+| family | h | ports | correctors per target | rank |
+|---|---|---|---|---|
+| complex p = 10 (#327) | 20 | 960 | 620 | 19 |
+| complex p = 11 (#193) | 22 | 1,320 | 884 | 21 |
+| bit, paired cube p = 4, 5, 10, 11 (ℚ, x ↦ xᵀ(I − J/9)y) | 8, 10, 20, 22 | 32 … 1,320 | 15, 39, 339, 435 | 7, 9, 19, 21 |
+| E8 bit dual (ℚ⁸, E8 form) | 8 | 120 | 63 | 7 |
+| E8 complex (unique h = 9 labelling, §6) | 9 | 120 | 56 | 8 |
+| #425's word, final lossless set of each target | 9 | 120 | | 8 |
+
+The complex p = 10, 11 rows and the #425 row are replayed drop-free correctors (final taint of every target). The bit
+and E8 rows are the in-cap neighbourhoods of the family, not replayed correctors: for the paired cube `paired_bit`
+computes port 0 only (the automorphism group of the paired cube is transitive on ports, not checked here), for E8 every
+line is computed.
+
+The p = 3 program is degenerate (its ports span a 4-dimensional subspace of F₂⁶; rank 3 ≠ h − 1 = 5) and is not used
+for Theorem B.
+
+Bounds from Cor. B1, against the widths in use: m ≥ 96 / 134 / 172 (p = 10, s = 5 / 7 / 9) against 100 / 140 / 180;
+106 (p = 11, s = 5) against 110; 36 (E8 bit dual, s = 5) against 40. For E8 complex at s = 9 the bound is
+1 + 9·8 = 73, and **#425's word has m = 73** (the `m` of its report): Theorem B is attained. For the E8 bit dual at
+s = 9 the bound is 64; the width of #386 is 64 (reported in #386, not checked here).
+
+**What (B1)–(B2) are in a layout [U].** In the five-stage layout of Sussman's repository at f010392c
+(`Work/Bridge/Net.lean`, `Work/BridgeGeom/Stage.lean`, `Work/BridgeGeom/Net.lean`) the target family of a pair is tied
+across the 2K + 1 stages by twin gates, and every stage has a common cut. These facts were read from those files; their
+consistency with (B1)–(B2) is not formally verified (see §12). #425's nine-stage word satisfies (B2) at its
+scatter (all 120 targets stand at frame 0 at the cut, checked in §9).
+
+**What Theorem B does not bound.** It bounds m below by a function of the neighbourhood rank. It says nothing about the
+helper count R, and nothing about a port family whose correctors span less than q^⊥.
+
+## 4. Theorem C (drop pricing) [P][C]
+
+**Statement.** Let a retained total be gathered drop-free from sources P, with root frame f ⊇ V := Σ_{T∈P} U_T
+(Theorem A). Its content is delivered to readers Q through a rooted tree of arrays: each non-root array is a copy
+taken by a gate at its parent's current frame, and every array may take any steps (drops and climbs, degenerate
+frames included). Every reader b reads at a frame ⊆ U_b^⊥. Then the total cost of the drops in the tree is at least
+rk(V × U(Q)) = dim V − dim(V ∩ U(Q)^⊥), the rank of the pairing V × U(Q) → F, with U(Q) = Σ_{b∈Q} U_b.
+
+**Proof.** Attach to every array e a *survivor* S_e ⊆ V, defined along time: at the root S = V; a nested step leaves
+S unchanged; a drop f_e → g_e replaces S by S ∩ g_e; a copy starts with the survivor of its parent at the copy time.
+By induction S ⊆ frame(e) at all times (at the root V ⊆ f; a nested step enlarges the frame; after a drop S ⊆ g_e;
+a copy stands at the parent's frame). A drop costs d(f_e, g_e) ≥ dim f_e − dim(f_e ∩ g_e) ≥ dim S − dim(S ∩ g_e),
+because S ⊆ f_e and the map S → f_e/(f_e ∩ g_e) has kernel S ∩ g_e.
+
+Let a reader b read array e at time t. Its common frame F_b with e lies in U_b^⊥ and contains S_e(t); write S_b = S_e(t),
+so S_b ⊥ U_b. For an array e define σ(e) as its survivor at creation and ρ(e) as the intersection of the S_b over
+all readers in the subtree of e (ρ(e) = σ(e) if there are none). Claim: cost(subtree of e) ≥ dim σ(e) − dim ρ(e).
+Induct over the tree. Along e the survivors form a decreasing chain S₀ = σ(e) ⊇ S₁ ⊇ … ⊇ S_last; e's own drops cost at
+least dim S₀ − dim S_last. A child c_i created when the survivor is S(i) ⊇ S_last has cost ≥ dim S(i) − dim ρ(c_i), with
+ρ(c_i) ⊆ S(i). Direct readers see survivors ⊇ S_last. Then ρ(e) ⊇ S_last ∩ ⋂ρ(c_i), and the codimension of S_last ∩ ρ(c_i) in
+S_last is at most that of ρ(c_i) in S(i); so dim ρ(e) ≥ dim S_last − Σ(dim S(i) − dim ρ(c_i)). Adding up gives the claim.
+At the root, ρ ⊆ V ∩ ⋂_b U_b^⊥ = V ∩ U(Q)^⊥. ∎
+
+**Corollary C1 (read discipline is worthless when the readers span the window).** If U(Q) is the whole nondegenerate
+window then V ∩ U(Q)^⊥ = 0 and the bound is dim V: splitting the readers into classes, reading at nonzero frames,
+climbing copies, or any tree is never cheaper than one copy dropped from f to 0 when f = V. The bound is subadditive over
+column blocks: rk(V × ΣU(Q_i)) ≤ Σ rk(V × U(Q_i)).
+
+**Equality [C].** In all three gcert/1 programs every retained total has frame f = V, and the readers (all ports)
+span the window, so the star copy meets the bound: p = 10: 20 totals × 18 = 360 = cst; p = 11: 440 = cst. On #425's word
+each of the eight centres gathers 57 sources (the drop-free taint before the cut, equal to the support of its root-line
+coordinate), which span F₂⁹; the pairing ranks are 9 each, 72 in total, equal to the frames paid (8 × 9). For the E8 bit-dual
+decoder M = UUᵀ (9 totals, each read by the 56 ports of its bit, rank_ℚ of the Gram block = 8) the package computes only the
+bound, 72; it equals the copy loss #370 reports (nine retained totals at the full frame), which is not replayed here.
+For the p = 3 program the port span is degenerate: the bound is 12 and the paid cost 18.
+
+**The one gain (p = 3 toy) [C]** (`checks/toy.py`). The ports of the p = 3 program span W of dimension 4 with W^⊥ ⊆ W
+(radical of dimension 2, nonzero vectors 001111, 110011, 111100). Allow the scatter in cuts; at a cut all y stand at
+one common frame g ⊆ ⋂ q^⊥ = W^⊥, copy k costs d(f_k, g), and the cut frames form a chain. Cut 1 at g = 0 for the totals
+2–5, the y climb to l = 111100, cut 2 at g = ⟨l⟩ for the totals 0, 1. Result: cst 18 → 16, moves 194 → 192, every
+register chain nested, equal frames at every gate, every y inside q^⊥ and avoiding q, block mass R·h + 2v(h − 1),
+and `gx.scal` passes (gates and coefficients unchanged). An exhaustive search over all chains of cut frames in W^⊥
+gives optimum 16 (the old format: 18). Control: a common frame outside the radical puts some target outside its q^⊥.
+The gain needs W^⊥ ≠ 0, a degenerate port span. In the p = 10 and p = 11 programs (and every family above) the ports
+span the nondegenerate window (`window_rank = h`), so g = 0 is forced and the gain is 0.
+
+## 5. The clique bound [P][C]
+
+**Statement.** Let K be a set of ports such that for all T ≠ S in K there is a drop-free path from x_T to y_S. Then the
+U_T (T ∈ K) are pairwise orthogonal and nondegenerate, so they form an orthogonal direct sum and
+Σ_{T∈K} k_T ≤ r, where r is the dimension of the window that contains the labels. If H is the graph of ports joined
+by a drop-free route, k_T = k for all T, then k·ω(H) ≤ r.
+
+**Proof.** Corollary A1 gives pairwise orthogonality; each U_T is nondegenerate, so U_T ∩ Σ_{S≠T} U_S ⊆ U_T ∩ U_T^⊥ = 0. ∎
+The statement is independent of the field, the form and any symmetry of the labels.
+
+**In the five-stage ledger** (D = 4kv − 5·cst, W = 4v + R, m = 5h): δ = D/(mW) < k/(5h) ≤ 1/(5ω(H)).
+
+**Instances [C]** (`checks/families.py`, `checks/sat_e8.py`; ω of the in-cap graph of the paired cube, exact maximum
+clique): bit side (|S ∩ T| = 1) ω = p for p = 4, 5, 6, 7; complex side (|S ∩ T| ∈ {0, 2}) ω = 8, 8, 12, 12 for p = 4, 5, 6, 7.
+For p = 10 complex, 20 ports (five pairwise disjoint transversal 4-sets from the K₅ stars, all four triples each) are
+pairwise orthogonal with q·q = 1 and every pair of them is a drop-free pair inside the #327 program itself, so
+Σ k_T ≤ 20 = h and k = 1. The E8 in-cap graph (3,360 edges, degree 56) has ω = 8 (8,640 maximal 8-cliques, each an
+orthonormal basis): k = 1 needs r ≥ 8, k = 2 needs r ≥ 16 by the bound alone and r ≥ 18 by §6.
+
+On the bit side ω = p = h/2, so the bound allows k = 2 at r = h; it says nothing against k = 2 there.
+
+## 6. Labels on the E8 in-cap graph: exact SAT results [C]
+
+Setting (`checks/sat_e8.py`; CaDiCaL 1.5.3 (`Cadical153`, uniqueness) and 1.9.5 (`cadical195`, planes) through python-sat 1.9.dev15). H is the graph on the 120 E8 root lines
+(Hamming coordinates of #370) joining lines with ⟨a, b⟩ = ±1. A *k-dimensional labelling in dimension r* assigns
+to every line a nondegenerate k-dimensional subspace A_T of (F₂^r, dot product), with A_S ⊥ A_T on every edge.
+By Cor. A1 these are the labels of any word in which H is realised by drop-free correctors.
+
+**Reduction.** Fix an 8-clique C of H (§5). Its labels are pairwise orthogonal and nondegenerate, hence an
+orthogonal direct sum. The window dimension is r ≤ 17, so the complement of the span of C has dimension at most 1
+(k = 2: r − 16) or r − 8 ≤ 1 (k = 1, r ∈ {8, 9}), and is absent or I₁.
+- k = 1: the labels are orthonormal; as F₂^r = ⟨C⟩ ⊕ ⟨C⟩^⊥ with ⟨C⟩ of dimension 8 and ⟨C⟩^⊥ absent (r = 8) or a nondegenerate
+  line ⟨w⟩ with w·w = 1 (r = 9), an isometry of (F₂^r, ·) maps them to e₁, …, e₈. Every other line T has A_T orthogonal to the blocks of the clique members adjacent
+  to T, i.e. supported on the blocks of the non-adjacent members and on the complement coordinates.
+- k = 2: each plane of C is either I₂ (p·p = q·q = 1, p·q = 0) or the hyperbolic plane (every nondegenerate symmetric
+  plane over F₂ is one of the two); for r = 16 the 8 planes are the whole space, so not all can be hyperbolic (that form
+  is alternating, the dot product is not); for r = 17 the complement is I₁. Non-clique planes are free of type.
+  The SAT instance enumerates all 2⁸ type patterns of the clique (255 for r = 16).
+
+**6.1 Uniqueness at h = 9.** The CNF has the 9 coordinates of every line, u·u = 1, orthogonality on H, and injectivity
+of the labels (E0 of `gx.check1`). Enumerated to exhaustion: **exactly one solution** (blocking clause after the first
+solution is UNSAT). It is checked independently of the CNF: u·u = 1, orthogonal on all 3,360 edges, injective, and
+u_a·u_b = 1 on all 3,780 E8-orthogonal pairs, which is the Gram matrix of Sussman's E8 complex unit (decoder
+M = J + (G mod 2)). So every h = 9 labelling of H is, up to an isometry of F₂⁹, Sussman's. Its correctors span rank 8
+for every line. #425's ports satisfy exactly this Gram relation with the word's own root lines (all 120 × 120 entries),
+so #425 uses that labelling. Control: the same CNF with h = 8 is UNSAT.
+
+**6.2 Planes (k = 2).** **UNSAT for r = 16 (255 patterns) and r = 17 (256 patterns).** A labelling in dimension r ≤ 17
+would embed in dimension 17 by padding with zero coordinates (the dot product of F₂^{r+1} restricts to that of F₂^r), so
+there is no k = 2 labelling of H in any dimension r ≤ 17: k = 2 needs r ≥ 18. Dimension 18 is attained by the direct sum
+of two copies of the 9-dimensional labelling (A_T = ⟨(u_T, 0), (0, u_T)⟩), a split labelling. Controls: the k = 1
+encoder at r = 8 is UNSAT and at r = 9 SAT (rank 9); the k = 2 encoder at r = 18 accepts the direct sum (SAT under the
+assumptions that fix it), so the UNSAT results are not an artefact of the encoding.
+
+This closes the labelling lever only for the **full E8 in-cap graph**: it says nothing about subgraphs, other
+decoders or other port sets.
+
+## 7. Fixed-role B₂ windows: Theorem P [P][C], with its exact scope
+
+**Scope.** Theorem P concerns layouts in which one helper set runs the same program on the same slots in every stage
+and the stage windows are transports of one window (the fixed-role windows of the five-stage layout B₂ of Sussman's
+repository, as read). It is stated for F₂ dot-product windows and checked on the complex gcert/1 programs of #327
+(p = 10) and #193 (p = 11) only, and it is **not a general obstruction**. **Compressed windows circumvent it**: #425's
+nine-stage complex word has m = 73 = 1 + 9·8 (checked in §9), which attains the width bound of Theorem B and lies below
+the s·h = 81 that Theorem P forces for a fixed-role complex layout whose program's slot-carried pairs span, so its
+windows do not satisfy (P1)–(P2) together with a spanning program. PR #386's nine-stage bit word (reported m = 64 =
+1 + 9·7, h = 8) has a ℚ form; Theorem P is not stated or checked for ℚ forms and nothing is claimed about whether
+the analogue would force 72 there.
+
+**Hypotheses [U].** (P1) Stage j runs the program c on the window W_j = ι_j(F₂^h), where ι_j : (F₂^h, ·) → L is an
+isometric embedding (so W_j is nondegenerate of dimension h); in stage j the source T starts at ι_j(⟨q_T⟩) and every
+frame of y_S during stage j lies in ι_j(q_S)^⊥. (P2) One helper set runs the *same* program on the *same* slots in
+every stage, and slots never drop (nested steps only). Both are read from the layout, not re-derived here.
+
+**Statement.** For stages j < j′ let K = ι_jᵀι_j′ (h × h over F₂). Let the *slot-carried pairs* be the pairs (T, S) such
+that T is in the final set of sources that reached some slot z drop-free in the program and z writes the target y_S.
+If the tensors q_T ⊗ q_S over the slot-carried pairs span F₂^{h×h}, then K = 0. Consequently W₁, …, W_s are pairwise
+orthogonal, nondegenerate, and m ≥ s·h.
+
+**Proof.** After stage j the slot z stands at a frame containing ι_j(q_T) for every such T (Theorem A in stage j) and it
+never drops. In stage j′ it writes y_S at a frame inside ι_j′(q_S)^⊥ (P1). So ⟨ι_j q_T, ι_j′ q_S⟩ = q_Tᵀ K q_S = 0 for
+every slot-carried pair. This is linear in K; if the q_T ⊗ q_S span, K = 0. Pairwise orthogonal nondegenerate
+subspaces form a direct sum. ∎
+
+**Check [C]** (`checks/pivots.py`). The pairs with q_T·q_S = 0 alone span a space of dimension h² − 1 (their tensors lie
+in the hyperplane tr M = Σ M_ii = q_T·q_S = 0; this leaves K = λI), and the full set of slot-carried pairs spans h²:
+399 / 400 on the #327 program (3,708 writer slots) and 483 / 484 on #193 (4,125 writer slots). So K = 0; a
+slot-carried pair with q_T·q_S = 1 is what excludes K = λI, i.e. two stages with identified windows. By Cor. A1 such a
+pair cannot be drop-free within one stage, so T necessarily reached the slot only after the slot wrote y_S.
+
+A shorter route to K = 0 exists under (P1)–(P2) and rule E1 of `gx.check1` (every slot ends a stage at the full window):
+the frame of a slot in stage j′ contains ι_j(F₂^h), so ι_j(F₂^h) ⊥ ι_j′(q_S) for every slot-written S, and the ports of the
+slot-written targets span F₂^h (rank 20 and 22, checked). The taint computation above proves the stronger version that
+needs only the frames actually reached by taint, so the rank 399/400 should not be overweighted.
+
+**What the exact part gives, and what is priced [M]** (`checks/pricing.py`, pricer `inputs/pricing/cost.py`, histogram of
+#325's p = 10 bit word, pinned root reproduced). If the 2K block pivots were free (m = 1 + 5(h − 1) = 96 instead of
+100: each helper wire loses 4 levels, the pair-1 end idle block 2h + 2 → 2h − 2, the pair-2 end idle block of rank 4
+disappears), the saving of the p = 10 bit word would be larger by +3.56 % to +8.11 %, depending on which helper blocks
+shrink (largest blocks / mid blocks / 2 → 1 / rank-1 blocks removed: +3.56, +5.75, +6.39, +8.11), and by +1.53, +3.68,
++4.30, +5.99 with 40 extra copy ranks per unit (below). The saving would be unavailable to fixed-role B₂ windows if the
+analogue of Theorem P holds for the bit word's programs; that is **not claimed**, since Theorem P is stated and was
+computed on complex F₂ programs only and no bit certificate was replayed. Nothing here concerns compressed windows.
+
+**Measurements for one overlap [C][model]** (`checks/pivots.py`). Suppose two stage windows of a *shared-helper* layout
+overlap in one direction w = ι_{j−1}(e) = ι_j(e′) with e·e = e′·e′ = 1. A retained total whose frame is not inside e^⊥
+carries w into stage j and, in the model, pays one extra copy rank there unless e′ lies in its frame. Over all
+odd a, b ∈ F₂²⁰ and the 20 retained totals of the #327 program (frames of dimension 18): at most 1 total has a frame
+inside a^⊥ and at most 9 frames contain b, so ≥ 20 − 1 − 9 = **10 forced extra copy ranks per overlap**. On the helper
+side, for each of the 1,160 candidate directions tried (the 20 unit vectors and the 1,140 weight-3 vectors) at least
+5,984 of the 6,515 slots end at a frame not inside e^⊥, and the forward closure of those slots through the gates
+contains all 8,435 registers and all 960 targets (their ports have rank 20, so no e′ ⊥ all of them exists). These are
+counts in a stated model for one program with shared slots; they say nothing about layouts whose helpers are organised
+differently.
+
+## 8. Non-nested steps on registers [P][M]
+
+**Proposition N [P].** A register whose chain runs from f₀ to f_end with nested steps except one non-nested step f → g
+has total mass (dim f_end − dim f₀) + 2r with r = dim f − dim(f ∩ g). *Proof.* The step costs d(f, g) = r + r′ with
+r′ = dim g − dim(f ∩ g); the nested steps cost (dim f − dim f₀) + (dim f_end − dim g); and dim f − dim g = r − r′. ∎
+So a descent of rank r on a register that must still reach its sink costs 2r more than a nested chain with the same
+endpoints. A copy of rank r is charged r by the ledger (`gx.check1`: N + tails = R·h + 2v(h − 1) + cst, with the copy
+blocks H["c"][dim f] in `cst`).
+
+By Theorem C the drops that carry a total to its readers cost at least rk(V × U(Q)) however they are arranged, and the
+star copy attains it on every family checked; a register descent pays that and then the re-climb.
+
+**Pricing [M]** (same pricer and histogram as §7). One rank-1 copy in each of the five stages, per 12-normalised unit,
+is 60 extra blocks of rank 1 (5 × 12): **−0.2457 %**. One rank-1 register descent in each of the five stages is one
+non-nested step of rank d = 2 per stage, 60 extra blocks of rank 2: **−0.4911 %**. Quote ≈ −0.49 %: other blockings of
+the same mass give −0.4906 % to −0.4914 % (two rank-1 blocks: −0.4914 %), so the fourth digit is a modelling choice.
+
+**Two-level words.** A word whose gates are small verified words flattens to a nested-frame network; Theorems A–C apply
+to the flattened network unchanged.
+
+## 9. PR #425's mutable-basis word: Theorems A–C and Theorem S [C][P]
+
+Input: `inputs/pr425/FULL-WORD.json.gz`, #425's `FULL-WORD.json.gz` (920 registers: 120 sources, 120 targets,
+680 helpers; m = 73, h = 9), regenerated with #425's own `hamming_packets.py` and `hamming_full.py` under Python 3.12;
+its sha256 equals the one in #425's `EXPECTED.json`.
+
+**Replay [C]** (`checks/mutable425.py`, the same frame and taint detector as §2). 0 violations of: nested climbs,
+common frame at every gate, taint of both gate registers inside the gate frame (Theorem A), target frames inside
+q^⊥, target taint orthogonal to its port (Cor. A1), no target feeding a source or helper register. All 120 targets stand at
+frame 0 at the scatter ((B2)), and the final lossless set of every target has rank 8 = h − 1 (Theorem B: m ≥ 1 + 9·8 = 73
+= m of the word). On the eight centre copies the pairing-rank bound (Theorem C) is 72 and the frames paid are 72. The
+same detector in *immutable-source / no-target-read* mode flags 1,640 modifications of sources and 112 reads of
+targets in the word as written. Exact scalar replay over ℚ: the helpers and sources are restored, J = I, and with the
+frame-0 preconditioner y ← V⁻¹(y − K s) the word is exactly (x, s, y) ↦ (x, s, y + x); omitting the K part is rejected.
+
+**Theorem S (simulation) [P].** *Hypotheses.* Let 𝒲 be a word with (H1) nested climbs and common-frame gates and
+invertible scalings; (H2) no gate has a target as source and a source or helper as destination; (H3) after a frame-0
+preconditioner y ← V⁻¹(y − K s) it acts as (x, s, y) ↦ (x, s, y + x). Let b_x be the number of source registers that are
+destinations of some gate and b_y the number of targets that are sources of some gate.
+*Statement.* There is a word 𝒲′ with R + b_x + b_y helper registers in which no source is modified and no target is read,
+with the same frames at the corresponding gates, that acts as (x, s, g, y) ↦ (x, s, g, y + x) after a frame-0 preconditioner.
+*Proof.* For each modified source r add a helper g_r and a gate g_r += x_r at frame ⟨q_r⟩ (g_r climbs from 0, x_r
+is already there). For each read target t add a helper g_t, starting at frame 0. Rename every occurrence of x_r and y_t
+in every event of 𝒲 to g_r and g_t. Then g_r and g_t follow the frame sequences of x_r and y_t, so every gate has the
+frames it had; the real x_r are never modified and the real y_t (t read) never read. After the last event add y_t += g_t
+at the frame q_t^⊥ (g_t is inside it, as y_t was). Finally undo, at the full frame and in reverse order, every renamed event
+whose destination is not a real target, and add g_r −= x_r. By (H2) these events read, in 𝒲′, only source registers and
+helpers (the g's are helpers), so the reverse sequence restores every source and helper register exactly, whatever the
+targets hold. Every real target then ends at y_t + (J x)_t + ℓ_t(s, γ), where γ is the initial dirt of the g's and ℓ_t is
+linear (it collects the V and K responses, including those that unread targets receive from renamed g_u); the
+coefficients of x are those of 𝒲, so J = I. The dirty registers (s and g) stand at frame 0 at the start, so the
+preconditioner y ← y − ℓ(s, γ) removes all such terms. ∎
+
+*Consequence.* Relative to its own simulation in the immutable-source / no-target-read format, the mutable-basis format
+saves at most b_x + b_y ≤ 2v helper registers (at most one per source and one per target). It bounds the helper
+count of the simulation, not its price.
+
+**Check [C].** On #425's word b_x = b_y = 112; the simulation has 904 = 680 + 112 + 112 helpers. The same detector finds 0
+violations in it (immutable mode), z is restored exactly, and after the frame-0 compensation the map is exactly
+(x, s, y) ↦ (x, s, y + x); without the compensation it is not. Under the coarse formula of #425's `hamming_full.py`
+(reproduced: b = 1.049105 × 10⁻³ for #425's word) the simulation has b = 9.102 × 10⁻⁴ (−13.24 %), below the pre-#425
+word (9.994 × 10⁻⁴). The simulation was not run through `gx.check1`; it was replayed by the detector above.
+
+## 10. What the theorems bound, and what they do not
+
+Theorems A–C and the clique bound bound three things: the label dimension k (for a given in-cap graph), the width m
+(Theorem B, for a given corrector rank) and the copy rank (Theorem C, for given source and reader spans). **They do not
+bound the helper count R, nor the idle and stock costs (the block profile of the helpers),** which is where PR #425
+(mutable-basis packets), #433 (split centres) and #442 gained. #425 keeps m at the Theorem B width and the copy rank at
+the Theorem C bound (checked in §9); #433 and #442 are not replayed here. The only family-dependent term in Theorem B is the rank of the
+corrector span; it equals h − 1 in every family above, and a family with a smaller rank and a cheap decoder would
+change the width bound.
+
+## 11. Statement → check
+
+| statement | check (module, function) | pinned facts in `expected.json` |
+|---|---|---|
+| gx.check1, Thm A, Cor A1/A2, Thm B data, Thm C on three programs | `certs.stage` | `certificates` |
+| negative controls (shifted labels, corrupted port) | `certs.controls` | `certificates.p3.controls` |
+| toy: 18 → 16, exhaustive optimum, control | `toy.run` | `toy_p3` |
+| Theorem P | `pivots.thm_P` | `pivots.thm_P` |
+| overlap measurements | `pivots.copy_count`, `pivots.overlap_scan` | `pivots` |
+| #425: replay, scalar, Theorem S, controls | `mutable425.run` | `mutable_425` |
+| pricing, pivot-free variants, copy/descent units | `pricing.run` | `pricing` |
+| neighbourhood ranks, E8 bit dual, ω, 20-clique | `families.*` | `families`, `thmB_table` |
+| h = 9 uniqueness, k = 2 planes, calibrations | `sat_e8.*` | `e8_labels` |
+
+## 12. Scope and open points
+
+- The correspondence between the model of §1 and each checker is by reading `gx.py` (E0–E7) and #425's `PROOF.txt` §3;
+  there is no mechanical proof of it and no Lean check.
+- (B1)–(B2) and (P1)–(P2) are hypotheses about the layout [U].
+- Theorem P holds for fixed-role B₂ windows only (§7) and is computed on complex programs; the bit words are not replayed.
+- The simulation of Theorem S is checked on #425's word only.
+- `inputs/pricing/hist_p10b.json` is a copy of the five-stage histogram that #325's verification writes to `math.json` (a
+  run output at head 0eca9340, not a committed file; its `bit_root` is the pinned root the pricer reproduces). It was not
+  re-derived from #325's records here.
+- Lean: (P1)–(P2) and (B1)–(B2) were read against `Work/Bridge/Net.lean`, `Work/BridgeGeom/Stage.lean` and
+  `Work/BridgeGeom/Net.lean` (f010392c); the docstrings and lemma statements are consistent with them, but this is not
+  formally verified. Two points are open: whether the backward stages 1 and 3 count as "the same program", and whether the
+  stage-j target frames lie in ι_j(q_S)^⊥.
+- `inputs/gcert1-p3-cone-R16.json` is a local research program (not published upstream).
diff --git a/research/format-obstructions/README.md b/research/format-obstructions/README.md
new file mode 100644
index 0000000000..c70124ee07
--- /dev/null
+++ b/research/format-obstructions/README.md
@@ -0,0 +1,96 @@
+# format-obstructions
+
+**Exact bounds on window width, copy cost and label dimension (no new κ).** This package states and proves three
+theorems about frames and gates (A: drop-free contributions force orthogonal labels; B: stage windows add up; C: a drop
+that reaches the readers costs the pairing rank), a clique bound on the label dimension, and a simulation theorem for
+PR #425's mutable-basis words (S). Everything that can be checked exactly is checked by one offline driver on pinned
+inputs: Sussman's `gx.check1` programs, #425's own word, and exact SAT instances on the E8 in-cap graph.
+[PROOF.md](PROOF.md) has the statements, the proofs and the scope of each.
+
+## Scope
+
+- **Model.** Registers stand at frames (subspaces of a label space), climb by nested steps, drop at a price
+  d(f, g), and gates add registers that stand at one common frame. Sources start at their label, targets stay inside
+  their label's orthogonal. This is the content of the rules E0–E3 of `gx.check1` and of the common-label rule of
+  #425's conjugation lemma.
+- **Checked semantics.** `gx.check1` (gcert/1, vendored unchanged from Jacob Sussman) on three programs: the p = 3 toy,
+  the p = 10 complex program of #327 and the p = 11 program of #193; and #425's mutable-basis semantics on its own
+  920-register word.
+- **Not covered.** No bit word is replayed (the bit words use the community's paired-cube checker). No Lean. The multi-stage
+  statements use hypotheses about the layout that are read from upstream, not re-derived: (B1)–(B2) of Theorem B and
+  (P1)–(P2) of Theorem P (PROOF.md §3, §7).
+
+## Results
+
+| | statement | status |
+|---|---|---|
+| **Thm A** | A contribution of x_T that reaches a register by a drop-free path forces the register's frame to contain U_T. So a lossless x_T → y_S route needs U_T ⊥ U_S, and there is no lossless x_S → y_S route. | proved; replay of every gate of three programs, 0 violations (1.08 M and 2.06 M gate checks at p = 10, 11) |
+| **Thm B** | For a pair a with stages j = 1..s: k_a + Σ_j rk C_j ≤ m (C_j = span of the labels of a's correctors in stage j). With corrector rank h − 1 this gives m ≥ 1 + s(h − 1). | proved under the layout hypotheses (B1)–(B2); rank h − 1 measured on the p = 10, 11 complex programs, the bit paired cubes p = 4, 5, 10, 11, the E8 bit dual (7) and the E8 complex labelling (8); #425's word has m = 73 = 1 + 9·8, so the bound is attained |
+| **Thm C** | The drops that carry a total gathered from sources P to readers Q cost at least rk(U(P) × U(Q)), whatever the tree of copies, frames and drops. | proved; the star copy meets it exactly on the p = 10 (360), p = 11 (440) programs, on #425's centres (72, 57 gathered sources per centre). For the E8 bit dual the package computes the bound 72, which equals the copy loss #370 reports (not replayed) |
+| **Toy** | The p = 3 program (degenerate port span) admits a split scatter, cst 18 → 16 (exhaustive optimum 16). | exact; gain needs a degenerate port span, 0 on the nondegenerate families |
+| **Clique bound** | If every pair in a set K has a drop-free route, the labels on K are orthogonal and nondegenerate: Σ_K k_T ≤ r. | proved; ω = p (bit) and 8, 8, 12, 12 (complex, p = 4..7); 20 ports pairwise lossless inside the p = 10 program, so k = 1; ω = 8 on the E8 in-cap graph |
+| **E8 labels** | The h = 9 labelling of the E8 in-cap graph is unique (Sussman's). No k = 2 labelling exists in dimension r ≤ 17 (r = 16: 255 plane-type patterns, r = 17: 256, all UNSAT). | exact SAT (CaDiCaL) with calibrations and a positive control; for the full in-cap graph only |
+| **Thm P** | For **fixed-role B₂ windows over F₂** (one helper set, same program, same slots in every stage, slot-carried pairs spanning) the cross-Gram of two stage windows is zero, so m ≥ s·h. | proved under (P1)–(P2); computed on the complex p = 10 and #193 programs only (399/400 and 483/484). **Compressed windows circumvent it**: #425's nine-stage complex word has m = 73 = 1 + 9·8 < 81, at the Theorem B bound. #386's bit word (m = 64) has a ℚ form; Theorem P is not claimed for it. Not a general obstruction. |
+| **Prop. N** | A non-nested step of rank r on a register that must still reach its sink costs 2r over the nested chain; a copy of rank r is charged r. | proved; pricing context for the p = 10 bit word: ≈ −0.49 % for a rank-1 register descent in each of the five stages (the fourth digit is a modelling choice) against −0.2457 % for a rank-1 copy in each stage |
+| **Thm S** | A mutable-basis word with b_x modified sources and b_y read targets is simulated, in the immutable-source / no-target-read format, with R + b_x + b_y helpers. | proved; exact on #425's word (680 + 112 + 112 = 904), with controls. A bound on helper count, not on price. |
+| #425 | Thms A and C and the clique bound hold on #425's word, and its width attains Theorem B. | replay: 0 violations, centre copies 72 = 72 (57 gathered sources per centre), all 120 targets at frame 0 at the cut, rank 8 for every target |
+
+## What this means for the swarm
+
+A–C and the clique bound bound **m and the copy rank**, the label dimension for a given in-cap graph, and (through the
+five-stage ledger D ≤ N_data·k, read from upstream) **D**. They do
+**not** bound the helper count R, nor the idle and stock costs (the block profile of the helpers), which is where
+#425, #433 and #442 gained (#425, the one replayed here, keeps m at the Theorem B width and the copy rank at the
+Theorem C bound). In the
+formats covered here a further κ gain therefore has to come from R or the stock, or from a new port family whose
+corrector rank is below h − 1 with a cheap decoder (the only family-dependent term of Theorem B). Nothing here is
+a statement about the size of R.
+
+## Verify
+
+    python -m pip install -r research/format-obstructions/requirements.txt   # python-sat 1.9.dev15, networkx 3.7, mpmath 1.3.0, numpy 2.3.5
+    python3 -B research/format-obstructions/verify.py --output /tmp/format-obstructions-out   # Python 3.11+, tested 3.12
+
+No network; `-O` is refused. The driver checks the sha256 pins of `SOURCE.json`, runs 91 independent tasks in a
+process pool (`--jobs N`, default the core count), compares the facts with `expected.json`, and requires an unchanged
+tree. Runtime: about 90 s wall on 12 cores, about 15 CPU-minutes in total (the CaDiCaL instances are 70 % of it).
+It runs:
+
+- `gx.check1` (with the exact scalar identity) on the three programs; the taint replay of Theorem A at every gate;
+  Cor. A1, the corrector ranks of Theorem B and the copy bounds of Theorem C;
+- the p = 3 toy (nestedness, equal frames, scalar half, exhaustive optimum);
+- Theorem P (rank of the cross-Gram constraints), the forced-copy count and the contamination scan;
+- #425's word: frame and taint replay, scalar identity over ℚ, the Theorem S conversion and its replay;
+- the pinned pricer on the p = 10 bit histogram (root, pivot-free variants, copy and descent units);
+- the neighbourhood ranks, the E8 bit-dual decoder, the clique numbers, the 20-clique;
+- the SAT checks: uniqueness at h = 9, k = 2 at r = 16 and 17.
+
+**Negative controls.** Shifted labels (12 non-orthogonal lossless pairs) and a corrupted port (`gx.check1` rejects it at
+E1; 14 replay violations); a toy cut frame outside the radical; the immutable/no-target-read detector flags 1,640
+source modifications and 112 target reads on #425's word as written and 0 on the converted word; omitting the K part
+of #425's preconditioner and the compensation of the converted word are rejected; k = 1 at r = 8 UNSAT and r = 9 SAT,
+h = 8 UNSAT, and the k = 2 encoder at r = 18 accepts the direct sum of two k = 1 solutions.
+
+## Not checked, or checked only in part
+
+- The bit side: Theorem P (stated for F₂ windows) and the copy/overlap measurements were computed on complex programs;
+  the pricing context uses the p = 10 bit histogram. No bit certificate is replayed, and nothing is claimed about
+  Theorem P for ℚ forms. The bit and E8 rows of the Theorem B table are in-cap neighbourhoods of the family, not replayed
+  correctors.
+- The layout hypotheses (B1)–(B2), (P1)–(P2) and the width m = 64 (#386) are taken from upstream. They were read against
+  Sussman's `Work/Bridge/Net.lean`, `Work/BridgeGeom/Stage.lean` and `Work/BridgeGeom/Net.lean` (f010392c), whose
+  statements are consistent with them; **this was read, not formally verified.** Two points are open: whether the
+  backward stages 1 and 3 count as "the same program", and whether the stage-j target frames lie in ι_j(q_S)^⊥.
+- The Theorem S simulation was replayed by the detector of `checks/mutable425.py`, not by `gx.check1`.
+- `inputs/pricing/hist_p10b.json` is a copy of the five-stage histogram that #325's verification writes to `math.json` (a
+  run output at head 0eca9340, not a committed file); the pricing numbers are pricer-model outputs for the p = 10 bit
+  word, not theorems.
+- The overlap measurements are for one program and 1,160 candidate directions.
+- `inputs/gcert1-p3-cone-R16.json` is a local research program that is not published upstream.
+
+## Files
+
+`verify.py`, `expected.json`, `SOURCE.json`, `requirements.txt`, `PROOF.md`, `NOTICE`; `checks/` (`certs.py`, `toy.py`,
+`pivots.py`, `mutable425.py`, `pricing.py`, `families.py`, `sat_e8.py`, `e8lib.py`, `f2.py`); `gx/gx.py` (Sussman,
+unchanged); `inputs/` (three gcert/1 programs, `pr425/FULL-WORD.json.gz`, `pricing/cost.py`, `pricing/hist_p10b.json`).
+`.github/workflows/format-obstructions.yml` runs the driver on Python 3.12.
diff --git a/research/format-obstructions/SOURCE.json b/research/format-obstructions/SOURCE.json
new file mode 100644
index 0000000000..bf19b313c9
--- /dev/null
+++ b/research/format-obstructions/SOURCE.json
@@ -0,0 +1,12 @@
+{
+ "description": "Pinned inputs of research/format-obstructions (sha256). Vendored byte for byte; nothing is fetched at run time.",
+ "files": {
+  "gx/gx.py": "6f3dc8bb639933227a8dae181ec4a542e85c7276be9c9e330f49518e7b630b95",
+  "inputs/gcert1-p3-cone-R16.json": "fe3ad604eec43845747603414222fc6a82ed16168ef64909440c0a72eba97dc9",
+  "inputs/gcert1-p10-anneal-mw-flow.json.gz": "117d1cf1ad9c0ab3fac42557cf4aa4f7166700e8c005eb3f928b187544f23f49",
+  "inputs/gcert1-p11-pr193.json.gz": "15251deb0c11e0504d7b297c9786818888a93f08b92b03c6c122784b70488633",
+  "inputs/pr425/FULL-WORD.json.gz": "e55a0d4a1f1e3daaa082749489c9df2025f6a4fe9576aad2726cf881280f16fb",
+  "inputs/pricing/cost.py": "d55d2b865d1c3d35e343e1a38c592b9172ed8e2596d1107e34d92e2a80c7d03f",
+  "inputs/pricing/hist_p10b.json": "d70c58e539141c49453a684d0b01324a08d49f7c28dccd3b5b82472b5a8594a4"
+ }
+}
diff --git a/research/format-obstructions/checks/certs.py b/research/format-obstructions/checks/certs.py
new file mode 100644
index 0000000000..8966b44c0d
--- /dev/null
+++ b/research/format-obstructions/checks/certs.py
@@ -0,0 +1,157 @@
+"""Theorems A, B, C on gcert/1 certificates (one invocation, window F_2^h, standard dot product).
+
+For a gcert/1 file:
+  * gx.check1 (Sussman's reference checker, label half + exact scalar half) must accept it;
+  * Thm A: replay the gates and carry, for every register, the set of x-sources whose content entered it along
+    drop-free paths ("taint"; the scatter cut, a drop to frame 0, passes no taint). At EVERY gate each newly
+    received source port must lie in the gate frame (exact F_2 reduction);
+  * Cor A1: the lossless set of every target y_S (its final taint) is orthogonal to q_S, and does not contain S;
+  * Thm B data: N(S) = span{q_T : T in taint(y_S)} and its rank; every y reads every total (co-read);
+  * Thm C: for every retained total, A_k = span of the ports of its taint; copy price paid = dim f_k; lower bound =
+    rank of the pairing A_k x span(readers).
+"""
+import gzip
+import json
+import sys
+import time
+from fractions import Fraction as Fr
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parent.parent
+sys.path.insert(0, str(ROOT / "gx"))
+sys.path.insert(0, str(ROOT / "checks"))
+import gx  # noqa: E402  (Jacob Sussman's reference checker, vendored unchanged)
+from f2 import dot, pairing_rank, rank  # noqa: E402
+
+
+def load(path):
+    path = str(path)
+    op = gzip.open if path.endswith(".gz") else open
+    with op(path, "rt") as f:
+        return json.load(f)
+
+
+def replay(c, gate_check=True):
+    """Taint replay (Thm A). Returns (taint, tot_taint, tot_frame, stats)."""
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    ports = c["ports"]
+    fr = [tuple(f) for f in c["frames"]]
+    taint = [(1 << r) if r < v else 0 for r in range(nreg)]
+    stats = dict(gate_checks=0, violations=0, pairs=0)
+
+    def portlist(bits):
+        while bits:
+            low = bits & -bits
+            yield ports[low.bit_length() - 1]
+            bits ^= low
+
+    def run(gates):
+        for g in gates:
+            regs, pairs, _ = gx.gate_regs(g)
+            for t, s, co in pairs:
+                stats["pairs"] += 1
+                new = taint[s] & ~taint[t]
+                if gate_check and new:
+                    for q in portlist(new):
+                        stats["gate_checks"] += 1
+                        if gx.reduce(q, fr[g[1]]) != 0:
+                            stats["violations"] += 1
+                taint[t] |= taint[s]
+
+    run(c["A"])
+    ret = sorted(c["ret"])
+    tot_taint = {k: taint[s] for k, s, f in ret}
+    tot_frame = {k: f for k, s, f in ret}
+    run(c["B"])
+    return taint, tot_taint, tot_frame, stats
+
+
+def analyse(path, gate_check=True):
+    c = load(path)
+    t0 = time.monotonic()
+    gx.check1(c, scalar=True)
+    t_check = time.monotonic() - t0
+    h, v, R = c["h"], c["v"], c["R"]
+    ports = c["ports"]
+    fr = [tuple(f) for f in c["frames"]]
+    t0 = time.monotonic()
+    taint, tot_taint, tot_frame, stats = replay(c, gate_check)
+    t_replay = time.monotonic() - t0
+    coread_all = True
+    for t in range(v):
+        ks = {k for k, co in gx.scat_row(c, t) if co != 0}
+        if ks != set(tot_taint):
+            coread_all = False
+
+    def portlist(bits):
+        out = []
+        while bits:
+            low = bits & -bits
+            out.append(ports[low.bit_length() - 1])
+            bits ^= low
+        return out
+
+    perp_ok, self_hit = True, False
+    ranks, degs = [], []
+    for S in range(v):
+        tb = taint[v + S]
+        self_hit |= bool(tb >> S & 1)
+        Ts = [T for T in range(v) if tb >> T & 1]
+        degs.append(len(Ts))
+        perp_ok &= all(dot(ports[T], ports[S]) == 0 for T in Ts)
+        ranks.append(rank([ports[T] for T in Ts]))
+    lb = paid = 0
+    for k in sorted(tot_taint):
+        A = portlist(tot_taint[k])
+        lb += pairing_rank(A, ports)
+        paid += len(fr[tot_frame[k]])
+    res = dict(file=Path(str(path)).name, h=h, v=v, R=R, cst=c["cst"], check1="PASS",
+               thmA_gate_checks=stats["gate_checks"], thmA_violations=stats["violations"],
+               incap_perp=perp_ok, self_lossless=self_hit, incap_deg=[min(degs), max(degs)],
+               N_rank=[min(ranks), max(ranks)], coread_all=coread_all, window_rank=rank(ports),
+               copy_lb=lb, copy_paid=paid, seconds=dict(check1=round(t_check, 1), replay=round(t_replay, 1)))
+    return res, c, taint
+
+
+def controls(c, taint):
+    """Negative controls on the small certificate c (p = 3)."""
+    import copy
+    v = c["v"]
+    ports = c["ports"]
+    out = {}
+    # (1) cyclically shifted labels: the lossless sets are no longer orthogonal -> the Cor A1 test fires
+    sh = ports[1:] + ports[:1]
+    bad = sum(1 for S in range(v) for T in range(v)
+              if taint[v + S] >> T & 1 and dot(sh[T], sh[S]))
+    out["shifted_labels_nonorthogonal_pairs"] = bad
+    assert bad > 0, "control failed: shifted labels still orthogonal"
+    # (2) a corrupted port that still passes E0 (odd weight, distinct): gx.check1 rejects it at E1 and the taint replay
+    #     reports source ports outside the gate frame
+    used = set(ports)
+    mask = next(m for m in range(3, 1 << c["h"]) if not bin(m).count("1") & 1 and (ports[0] ^ m) not in used
+                and (ports[0] ^ m) > 0)
+    c2 = copy.deepcopy(c)
+    c2["ports"][0] ^= mask
+    try:
+        gx.check1(c2, scalar=True)
+        raise AssertionError("control failed: gx.check1 accepted a corrupted port")
+    except AssertionError as e:
+        if "control failed" in str(e):
+            raise
+        out["corrupted_port_check1_rejects"] = str(e).split(":")[0]
+    _, _, _, st = replay(c2, True)
+    out["corrupted_port_replay_violations"] = st["violations"]
+    assert st["violations"] > 0, "control failed: replay saw no violation for a corrupted port"
+    return out
+
+
+FILES = {"p3": "gcert1-p3-cone-R16.json", "p10": "gcert1-p10-anneal-mw-flow.json.gz", "p11": "gcert1-p11-pr193.json.gz"}
+
+
+def stage(name):
+    """One certificate: gx.check1, Thm A/B/C data, and (for the small one) the negative controls."""
+    res, c, taint = analyse(ROOT / "inputs" / FILES[name], gate_check=True)
+    if name == "p3":
+        res["controls"] = controls(c, taint)
+    return res
diff --git a/research/format-obstructions/checks/e8lib.py b/research/format-obstructions/checks/e8lib.py
new file mode 100644
index 0000000000..7f982baf0a
--- /dev/null
+++ b/research/format-obstructions/checks/e8lib.py
@@ -0,0 +1,50 @@
+"""The 120 E8 root lines in Construction-A (Hamming) coordinates, and the graphs on them.
+
+Excerpt of the local research file cxlib.py (DreamingOfClouds, with Anthropic Claude assistance). The coordinates
+are those of PR #370 (Dylan Crow): 8 coordinate lines 2e_i and 112 sign lines on the 14 weight-4 blocks of the
+extended Hamming code H8. With <a, b> = (a . b)/2 on the integer representatives:
+  * the 'in-cap' (lossless-helper) graph of the E8 complex word joins lines with <a, b> = +-1 (degree 56);
+  * the E8-orthogonal pairs (<a, b> = 0) have degree 63.
+"""
+import itertools
+
+V = 120
+
+
+def _ham():
+    g = [(1, 1, 1, 1, 0, 0, 0, 0), (1, 1, 0, 0, 1, 1, 0, 0), (1, 0, 1, 0, 1, 0, 1, 0), (0, 1, 1, 0, 1, 0, 0, 1)]
+    H = set()
+    for a in itertools.product((0, 1), repeat=4):
+        H.add(tuple(sum(a[k] * g[k][i] for k in range(4)) % 2 for i in range(8)))
+    return sorted(H)
+
+
+H8 = _ham()
+BLOCKS = sorted(tuple(i for i in range(8) if w[i]) for w in H8 if sum(w) == 4)
+assert len(BLOCKS) == 14
+LINES = []
+for i in range(8):
+    v = [0] * 8
+    v[i] = 2
+    LINES.append(tuple(v))
+for c in BLOCKS:
+    for sg in itertools.product((1, -1), repeat=3):
+        v = [0] * 8
+        v[c[0]] = 1
+        for k, s in zip(c[1:], sg):
+            v[k] = s
+        LINES.append(tuple(v))
+assert len(LINES) == V
+
+
+def ip(u, v):
+    return sum(a * b for a, b in zip(u, v)) // 2
+
+
+G = [[ip(LINES[a], LINES[b]) for b in range(V)] for a in range(V)]
+NB = [[b for b in range(V) if abs(G[a][b]) == 1] for a in range(V)]          # in-cap (non-orthogonal) neighbours
+OR = [[b for b in range(V) if a != b and G[a][b] == 0] for a in range(V)]     # E8-orthogonal pairs
+assert all(len(x) == 56 for x in NB) and all(len(x) == 63 for x in OR)
+assert all(G[a][a] == 2 for a in range(V))
+ORTH_PAIRS = [(a, b) for a in range(V) for b in range(a + 1, V) if G[a][b] == 0]
+assert len(ORTH_PAIRS) == V * 63 // 2
diff --git a/research/format-obstructions/checks/f2.py b/research/format-obstructions/checks/f2.py
new file mode 100644
index 0000000000..e0cd86171b
--- /dev/null
+++ b/research/format-obstructions/checks/f2.py
@@ -0,0 +1,39 @@
+"""Exact F_2 helpers (vectors are Python ints, the form is the standard dot product mod 2)."""
+
+
+def par(x):
+    return bin(x).count("1") & 1
+
+
+def dot(a, b):
+    return bin(a & b).count("1") & 1
+
+
+def span_basis(vecs):
+    """Echelon basis (list of ints, descending pivots) of the F_2 span."""
+    basis = []
+    for u in vecs:
+        for b in basis:
+            if u ^ b < u:
+                u ^= b
+        if u:
+            basis.append(u)
+            basis.sort(reverse=True)
+    return basis
+
+
+def rank(vecs):
+    return len(span_basis(vecs))
+
+
+def pairing_rank(A, B):
+    """Rank of the bilinear pairing span(A) x span(B) -> F_2 (dot product)."""
+    a, b = span_basis(A), span_basis(B)
+    rows = []
+    for x in a:
+        r = 0
+        for j, y in enumerate(b):
+            if dot(x, y):
+                r |= 1 << j
+        rows.append(r)
+    return rank(rows)
diff --git a/research/format-obstructions/checks/families.py b/research/format-obstructions/checks/families.py
new file mode 100644
index 0000000000..fae63dfcf0
--- /dev/null
+++ b/research/format-obstructions/checks/families.py
@@ -0,0 +1,153 @@
+"""Port-family data for Thm B (neighbourhood rank), Thm C (copy lower bound) and the clique bound.
+
+paired_bit(p)   paired cube, bit side: Q labels t_T (indicators of the 3-subset), form x^T (I - J/9) y, so
+                form(S, T) = |S cap T| - 1 and 'in-cap' = |S cap T| = 1.
+e8_bitdual()    120 E8 root lines in Q^8 with the E8 form; in-cap = orthogonal roots; F_2 decoder M = (J + Gram) mod 2
+                = U U^T for the 9-bit words U (isomorphism search), its copy lower bound sum_i rank_Q Gram[S_i, S_i].
+e8_complex(u)   the unique h = 9 complex labelling (from sat_e8.uniqueness_h9): neighbourhood ranks.
+incap_omega     clique numbers of the paired-cube in-cap graphs by exact max-clique search.
+p10_clique      the explicit 20-clique of the complex in-cap graph of the p = 10 program, checked inside the program's
+                own realised lossless graph.
+A neighbourhood N(S) of rank h - 1 equals q_S^perp, which is nondegenerate (its complement q_S is), so its form rank
+equals its dimension; we assert this wherever the table uses it.
+"""
+import itertools
+import sys
+import time
+from fractions import Fraction as Fr
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parent.parent
+sys.path.insert(0, str(ROOT / "checks"))
+from certs import load, replay  # noqa: E402
+from f2 import dot, span_basis  # noqa: E402
+
+
+def qrank(rows):
+    M = [[Fr(x) for x in r] for r in rows]
+    rk = 0
+    ncol = len(M[0]) if M else 0
+    for col in range(ncol):
+        piv = next((i for i in range(rk, len(M)) if M[i][col] != 0), None)
+        if piv is None:
+            continue
+        M[rk], M[piv] = M[piv], M[rk]
+        for i in range(len(M)):
+            if i != rk and M[i][col] != 0:
+                f = M[i][col] / M[rk][col]
+                M[i] = [a - f * b for a, b in zip(M[i], M[rk])]
+        rk += 1
+    return rk
+
+
+def paired_bit(p):
+    P = [frozenset(2 * i + b for i, b in zip(I, bb)) for I in itertools.combinations(range(p), 3)
+         for bb in itertools.product((0, 1), repeat=3)]
+    h = 2 * p
+    T = P[0]
+    nb = [S for S in P if len(S & T) == 1]
+    form = lambda S, U: Fr(len(S & U)) - Fr(len(S) * len(U), 9)   # noqa: E731
+    assert all(form(S, T) == 0 for S in nb) and form(T, T) == 2
+    r = qrank([[1 if i in S else 0 for i in range(h)] for S in nb])
+    assert r == h - 1                       # N = U^perp, nondegenerate
+    return dict(p=p, h=h, v=len(P), incap_deg=len(nb), N_rank=r)
+
+
+def _e8_roots():
+    Rt = []
+    for i, j in itertools.combinations(range(8), 2):
+        for si in (2, -2):
+            for sj in (2, -2):
+                r = [0] * 8
+                r[i] = si
+                r[j] = sj
+                Rt.append(tuple(r))
+    for s in itertools.product((1, -1), repeat=8):
+        if s.count(-1) % 2 == 0:
+            Rt.append(s)
+    lines, seen = [], set()
+    for r in Rt:
+        if tuple(-x for x in r) in seen:
+            continue
+        seen.add(r)
+        lines.append(r)
+    return lines
+
+
+def e8_bitdual():
+    import networkx as nx
+    lines = _e8_roots()
+    v = len(lines)
+    assert v == 120
+    G = [[sum(a * b for a, b in zip(x, y)) // 4 for y in lines] for x in lines]
+    assert all(G[i][i] == 2 for i in range(v))
+    nb_ranks = set()
+    for a in range(v):
+        nb = [lines[b] for b in range(v) if b != a and G[a][b] == 0]
+        nb_ranks.add((len(nb), qrank(nb)))
+    off = [[(G[i][j] % 2 == 1) for j in range(v)] for i in range(v)]
+    lab9 = [frozenset(t) for t in itertools.combinations(range(9), 3)] + \
+           [frozenset(range(9)) - frozenset(p) for p in itertools.combinations(range(9), 2)]
+    G9 = nx.Graph()
+    G9.add_edges_from((i, j) for i in range(v) for j in range(i + 1, v) if len(lab9[i] & lab9[j]) % 2 == 0)
+    Goff = nx.Graph()
+    Goff.add_edges_from((i, j) for i in range(v) for j in range(i + 1, v) if off[i][j])
+    iso = next(nx.algorithms.isomorphism.GraphMatcher(Goff, G9).isomorphisms_iter())
+    U = [[1 if i in lab9[iso[s]] else 0 for i in range(9)] for s in range(v)]
+    ok = all((sum(U[i][t] * U[j][t] for t in range(9)) % 2) == ((1 + G[i][j]) % 2) for i in range(v) for j in range(v))
+    assert ok
+    lb, betas = 0, []
+    for t in range(9):
+        S = [s for s in range(v) if U[s][t]]
+        b = qrank([[G[a][c] for c in S] for a in S])      # pairing rank span(src) x span(readers), src = readers = S
+        betas.append(b)
+        lb += b
+    assert sorted(nb_ranks) == [(63, 7)]
+    return dict(v=v, h=8, neighbourhood=sorted(nb_ranks), decoder_equals_UUt=ok, totals=9, betas=betas,
+                copy_lb=lb)
+
+
+def e8_complex(u):
+    from e8lib import NB, V
+    ranks = set()
+    for S in range(V):
+        N = span_basis([u[T] for T in NB[S]])
+        assert all(dot(x, u[S]) == 0 for x in N)
+        ranks.add(len(N))
+    assert ranks == {8}
+    return dict(v=V, h=9, incap_deg=56, N_rank=8)
+
+
+def incap_omega(p, kind):
+    import networkx as nx
+    P = [frozenset(2 * i + b for i, b in zip(I, bb)) for I in itertools.combinations(range(p), 3)
+         for bb in itertools.product((0, 1), repeat=3)]
+    pred = (lambda j: j == 1) if kind == "bit" else (lambda j: j in (0, 2))
+    G = nx.Graph()
+    G.add_nodes_from(range(len(P)))
+    G.add_edges_from((a, b) for a in range(len(P)) for b in range(a + 1, len(P)) if pred(len(P[a] & P[b])))
+    t0 = time.monotonic()
+    cl, w = nx.max_weight_clique(G, weight=None)
+    return dict(p=p, h=2 * p, kind=kind, omega=w, seconds=round(time.monotonic() - t0, 1))
+
+
+def p10_clique(path):
+    """5 pairwise-disjoint transversal 4-sets from the K5 stars, all 4 triples each: 20 ports, pairwise even overlap."""
+    c = load(path)
+    h, v = c["h"], c["v"]
+    assert h == 20
+    ports = c["ports"]
+    index = {q: i for i, q in enumerate(ports)}
+    E = list(itertools.combinations(range(5), 2))
+    assert len(E) == 10
+    sets = [[2 * i + (0 if e[0] == x else 1) for i, e in enumerate(E) if x in e] for x in range(5)]
+    tri = [sum(1 << c_ for c_ in t) for s in sets for t in itertools.combinations(s, 3)]
+    assert len(tri) == 20 and len(set(tri)) == 20
+    T = [index[q] for q in tri]                                  # all 20 are ports of the program
+    assert all(dot(a, a) == 1 for a in tri)
+    assert all(dot(a, b) == 0 for a, b in itertools.combinations(tri, 2))
+    taint, _, _, _ = replay(c, gate_check=False)
+    realised = all(taint[v + S] >> Tt & 1 for S in T for Tt in T if S != Tt)
+    assert realised
+    return dict(size=len(T), orthonormal=True, all_pairs_lossless_in_program=realised, window_dim=h,
+                label_dim_sum_le=h)
diff --git a/research/format-obstructions/checks/mutable425.py b/research/format-obstructions/checks/mutable425.py
new file mode 100644
index 0000000000..5ac0f815e9
--- /dev/null
+++ b/research/format-obstructions/checks/mutable425.py
@@ -0,0 +1,360 @@
+"""PR #425's mutable-basis word (jacklightChen / Zhihao Chen): Thms A, B, C and Theorem S, checked exactly.
+
+Input: inputs/pr425/FULL-WORD.json.gz = #425's FULL-WORD.json.gz (920 registers: 120 sources, 120 targets, 680 helper
+registers), regenerated by #425's own hamming_packets.py + hamming_full.py; sha256 equals #425's EXPECTED.json.
+Events: ['add', t, s, num, den, frame] (t += (num/den) s with both at 'frame'; frame None marks the centre copy that
+drops full -> 0) and ['scale', t, num, den, frame].  Phases: A, cut (8 centre copies), B, cleanup (at the full frame).
+
+(1) frame + taint replay on the word as written. Thm A (taint of each gate register inside the gate frame), Cor A1
+    (target taint orthogonal to its port, target frames inside the hyperplane), no target feeds a z-register
+    (z = sources + helpers), nested climbs, every gate at one common frame. The same detector, run in 'immutable'
+    mode, counts the two things the immutable-source / no-target-read format forbids.
+    Thm C on the 8 centre copies (pairing ranks vs frames paid). Thm B data (neighbourhood rank of every target) and
+    the common cut (all targets at frame 0 at the scatter).
+(2) exact scalar replay over Q: z restored, J = I, V and K; with the frame-0 preconditioner y <- V^-1 (y - K s) the
+    word is exactly (x, s, y) -> (x, s, y + x). Control: omitting the K part is rejected.
+(3) Theorem S: the conversion to the immutable-source / no-target-read format with one helper per modified source and
+    one per read target. Replayed with the same detector (zero violations) and exactly over Q (after the frame-0
+    compensation of the response to the dirty registers). Controls: the same detector flags the unconverted word;
+    without the compensation the converted word is not (x, s, y) -> (x, s, y + x).
+(4) coarse price (the moment of #425's hamming_full.py) of the word and of the converted word.
+"""
+import copy
+import gzip
+import json
+import sys
+from collections import Counter, defaultdict
+from fractions import Fraction as Q
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parent.parent
+sys.path.insert(0, str(ROOT / "checks"))
+from f2 import pairing_rank, par  # noqa: E402
+
+V = 120
+
+
+def basis(vectors):
+    """Reduced echelon basis (tuple, descending pivots) of the F_2 span of ints."""
+    b = {}
+    for x in vectors:
+        for p in sorted(b, reverse=True):
+            if x >> p & 1:
+                x ^= b[p]
+        if x:
+            p = x.bit_length() - 1
+            for q in b:
+                if b[q] >> p & 1:
+                    b[q] ^= x
+            b[p] = x
+    return tuple(b[p] for p in sorted(b, reverse=True))
+
+
+def inside(a, b):
+    return basis(list(a) + list(b)) == tuple(b) if b else not a
+
+
+def perp(p, F):
+    return all(par(p & x) == 0 for x in F)
+
+
+def hyper(p):
+    return basis([x for x in range(1, 512) if par(x & p) == 0])
+
+
+FULL = basis([1 << i for i in range(9)])
+
+
+def is_tgt(r):
+    return V <= r < 2 * V
+
+
+def frame_replay(phases, n, P, immutable, final_frames=True, collect=None):
+    """phases: [(name, events)]. Returns (cur, taint, kids, viol, stats)."""
+    cur = [((P[r],) if r < V else ()) for r in range(n)]
+    taint = [((P[r],) if r < V else ()) for r in range(n)]
+    kids = defaultdict(list)
+    viol, stats = Counter(), Counter()
+
+    def climb(r, F):
+        if not inside(cur[r], F):
+            viol["non-nested"] += 1
+        dr = len(F) - len(cur[r])
+        if dr:
+            kids[r].append(dr)
+        cur[r] = F
+
+    for ph, evs in phases:
+        if ph == "cut" and collect is not None:
+            collect["frames_at_cut"] = {r: cur[r] for r in range(n)}
+        for e in evs:
+            if e[0] == "scale":
+                t = e[1]
+                F = tuple(e[4]) if e[4] is not None else cur[t]
+                climb(t, F)
+                stats["scale"] += 1
+                if t < V:
+                    stats["src_modified"] += 1
+                    if immutable:
+                        viol["source modified"] += 1
+                continue
+            _, t, s = e[:3]
+            F = e[5]
+            if F is None:                       # centre copy: drop full -> 0, the target must stand at frame 0
+                if cur[t] != ():
+                    viol["cut-target-not-at-0"] += 1
+                stats["cut"] += 1
+                continue
+            F = tuple(F)
+            climb(t, F)
+            climb(s, F)
+            if is_tgt(s) and not is_tgt(t):
+                viol["target-feeds-z"] += 1
+            if is_tgt(s):
+                stats["target_read"] += 1
+                if immutable:
+                    viol["target read"] += 1
+            if t < V:
+                stats["src_modified"] += 1
+                if immutable:
+                    viol["source modified"] += 1
+            taint[t] = basis(list(taint[t]) + list(taint[s]))
+            for r in (t, s):
+                if not inside(taint[r], cur[r]):
+                    viol["ThmA taint not in frame"] += 1
+                if is_tgt(r):
+                    if not perp(P[r - V], cur[r]):
+                        viol["target frame not in hyperplane"] += 1
+                    if not perp(P[r - V], taint[r]):
+                        viol["CorA1 target taint not perp"] += 1
+        if ph == "B" and final_frames:
+            for r in range(n):
+                climb(r, hyper(P[r - V]) if is_tgt(r) else FULL)
+    return cur, taint, kids, viol, stats
+
+
+def scalar_replay(evs, n, rows=None):
+    rows = rows if rows is not None else [{r: Q(1)} for r in range(n)]
+    for e in evs:
+        if e[0] == "scale":
+            q = Q(e[2], e[3])
+            rows[e[1]] = {j: q * x for j, x in rows[e[1]].items()}
+            continue
+        t, s, q = e[1], e[2], Q(e[3], e[4])
+        for j, x in list(rows[s].items()):
+            z = rows[t].get(j, Q(0)) + q * x
+            if z:
+                rows[t][j] = z
+            else:
+                rows[t].pop(j, None)
+    return rows
+
+
+def inv_mat(M):
+    idx = sorted(M)
+    k = len(idx)
+    pos = {r: i for i, r in enumerate(idx)}
+    a = [[M[r].get(c, Q(0)) for c in idx] + [Q(int(i == j)) for j in range(k)] for i, r in enumerate(idx)]
+    for col in range(k):
+        piv = next(i for i in range(col, k) if a[i][col])
+        a[col], a[piv] = a[piv], a[col]
+        q = a[col][col]
+        a[col] = [x / q for x in a[col]]
+        for i in range(k):
+            if i != col and a[i][col]:
+                f = a[i][col]
+                a[i] = [x - f * y for x, y in zip(a[i], a[col])]
+    return {r: {c: a[pos[r]][k + pos[c]] for c in idx if a[pos[r]][k + pos[c]]} for r in idx}
+
+
+def price(local, Rc, m=73, T=8, L=9):
+    """Coarse complex price of hamming_full.py: root of the moment sum (floats, 80 bisections)."""
+    W = 7680 + L * Rc
+    g = Counter({r: z * L * T for r, z in local.items()})
+    for r in (16, 8, 32, 8, 48, 8, 48, 32, 16):
+        g[r] += 240 * T
+    g[1] += L * Rc
+    N = sum(g.values())
+    lo, hi = 0., .003
+    for _ in range(80):
+        q = (lo + hi) / 2
+        mom = sum(r * z / (m * W) * (m / r) ** q for r, z in g.items()) + 32 * m * N / (10 ** 16 * W) * m ** q
+        if mom < 1:
+            lo = q
+        else:
+            hi = q
+    return (lo + hi) / 2
+
+
+def local_hist(kids):
+    H = Counter()
+    for r, ks in kids.items():
+        for d in ks:
+            H[d] += 1
+    H[9] += 8      # centre copies
+    return H
+
+
+def run():
+    WD = json.load(gzip.open(ROOT / "inputs" / "pr425" / "FULL-WORD.json.gz"))
+    P, Rr = WD["ports"], WD["root_lines"]
+    assert len(P) == V and all(par(q) for q in P)
+    A, B, CUT, CLEAN = WD["A"], WD["B"], WD["cut"], WD["cleanup"]
+    n = 1 + max(max(e[1], e[2] if e[0] == "add" else 0) for e in A + B + CUT + CLEAN)
+    R0 = n - 2 * V
+    centers = WD["centers"]
+    out = dict(registers=n, helpers=R0, m=WD["report"]["m"], h=WD["report"]["h"], v=WD["report"]["v"])
+    assert (n, R0, out["m"], out["h"]) == (920, 680, 73, 9)
+    # the ports are an admissible E8 labelling in the sense of sat_e8 (u.u = 1, orthogonal on the 3,360 non-orthogonal
+    # root pairs, Gram 1 on the 3,780 orthogonal pairs), with the word's own root_lines: by the uniqueness result
+    # of sat_e8.uniqueness_h9 it is Sussman's labelling up to isometry
+    ip = lambda a, b: sum(int(x) * int(y) for x, y in zip(a, b)) // 2     # noqa: E731
+    assert all(sum(int(x) * int(x) for x in Rr[s]) == 4 for s in range(V))
+    gram_ok = all(bin(P[s] & P[t]).count("1") % 2 == (1 if s == t else (1 + ip(Rr[s], Rr[t])) % 2)
+                  for s in range(V) for t in range(V))
+    assert gram_ok
+    out["ports_realise_E8_gram"] = gram_ok
+
+    # ---- (1) frame + taint replay, Thm A, Cor A1, Thm B data, Thm C on the centres
+    cleanFULL = [[e[0], *e[1:-1], list(FULL)] for e in CLEAN]
+    phases = [("A", A), ("cut", CUT), ("B", B), ("clean", cleanFULL)]
+    coll = {}
+    cur, taint, kids, viol, stats = frame_replay(phases, n, P, immutable=False, collect=coll)
+    assert not viol, dict(viol)
+    _, _, _, viol_imm, stats_imm = frame_replay(phases, n, P, immutable=True)
+    out["detector_on_word_as_written"] = dict(sorted(viol_imm.items()))
+    assert viol_imm == Counter({"source modified": 1640, "target read": 112}), dict(viol_imm)
+    modsrc = sorted({e[1] for e in A + B if e[1] < V})
+    readtg = sorted({e[2] for e in A + B if e[0] == "add" and is_tgt(e[2])})
+    out.update(thmA_violations=0, modified_sources=len(modsrc), read_targets=len(readtg),
+               stats={k: stats[k] for k in ("scale", "cut", "src_modified", "target_read")})
+    assert (len(modsrc), len(readtg)) == (112, 112)
+    # common cut (R2 of Thm B): every target stands at frame 0 at the scatter
+    at_cut = coll["frames_at_cut"]
+    out["all_targets_at_frame0_at_cut"] = all(at_cut[V + t] == () for t in range(V))
+    assert out["all_targets_at_frame0_at_cut"]
+    # Thm B data: lossless set of each target = final taint; its rank (= h-1 means N = q^perp)
+    nr = sorted({len(taint[V + t]) for t in range(V)})
+    out["target_neighbourhood_ranks"] = nr
+    assert nr == [8] and all(perp(P[t], taint[V + t]) for t in range(V))
+    assert all(len(basis(list(taint[V + t]) + [P[t]])) == 9 for t in range(V))     # q_t is not in its own lossless span
+    out["m_min_nine_stages"] = 1 + 9 * max(nr)
+    assert out["m_min_nine_stages"] == out["m"] == 73
+    # Thm C on the eight centre copies: pairing rank of (sources gathered drop-free into the centre before the cut)
+    # x (targets reading it) against the frames paid. The gathered sources come from a bit-set taint replay of phase A
+    # and must equal the support of the centre's root-line coordinate.
+    tm = [(1 << r) if r < V else 0 for r in range(n)]
+    for e in A:
+        if e[0] == "add":
+            tm[e[1]] |= tm[e[2]]
+    cst_bound = cst_paid = 0
+    gathered = []
+    for i, c in enumerate(centers):
+        srcs = [s for s in range(V) if tm[c] >> s & 1]
+        assert srcs == [s for s in range(V) if int(Rr[s][i]) != 0], "centre taint differs from its root-line support"
+        assert basis([P[s] for s in srcs]) == coll["frames_at_cut"][c], "sources do not span the centre's frame"
+        readers = sorted({e[1] - V for e in CUT if e[2] == c})
+        cst_bound += pairing_rank([P[s] for s in srcs], [P[t] for t in readers])
+        cst_paid += len(coll["frames_at_cut"][c])
+        gathered.append(len(srcs))
+    out["sources_gathered_per_centre"] = gathered
+    out.update(cst_bound=cst_bound, cst_paid=cst_paid)
+    assert cst_bound == cst_paid == WD["report"]["cst"] == 72
+
+    # ---- (2) exact scalar replay
+    rows = scalar_replay(A + CUT + B + CLEAN, n)
+    zok = all(rows[r] == {r: Q(1)} for r in range(n) if not is_tgt(r))
+    J_ok = all({j: x for j, x in rows[V + t].items() if j < V} == {t: Q(1)} for t in range(V))
+    Vnon = sum(1 for t in range(V) for j, x in rows[V + t].items() if is_tgt(j) and j != V + t)
+    Knz = sum(1 for t in range(V) for j in rows[V + t] if j >= 2 * V)
+    Vm = {V + t: {j: x for j, x in rows[V + t].items() if is_tgt(j)} for t in range(V)}
+    Vi = inv_mat(Vm)
+    pre_rows = []
+    for t in range(V):
+        row = dict(Vi[V + t])
+        for u, a in Vi[V + t].items():
+            for j, x in rows[u].items():
+                if j >= 2 * V:
+                    row[j] = row.get(j, Q(0)) - a * x
+        pre_rows.append({j: x for j, x in row.items() if x})
+    init = [{r: Q(1)} for r in range(n)]
+    for t in range(V):
+        init[V + t] = pre_rows[t]
+    rows2 = scalar_replay(A + CUT + B + CLEAN, n, init)
+    ok2 = all(rows2[r] == {r: Q(1)} for r in range(n) if not is_tgt(r)) and \
+        all(rows2[V + t] == {V + t: Q(1), t: Q(1)} for t in range(V))
+    init3 = [{r: Q(1)} for r in range(n)]
+    for t in range(V):
+        init3[V + t] = dict(Vi[V + t])
+    rows3 = scalar_replay(A + CUT + B + CLEAN, n, init3)
+    neg = any(rows3[V + t] != {V + t: Q(1), t: Q(1)} for t in range(V))
+    out.update(z_restored=zok, J_is_identity=J_ok, V_offdiag=Vnon, K_nonzeros=Knz,
+               preconditioned_map_exact=ok2, omit_K_control_rejected=neg)
+    assert zok and J_ok and ok2 and neg
+
+    # ---- (3) Theorem S
+    sim, nn = {}, n
+    for r in modsrc + readtg:
+        sim[r] = nn
+        nn += 1
+
+    def ren(e):
+        e = list(e)
+        e[1] = sim.get(e[1], e[1])
+        if e[0] == "add":
+            e[2] = sim.get(e[2], e[2])
+        return e
+
+    pre = [["add", sim[r], r, 1, 1, [P[r]]] for r in modsrc]
+    A1, C1, B1 = [ren(e) for e in A], [ren(e) for e in CUT], [ren(e) for e in B]
+    final = [["add", t, sim[t], 1, 1, list(hyper(P[t - V]))] for t in readtg]
+    fwd_z = [e for e in pre + A1 + C1 + B1 if not is_tgt(e[1])][len(pre):]
+
+    def inv(e):
+        e = copy.deepcopy(e)
+        if e[0] == "add":
+            e[3] = -e[3]
+            e[5] = list(FULL)
+        else:
+            e[2], e[3] = e[3], e[2]
+            e[4] = list(FULL)
+        return e
+
+    clean1 = [inv(e) for e in reversed(fwd_z)] + [["add", sim[r], r, -1, 1, list(FULL)] for r in modsrc]
+    phasesS = [("A", pre + A1), ("cut", C1), ("B", B1 + final), ("clean", clean1)]
+    # the same detector on the converted word; the converted targets g_t must stay inside their hyperplane
+    # (the detector checks real targets; g_t is a helper, checked separately below)
+    # g_t follows y_t's frames; the final gate y_t += g_t is at hyper(P_t), so the detector's nestedness test also
+    # checks that g_t stands inside the hyperplane at that time
+    curS, taintS, kidsS, violS, statsS = frame_replay(phasesS, nn, P, immutable=True, final_frames=True)
+    out["detector_on_converted_word"] = dict(sorted(violS.items()))
+    assert not violS, dict(violS)
+    rowsS = scalar_replay(pre + A1 + C1 + B1 + final + clean1, nn)
+    zS = all(rowsS[r] == {r: Q(1)} for r in range(nn) if not is_tgt(r))
+    KS = sum(1 for t in range(V) for j in rowsS[V + t] if j >= 2 * V)
+    initS = [{r: Q(1)} for r in range(nn)]
+    for t in range(V):
+        initS[V + t] = {V + t: Q(1)}
+        for j, x in rowsS[V + t].items():
+            if j >= 2 * V:
+                initS[V + t][j] = -x
+    rowsS2 = scalar_replay(pre + A1 + C1 + B1 + final + clean1, nn, initS)
+    okS = all(rowsS2[r] == {r: Q(1)} for r in range(nn) if not is_tgt(r)) and \
+        all(rowsS2[V + t] == {V + t: Q(1), t: Q(1)} for t in range(V))
+    negS = any(rowsS[V + t] != {V + t: Q(1), t: Q(1)} for t in range(V))      # without the frame-0 compensation
+    RS = nn - 2 * V
+    out.update(S_helpers=RS, S_helpers_formula="R + b_x + b_y", S_z_restored=zS, S_K_nonzeros=KS,
+               S_exact_after_compensation=okS, S_uncompensated_rejected=negS)
+    assert RS == R0 + len(modsrc) + len(readtg) == 904 and zS and okS and negS
+
+    # ---- (4) coarse price
+    H0 = local_hist(kids)
+    b0 = price(H0, R0)
+    HS = local_hist(kidsS)
+    bS = price(HS, RS)
+    rep = WD["report"]
+    assert abs(b0 - rep["complex_coarse_float"]) < 1e-15
+    out.update(local_hist={str(k): v for k, v in sorted(H0.items())}, mass=sum(r * z for r, z in H0.items()),
+               b=b0, b_pre425=rep["baseline_b"], S_local_hist={str(k): v for k, v in sorted(HS.items())},
+               S_mass=sum(r * z for r, z in HS.items()), S_b=bS, S_b_vs_b_percent=round(100 * (bS / b0 - 1), 2))
+    return out
diff --git a/research/format-obstructions/checks/pivots.py b/research/format-obstructions/checks/pivots.py
new file mode 100644
index 0000000000..24c3a87bbb
--- /dev/null
+++ b/research/format-obstructions/checks/pivots.py
@@ -0,0 +1,146 @@
+"""Section 7 of PROOF.md: the 2K block pivots of a shared-helper B_K layout (F_2 windows, gcert/1 programs).
+
+thm_P (exact).  A helper set that runs the SAME gcert/1 program on the SAME slots in every stage: a slot never drops,
+so after stage j its frame contains iota_j(q_T) for every source T it touched (Thm A); when it writes y_S in stage
+j' > j its frame lies inside iota_j'(q_S)^perp. With K = iota_j^T iota_j' (h x h over F_2) this gives
+q_T^T K q_S = 0 for every slot-carried pair (T in taint(z), S in writes(z)). The condition is linear in K; we compute
+the rank of {q_T (x) q_S}. Rank h^2 means K = 0, i.e. the stage windows are pairwise orthogonal and m >= s*h.
+
+copy_count (exact, over ALL odd a, b in F_2^h).  Model of ONE overlap w between the helper windows of two stages,
+w = iota_{j-1}(e) = iota_j(e') with e.e = e'.e' = 1: a retained total whose frame is not inside e^perp carries w into
+stage j; in stage j such a total pays one extra copy rank unless e' lies in its frame. Hence the forced extra copy
+rank per overlap is >= (#totals) - (max #totals inside some e^perp) - (max #frames containing some e').
+
+overlap_scan (exact on the candidates).  The same model on the helper side: slots whose last frame is not inside e^perp
+carry w; their forward closure through the gates contaminates registers, and a contaminated target y_S needs
+e'.q_S = 0. Candidates: the h unit vectors and the 1,140 weight-3 vectors (h = 20), not all directions.
+"""
+import sys
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parent.parent
+sys.path.insert(0, str(ROOT / "gx"))
+sys.path.insert(0, str(ROOT / "checks"))
+import gx  # noqa: E402
+from certs import load  # noqa: E402
+from f2 import dot, span_basis  # noqa: E402
+
+
+def _taint_and_writes(c):
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    taint = [(1 << r) if r < v else 0 for r in range(nreg)]
+    writes = {}
+    for g in c["A"] + c["B"]:
+        regs, pairs, _ = gx.gate_regs(g)
+        for t, s, co in pairs:
+            taint[t] |= taint[s]
+            if 2 * v <= s and v <= t < 2 * v:
+                writes.setdefault(s, set()).add(t - v)
+    return taint, writes
+
+
+def thm_P(path):
+    c = load(path)
+    gx.check1(c, scalar=False)
+    h, v, R = c["h"], c["v"], c["R"]
+    ports = c["ports"]
+    taint, writes = _taint_and_writes(c)
+
+    def kron(a, b):
+        out = 0
+        for i in range(h):
+            if a >> i & 1:
+                out |= b << (h * i)
+        return out
+
+    def rank_of(only_orth, cap):
+        basis, used = [], 0
+        for z in sorted(writes):
+            Ts = [T for T in range(v) if taint[z] >> T & 1]
+            for S in writes[z]:
+                for T in Ts:
+                    if only_orth and dot(ports[T], ports[S]):
+                        continue
+                    vec = kron(ports[T], ports[S])
+                    used += 1
+                    for b_ in basis:
+                        if vec ^ b_ < vec:
+                            vec ^= b_
+                    if vec:
+                        basis.append(vec)
+                        basis.sort(reverse=True)
+                        if len(basis) == cap:
+                            return len(basis), used
+        return len(basis), used
+
+    # (a) pairs with q_T . q_S = 0 only: their tensors lie in the trace hyperplane (tr M = sum_i M_ii = q_T . q_S)
+    r_orth, used_orth = rank_of(True, h * h)
+    # (b) all slot-carried pairs
+    r_all, used_all = rank_of(False, h * h)
+    assert r_orth == h * h - 1 and r_all == h * h, (r_orth, r_all)       # K = lambda * I from (a); K = 0 from (b)
+    # cross-cube slots (descriptive; paired-cube ports only)
+    cube = [frozenset(i // 2 for i in range(h) if q >> i & 1) for q in ports]
+    written = sorted({S for z in writes for S in writes[z]})
+    written_rank = len(span_basis([ports[S] for S in written]))
+    assert written_rank == h                 # the slot-written targets' ports span the window
+    cross = sum(1 for z in range(2 * v, 2 * v + R) if len({cube[T] for T in range(v) if taint[z] >> T & 1}) > 1)
+    return dict(file=Path(str(path)).name, h=h, slots=R, writer_slots=len(writes),
+                rank_orthogonal_pairs_only=r_orth, rank_all_carried_pairs=r_all, h_squared=h * h,
+                K_solution_dim=h * h - r_all, slot_written_targets_port_rank=written_rank, K_zero_gives_m_ge="s*h, under (P1)-(P2)", cross_cube_slots=cross)
+
+
+def copy_count(path):
+    c = load(path)
+    h = c["h"]
+    fr = [tuple(f) for f in c["frames"]]
+    F = [fr[f] for k, s, f in sorted(c["ret"])]
+    best_nc = best_in = 0
+    for a in range(1, 1 << h):
+        if not (bin(a).count("1") & 1):
+            continue
+        nc = sum(1 for f in F if all(not dot(b, a) for b in f))
+        ins = sum(1 for f in F if gx.reduce(a, f) == 0)
+        best_nc = max(best_nc, nc)
+        best_in = max(best_in, ins)
+    return dict(totals=len(F), frame_dims=sorted({len(f) for f in F}), max_noncarrier_totals=best_nc,
+                max_totals_containing_b=best_in, forced_extra_copy_per_overlap=len(F) - best_nc - best_in)
+
+
+def overlap_scan(path):
+    c = load(path)
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    ports = c["ports"]
+    fr = [tuple(f) for f in c["frames"]]
+    gates = [("A", g) for g in c["A"]] + [("B", g) for g in c["B"]]
+    regsets = [gx.gate_regs(g)[0] for _, g in gates]
+    last = {}
+    for i, (_, g) in enumerate(gates):
+        for r in regsets[i]:
+            if r >= 2 * v:
+                last[r] = g[1]
+    slots = sorted(last)
+
+    def carriers(e):
+        return {z for z in slots if any(dot(b, e) for b in fr[last[z]])}
+
+    def closure(C):
+        bad = set(C)
+        for i, (_, g) in enumerate(gates):
+            rs = regsets[i]
+            if any(r in bad for r in rs):
+                bad.update(rs)
+        return bad
+
+    odd = [e for e in range(1, 1 << h) if bin(e).count("1") & 1]
+    cand = [1 << i for i in range(h)] + [e for e in odd if bin(e).count("1") == 3][:2000]
+    recs = []
+    for e in cand:
+        C = carriers(e)
+        bad = closure(C)
+        ys = [r - v for r in bad if v <= r < 2 * v]
+        recs.append((len(C), len(bad), len(ys), len(span_basis([ports[S] for S in ys]))))
+    return dict(h=h, slots=R, slots_with_gates=len(slots), registers=nreg, candidates=len(cand),
+                min_carriers=min(r[0] for r in recs), min_contaminated_registers=min(r[1] for r in recs),
+                min_contaminated_targets=min(r[2] for r in recs), min_port_rank_of_contaminated=min(r[3] for r in recs))
diff --git a/research/format-obstructions/checks/pricing.py b/research/format-obstructions/checks/pricing.py
new file mode 100644
index 0000000000..84b6a36899
--- /dev/null
+++ b/research/format-obstructions/checks/pricing.py
@@ -0,0 +1,97 @@
+"""Pricing context for the pivots (PROOF.md section 7) and for non-nested chains (section 8), with the pinned pricer.
+
+Pricer: inputs/pricing/cost.py (solve_moment), PR #346 (Louis Harrison) from PR #310 (Dylan Crow, utcorvusvolat-dotcom),
+vendored unchanged. Input: the normalised five-stage histogram of PR #325's p = 10 bit word (hist_p10b.json: m = 100,
+W = 131,760 = 12 * 10,980, deficit D = 24,480; every count is 12 x a per-invocation count) and its pinned bit root
+385138454781521/(5 * 10^17).
+
+ * baseline: the pricer reproduces the pinned root.
+ * pivot-free variants (m = 96 = 1 + 5 (h - 1)): each helper wire loses 4 levels (4 blocks of rank r -> r - 1), the
+   pair-1 end idle block goes 2h+2 -> 2h-2 and the pair-2 end idle block of rank 4 disappears. Four ways of choosing
+   which helper blocks shrink bracket the gain. 'forced copies': 40 extra copy rank per unit (4 overlaps x 10, see
+   pivots.copy_count), modelled as 40 x 12 blocks of rank 18 becoming rank 19.
+ * non-nested chain (section 8): one rank-1 copy in each of the five stages, per 12-normalised unit, is +60 blocks of rank 1 (5 stages
+   x 12); a rank-1 register descent is a single non-nested step of frame-lemma rank d = 2, +60 blocks of rank 2.
+These are pricer-model outputs for the p = 10 bit word, not theorems.
+"""
+import collections
+import importlib.util
+import json
+import sys
+from pathlib import Path
+
+from mpmath import mp, mpf
+
+ROOT = Path(__file__).resolve().parent.parent
+
+
+def run():
+    spec = importlib.util.spec_from_file_location("cost", ROOT / "inputs" / "pricing" / "cost.py")
+    C = importlib.util.module_from_spec(spec)
+    sys.modules.setdefault("cost", C)
+    spec.loader.exec_module(C)
+    mp.dps = 40
+    d = json.load(open(ROOT / "inputs" / "pricing" / "hist_p10b.json"))
+    H0 = collections.Counter({int(k): v for k, v in d["histogram"].items()})
+    W = mpf(d["W"])
+    m0 = d["m"]
+    v12 = 960 * 12
+    Rs = d["W"] - 4 * v12
+    mass0 = sum(r * n for r, n in H0.items())
+    D0 = d["W"] * m0 - mass0
+    assert (m0, d["W"], D0, d["deficit"]) == (100, 131760, 24480, 24480)
+    a0 = C.solve_moment(H0, W, m0)
+    num, den = d["bit_root"].split("/")
+    pinned = mpf(int(num)) / mpf(int(den))
+    # the pinned value is the certified grid point (step 1/den = 2e-18) at or just below the root
+    assert 0 <= a0 - pinned < mpf(1) / int(den), (a0, pinned)
+    out = dict(m=m0, W=d["W"], D=D0, root=format(float(a0), ".12e"), pinned_root=format(float(pinned), ".12e"))
+
+    def variant(shrink, extra_copy=0):
+        H = collections.Counter(H0)
+        H[42] -= 2 * v12
+        H[38] += 2 * v12
+        H[4] -= 2 * v12
+        for r, cnt in shrink:
+            H[r] -= cnt
+            if r > 1:
+                H[r - 1] += cnt
+        assert all(c >= 0 for c in H.values())
+        if extra_copy:
+            H[18] -= extra_copy
+            H[19] += extra_copy
+        m = 96
+        return W * m - sum(r * n for r, n in H.items()), C.solve_moment(H, W, m)
+
+    def spread(ranks):
+        need = 4 * Rs
+        res = []
+        for r in ranks:
+            take = min(need, H0[r] - (2 * v12 if r in (4,) else 0))
+            res.append((r, take))
+            need -= take
+            if not need:
+                break
+        assert need == 0
+        return res
+
+    rows = {}
+    for name, shrink in (("rank-1 blocks removed", [(1, 4 * Rs)]), ("blocks 2->1", [(2, 4 * Rs)]),
+                         ("mid blocks shortened", spread([3, 4, 5, 6, 7, 8, 9, 10, 12, 14])),
+                         ("largest blocks shortened", spread([19, 18, 17, 16, 14, 12, 11, 10]))):
+        D, a = variant(shrink)
+        D2, a2 = variant(shrink, extra_copy=40 * 12)
+        assert D == D0
+        rows[name] = dict(zero_cost_percent=format(float((a / a0 - 1) * 100), "+.2f"),
+                          with_40_forced_copy_percent=format(float((a2 / a0 - 1) * 100), "+.2f"))
+    out["pivot_free_m96"] = rows
+
+    def delta(H):
+        return format(float((C.solve_moment(H, W, m0) / a0 - 1) * 100), "+.4f")
+    Hc = collections.Counter(H0)
+    Hc[1] += 60
+    Hd = collections.Counter(H0)
+    Hd[2] += 60
+    out["per_stage_rank1_copy_percent"] = delta(Hc)
+    out["per_stage_rank1_register_descent_percent"] = delta(Hd)
+    return out
diff --git a/research/format-obstructions/checks/sat_e8.py b/research/format-obstructions/checks/sat_e8.py
new file mode 100644
index 0000000000..082d1cbba0
--- /dev/null
+++ b/research/format-obstructions/checks/sat_e8.py
@@ -0,0 +1,476 @@
+"""Exact SAT checks on the E8 in-cap graph (CaDiCaL through python-sat).
+
+Model. An 'in-cap labelling' of a graph H on the 120 E8 lines assigns to each line T a nondegenerate k-dimensional
+subspace A_T of (F_2^r, standard dot product), with A_S orthogonal to A_T for every edge {S, T} of H. H below is the
+E8 non-orthogonal graph (3360 edges, degree 56, clique number 8): the pairs the E8 decoder leaves to lossless
+helpers. By Cor. A1 of PROOF.md every such pair needs orthogonal labels.
+
+Reduction (WLOG, argued in PROOF.md section 6): fix an 8-clique C of H. Its labels are pairwise orthogonal and
+nondegenerate, so they form an orthogonal direct sum, and the window splits along them. For k = 1 the 8 labels are
+orthonormal and can be taken to be e_1..e_8; for k = 2 each plane is the standard I_2 or the hyperbolic H plane and
+the window form is the sum of the 8 plane forms and I on the complement (any non-alternating form of that rank is
+isometric to I_r). Every other line T then has A_T supported on the blocks of the clique members not adjacent to T,
+plus the complement coordinates.
+
+ (a) uniqueness_h9: k = 1, r = 9, with distinct labels (E0 of gx.check1): exactly ONE labelling, enumerated to
+     exhaustion, and it is Sussman's (u_S . u_T = 1 on every E8-orthogonal pair, 0 on every in-cap pair).
+ (b) k2_planes: k = 2, r = 16 and 17: UNSAT for every plane-type pattern of the clique (any port type).
+Controls: k = 1, r = 8 is UNSAT and r = 9 SAT (calibration); the k = 2 encoder at r = 18 accepts the direct sum of
+two copies of the k = 1 solution (positive control).
+"""
+import itertools
+import sys
+import time
+from pathlib import Path
+
+sys.path.insert(0, str(Path(__file__).resolve().parent))
+from e8lib import G, NB, ORTH_PAIRS, V  # noqa: E402
+from f2 import dot, span_basis  # noqa: E402
+
+
+# ----------------------------------------------------------------------------------------------------------------
+# (a) k = 1: the admissible labellings at h = 9 (from the local research file satlab.py)
+CL = [119, 118, 102, 98, 79, 72, 37, 34]   # an 8-clique (Hamming indices) of the in-cap graph
+
+
+class Model:
+    """CNF of: u: lines -> F_2^h, u.u = 1, u_S.u_T = 0 on in-cap pairs, labels distinct; CL = e_1..e_8."""
+
+    def __init__(s, h):
+        s.h = h
+        s.nv = 0
+        s.cl = []
+        s.T = s.var()
+        s.cl.append([s.T])
+        s.u = [[None] * h for _ in range(V)]
+        for S in range(V):
+            for k in range(h):
+                if S in CL:
+                    s.u[S][k] = s.T if k == CL.index(S) else -s.T
+                elif k < 8 and G[S][CL[k]] != 0:
+                    s.u[S][k] = -s.T
+                else:
+                    s.u[S][k] = s.var()
+        for S in range(V):
+            if S not in CL:
+                s.bil_eq(S, S, 1)
+        for S in range(V):
+            for T in NB[S]:
+                if S < T:
+                    s.bil_eq(S, T, 0)
+        for S in range(V):
+            for T in range(S + 1, V):
+                ds = [s.xor2(s.u[S][k], s.u[T][k]) for k in range(h)]
+                ds = [x for x in ds if x != -s.T]
+                s.cl.append(ds)
+
+    def var(s):
+        s.nv += 1
+        return s.nv
+
+    def and2(s, a, b):
+        if a == -s.T or b == -s.T:
+            return -s.T
+        if a == s.T:
+            return b
+        if b == s.T:
+            return a
+        y = s.var()
+        s.cl += [[-y, a], [-y, b], [y, -a, -b]]
+        return y
+
+    def xor2(s, a, b):
+        if a == -s.T:
+            return b
+        if b == -s.T:
+            return a
+        if a == s.T:
+            return -b
+        if b == s.T:
+            return -a
+        y = s.var()
+        s.cl += [[-y, a, b], [-y, -a, -b], [y, -a, b], [y, a, -b]]
+        return y
+
+    def xor_eq(s, lits, c):
+        lits = [x for x in lits if x != -s.T]
+        n_t = sum(1 for x in lits if x == s.T)
+        lits = [x for x in lits if x != s.T]
+        c ^= n_t & 1
+        if not lits:
+            if c:
+                s.cl.append([])
+            return
+        acc = lits[0]
+        for x in lits[1:]:
+            acc = s.xor2(acc, x)
+        s.cl.append([acc] if c else [-acc])
+
+    def bil_eq(s, S, T, c):
+        s.xor_eq([s.and2(s.u[S][k], s.u[T][k]) for k in range(s.h)], c)
+
+    def labels(s, m):
+        out = []
+        for S in range(V):
+            x = 0
+            for k in range(s.h):
+                lit = s.u[S][k]
+                b = (lit == s.T) if abs(lit) == s.T else ((m[abs(lit) - 1] > 0) == (lit > 0))
+                if b:
+                    x |= 1 << k
+            out.append(x)
+        return out
+
+
+def uniqueness_h9():
+    from pysat.solvers import Cadical153
+    t0 = time.monotonic()
+    m = Model(9)
+    sol = Cadical153(bootstrap_with=m.cl)
+    free = [m.u[S][k] for S in range(V) for k in range(9) if abs(m.u[S][k]) != m.T]
+    sols = []
+    while sol.solve():
+        mod = sol.get_model()
+        sols.append(m.labels(mod))
+        sol.add_clause([-x if (mod[abs(x) - 1] > 0) == (x > 0) else x for x in free])
+        assert len(sols) <= 5, "more than 5 solutions"
+    sol.delete()
+    assert len(sols) == 1, len(sols)
+    u = sols[0]
+    # independent verification of the labelling, not through the CNF
+    assert all(dot(u[S], u[S]) == 1 for S in range(V))
+    assert all(dot(u[S], u[T]) == 0 for S in range(V) for T in NB[S])
+    assert len(set(u)) == V
+    assert all(dot(u[a], u[b]) == 1 for a, b in ORTH_PAIRS)       # Sussman's Gram: 1 on all 3780 E8-orthogonal pairs
+    # neighbourhood rank (Thm B data): u_S is orthogonal to N(S) and u_S.u_S = 1, so rank N(S) <= 8
+    ranks = {len(span_basis([u[T] for T in NB[S]])) for S in range(V)}
+    # controls: h = 8 admits no labelling (calibration); the same CNF with the distinctness clauses is satisfiable at 9
+    t8 = Model(8)
+    s8 = Cadical153(bootstrap_with=t8.cl)
+    unsat8 = not s8.solve()
+    s8.delete()
+    assert unsat8, "control failed: an h = 8 labelling exists"
+    return dict(solutions=len(sols), gram_is_sussman=True, labels_distinct=True, incap_degree=56,
+                neighbourhood_ranks=sorted(ranks), h8_control_unsat=unsat8, vars=m.nv, clauses=len(m.cl),
+                seconds=round(time.monotonic() - t0, 1)), u
+
+
+# ----------------------------------------------------------------------------------------------------------------
+# (b) k-dimensional planes on the full in-cap graph (from the local research files plsat.py and plsat2.py)
+def _graph():
+    adj = [set(NB[a]) for a in range(V)]
+    return adj
+
+
+def max_clique8(adj):
+    import networkx as nx
+    g = nx.Graph()
+    g.add_nodes_from(range(V))
+    g.add_edges_from((a, b) for a in range(V) for b in adj[a] if a < b)
+    best = max(nx.find_cliques(g), key=len)
+    return sorted(best)
+
+
+class PlaneModel:
+    """k-dim planes, form I_r, clique C fixed to coordinate blocks. usage as plsat.py (local research)."""
+
+    def __init__(s, k, r, adj, C, dist=False):
+        s.k, s.r, s.nv, s.cl = k, r, 1, [[1]]
+        s.T = 1
+        n = len(adj)
+        s.n = n
+        idx = {c: i for i, c in enumerate(C)}
+        s.vec = {}
+        for T in range(n):
+            for j in range(k):
+                if T in idx:
+                    s.vec[T, j] = [s.T if t == k * idx[T] + j else -s.T for t in range(r)]
+                else:
+                    allowed = set(range(k * len(C), r))
+                    for c, i in idx.items():
+                        if c not in adj[T]:
+                            allowed |= set(range(k * i, k * i + k))
+                    s.vec[T, j] = [s.new() if t in allowed else -s.T for t in range(r)]
+        for T in range(n):
+            if T in idx:
+                continue
+            for a in range(k):
+                for b in range(a, k):
+                    s.bil(s.vec[T, a], s.vec[T, b], 1 if a == b else 0)
+        ne = 0
+        for S in range(n):
+            for T in adj[S]:
+                if S < T:
+                    for a in range(k):
+                        for b in range(k):
+                            s.bil(s.vec[S, a], s.vec[T, b], 0)
+                    ne += 1
+        s.ne = ne
+        if dist:
+            assert k == 1
+            for S in range(n):
+                for T in range(S + 1, n):
+                    d = [s.xor2(x, y) for x, y in zip(s.vec[S, 0], s.vec[T, 0])]
+                    d = [x for x in d if x != -s.T]
+                    if s.T in d:
+                        continue
+                    s.cl.append(d)
+
+    def new(s):
+        s.nv += 1
+        return s.nv
+
+    def and2(s, a, b):
+        T = s.T
+        if a == -T or b == -T:
+            return -T
+        if a == T:
+            return b
+        if b == T:
+            return a
+        y = s.new()
+        s.cl += [[-y, a], [-y, b], [y, -a, -b]]
+        return y
+
+    def xor2(s, a, b):
+        T = s.T
+        if a == -T:
+            return b
+        if b == -T:
+            return a
+        if a == T:
+            return -b
+        if b == T:
+            return -a
+        y = s.new()
+        s.cl += [[-y, a, b], [-y, -a, -b], [y, -a, b], [y, a, -b]]
+        return y
+
+    def xorl(s, lits):
+        acc = -s.T
+        for x in lits:
+            acc = s.xor2(acc, x)
+        return acc
+
+    def bil(s, u, w, c):
+        terms = [s.and2(a, b) for a, b in zip(u, w)]
+        terms = [t for t in terms if t != -s.T]
+        acc = s.xorl(terms)
+        if acc == s.T:
+            if c == 0:
+                s.cl.append([])
+        elif acc == -s.T:
+            if c == 1:
+                s.cl.append([])
+        else:
+            s.cl.append([acc] if c else [-acc])
+
+    def labels(s, mod):
+        val = lambda x: (x == s.T) if abs(x) == s.T else (mod[abs(x) - 1] > 0) == (x > 0)   # noqa: E731
+        return {T: [sum(1 << t for t, x in enumerate(s.vec[T, j]) if val(x)) for j in range(s.k)]
+                for T in range(s.n)}
+
+
+def calibrations():
+    """Controls of the encoders: k = 1 (r = 8 UNSAT, r = 9 SAT, rank 9) and the k = 2 positive control at r = 18."""
+    cal, C, L1 = _k1_calibration()
+    cal["k2_positive_control"] = _k2_positive_control(C, L1)
+    return cal
+
+
+def _k1_calibration():
+    """k = 1: r = 8 UNSAT, r = 9 SAT (rank 9, verified independently)."""
+    from pysat.solvers import Solver
+    adj = _graph()
+    C = max_clique8(adj)
+    out = {}
+    for r in (8, 9):
+        m = PlaneModel(1, r, adj, C, dist=False)
+        with Solver(name="cadical195", bootstrap_with=m.cl) as sol:
+            ok = sol.solve()
+            if ok:
+                L = {T: x[0] for T, x in m.labels(sol.get_model()).items()}
+                assert all(dot(L[T], L[T]) == 1 for T in range(V))
+                assert all(dot(L[S], L[T]) == 0 for S in range(V) for T in adj[S])
+                out[f"r{r}"] = dict(sat=True, rank=len(span_basis(list(L.values()))))
+                sol1 = L
+            else:
+                out[f"r{r}"] = dict(sat=False)
+    assert out["r8"] == dict(sat=False) and out["r9"] == dict(sat=True, rank=9)
+    return out, C, sol1
+
+
+def _k2_positive_control(C, L1):
+    """The k = 2 encoder at r = 18 accepts the direct sum of two copies of a k = 1, r = 9 labelling."""
+    from pysat.solvers import Solver
+    adj = _graph()
+    m = PlaneModel(2, 18, adj, C)
+
+    def emb(x, copy):
+        y = 0
+        for t in range(9):
+            if x >> t & 1:
+                c = 2 * t + copy if t < 8 else 16 + copy
+                y |= 1 << c
+        return y
+
+    L2 = {T: [emb(L1[T], 0), emb(L1[T], 1)] for T in range(V)}
+    # independent check of the direct sum: planes nondegenerate (I_2), orthogonal on all in-cap edges
+    assert all(dot(L2[T][0], L2[T][0]) == 1 and dot(L2[T][1], L2[T][1]) == 1 and dot(L2[T][0], L2[T][1]) == 0
+               for T in range(V))
+    assert all(dot(a, b) == 0 for S in range(V) for T in adj[S] for a in L2[S] for b in L2[T])
+    assume = []
+    for T in range(V):
+        for j in range(2):
+            for t, lit in enumerate(m.vec[T, j]):
+                bit = L2[T][j] >> t & 1
+                if abs(lit) == m.T:
+                    assert (lit > 0) == bool(bit), "direct sum violates the support rule of the encoder"
+                else:
+                    assume.append(lit if bit else -lit)
+    with Solver(name="cadical195", bootstrap_with=m.cl) as sol:
+        ok = sol.solve(assumptions=assume)
+    assert ok, "control failed: encoder rejects the r = 18 direct sum"
+    return dict(r18_direct_sum_accepted=True, vars=m.nv, clauses=len(m.cl))
+
+
+
+# any-type k = 2 encoder, one pattern of clique plane types (from plsat2.py)
+_STATE = {}
+
+
+def _init():
+    if not _STATE:
+        adj = _graph()
+        _STATE["adj"] = adj
+        _STATE["C"] = max_clique8(adj)
+
+
+class AnyTypePlanes:
+    def __init__(s, r, pat):
+        adj, C = _STATE["adj"], _STATE["C"]
+        idx = {c: i for i, c in enumerate(C)}
+        s.r = r
+        s.nv, s.cl, s.T, s.pat = 1, [[1]], 1, pat
+        s.vec, s.tau = {}, {}
+        for T in range(V):
+            if T in idx:
+                i = idx[T]
+                s.vec[T, 0] = [s.T if t == 2 * i else -s.T for t in range(r)]
+                s.vec[T, 1] = [s.T if t == 2 * i + 1 else -s.T for t in range(r)]
+            else:
+                al = set(range(16, r))
+                for c, i in idx.items():
+                    if c not in adj[T]:
+                        al |= {2 * i, 2 * i + 1}
+                for j in range(2):
+                    s.vec[T, j] = [s.new() if t in al else -s.T for t in range(r)]
+                s.tau[T] = s.new()
+                p, q = s.vec[T, 0], s.vec[T, 1]
+                s.eq(s.form(p, p), -s.tau[T])      # tau = 0: plane I_2 (p.p = q.q = 1, p.q = 0)
+                s.eq(s.form(q, q), -s.tau[T])      # tau = 1: hyperbolic plane (p.p = q.q = 0, p.q = 1)
+                s.eq(s.form(p, q), s.tau[T])
+        for S in range(V):
+            for T in adj[S]:
+                if S < T:
+                    for a in range(2):
+                        for b in range(2):
+                            s.eq(s.form(s.vec[S, a], s.vec[T, b]), -s.T)
+
+    def new(s):
+        s.nv += 1
+        return s.nv
+
+    def and2(s, a, b):
+        T = s.T
+        if a == -T or b == -T:
+            return -T
+        if a == T:
+            return b
+        if b == T:
+            return a
+        y = s.new()
+        s.cl += [[-y, a], [-y, b], [y, -a, -b]]
+        return y
+
+    def xor2(s, a, b):
+        T = s.T
+        if a == -T:
+            return b
+        if b == -T:
+            return a
+        if a == T:
+            return -b
+        if b == T:
+            return -a
+        y = s.new()
+        s.cl += [[-y, a, b], [-y, -a, -b], [y, -a, b], [y, a, -b]]
+        return y
+
+    def form(s, u, w):
+        terms = []
+        for t in range(s.r):
+            if t < 16 and s.pat[t // 2] == "H":
+                terms.append(s.and2(u[t], w[t ^ 1]))
+            else:
+                terms.append(s.and2(u[t], w[t]))
+        acc = -s.T
+        for x in terms:
+            acc = s.xor2(acc, x)
+        return acc
+
+    def eq(s, a, b):
+        x = s.xor2(a, b)
+        if x == s.T:
+            s.cl.append([])
+        elif x != -s.T:
+            s.cl.append([-x])
+
+
+def _solve_patterns(args):
+    from pysat.solvers import Solver
+    r, pats = args
+    _init()
+    out = {}
+    for pat in pats:
+        f = AnyTypePlanes(r, pat)
+        with Solver(name="cadical195", bootstrap_with=f.cl) as sol:
+            out[pat] = sol.solve()
+    return out
+
+
+def k2_tasks(r, n_chunks=32):
+    """Split the plane-type patterns of the clique into chunks (one CaDiCaL instance per pattern)."""
+    assert r in (16, 17)
+    pats = ["".join(b) for b in itertools.product("IH", repeat=8)]
+    if r == 16:
+        pats.remove("H" * 8)     # an alternating window form is not the dot product
+    return [(r, pats[i::n_chunks]) for i in range(n_chunks)], len(pats)
+
+
+def k2_solve(args):
+    return _solve_patterns(args)
+
+
+def k2_collect(r, parts, npats):
+    res = {}
+    for part in parts:
+        res.update(part)
+    assert len(res) == npats
+    assert sum(res.values()) == 0, f"k = 2 planes found at r = {r}"
+    return dict(r=r, patterns=npats, sat=0)
+
+
+def e8_clique_facts():
+    """Clique structure of the in-cap graph: omega = 8, 8640 maximal cliques of size 8 (each is an orthonormal basis)."""
+    import networkx as nx
+    adj = _graph()
+    g = nx.Graph()
+    g.add_nodes_from(range(V))
+    g.add_edges_from((a, b) for a in range(V) for b in adj[a] if a < b)
+    sizes = {}
+    for q in nx.find_cliques(g):
+        sizes[len(q)] = sizes.get(len(q), 0) + 1
+    assert max(sizes) == 8
+    return dict(vertices=V, edges=g.number_of_edges(), degree=56, omega=max(sizes), maximal_cliques_by_size=
+                {str(k): sizes[k] for k in sorted(sizes)})
diff --git a/research/format-obstructions/checks/toy.py b/research/format-obstructions/checks/toy.py
new file mode 100644
index 0000000000..2d7765dbe8
--- /dev/null
+++ b/research/format-obstructions/checks/toy.py
@@ -0,0 +1,128 @@
+"""The p = 3 toy: the one place where a read-discipline change beats gx's E2 (all targets at frame 0 at the scatter).
+
+Instance: inputs/gcert1-p3-cone-R16.json (paired cube p = 3, h = 6, v = 8, R = 16, cst = 18), accepted by gx.check1.
+Its 8 ports span W of dimension 4, and the window form restricted to W is DEGENERATE: W^perp (dim 2) lies in W.
+Modified rule E2': the scatter is done in cuts. At a cut, all targets stand at ONE common frame g contained in every
+target's final q^perp (so g <= W^perp), and copying total k costs d(f_k, g) = dim f + dim g - 2 dim(f cap g).
+Exact checks (all F_2 / integer): every register chain is nested, equal frames at every gate, targets stay inside
+q^perp and avoid q, the block-mass identity, the scalar half (gates and coefficients unchanged), and the exhaustive
+optimum over all chains of cut frames. Control: a cut frame outside the radical is rejected.
+"""
+import itertools
+import sys
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parent.parent
+sys.path.insert(0, str(ROOT / "gx"))
+sys.path.insert(0, str(ROOT / "checks"))
+import gx  # noqa: E402
+from certs import load  # noqa: E402
+from f2 import dot, rank  # noqa: E402
+
+
+def span(gens):
+    b = []
+    for u in gens:
+        for x in b:
+            if u ^ x < u:
+                u ^= x
+        if u:
+            b.append(u)
+            b.sort(reverse=True)
+    return tuple(b)
+
+
+def inside(a, b):
+    return all(gx.reduce(u, b) == 0 for u in a)
+
+
+def run():
+    c = load(ROOT / "inputs" / "gcert1-p3-cone-R16.json")
+    gx.check1(c)
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    ports = c["ports"]
+    fr = [tuple(f) for f in c["frames"]]
+    assert not c["ext"]
+
+    def d(f, g):
+        both = span([x for x in range(1, 1 << h) if gx.reduce(x, f) == 0 and gx.reduce(x, g) == 0])
+        return len(f) + len(g) - 2 * len(both)
+
+    # geometry of the ports: W = span, W^perp, radical
+    W = span(ports)
+    perpW = [x for x in range(1 << h) if all(dot(x, q) == 0 for q in ports)]
+    rad_nonzero = sorted(x for x in perpW if x)
+    assert len(W) == 4 and len(perpW) == 4 and all(gx.reduce(x, W) == 0 for x in perpW)   # W^perp lies in W: radical
+    L = 0b111100
+    assert L in rad_nonzero and all(dot(L, q) == 0 for q in ports)
+
+    cur = [fr[s] for s in c["start"]]
+    mass = 0
+
+    def climb(r, f):
+        nonlocal mass
+        if cur[r] != f:
+            assert inside(cur[r], f), ("not nested", r)
+            mass += len(f) - len(cur[r])
+            cur[r] = f
+
+    for g in c["A"]:
+        regs, _, _ = gx.gate_regs(g)
+        for r in regs:
+            climb(r, fr[g[1]])
+    tot = {k: (s, fr[f]) for k, s, f in c["ret"]}
+    for k, (s, f) in tot.items():
+        assert cur[s] == f
+    assert all(cur[v + t] == () for t in range(v))
+    old_cst = sum(len(f) for _, f in tot.values())
+    assert old_cst == c["cst"] == 18
+    cut1 = [k for k in tot if gx.reduce(L, tot[k][1]) != 0]     # totals not containing l: read at g = 0
+    cut2 = [k for k in tot if gx.reduce(L, tot[k][1]) == 0]     # totals containing l: read at g = <l>
+    cst = sum(d(tot[k][1], ()) for k in cut1)
+    for t in range(v):
+        climb(v + t, span([L]))                                 # the targets climb to the common frame l
+    cst += sum(d(tot[k][1], span([L])) for k in cut2)
+    for g in c["B"]:
+        regs, _, _ = gx.gate_regs(g)
+        f = span(list(fr[g[1]]) + [L])
+        for r in regs:
+            climb(r, f)
+    for r in range(nreg):
+        fin = fr[c["final"][r]]
+        climb(r, span(list(fin) + ([L] if r >= v else [])) if r < v or r >= 2 * v else fin)
+    for t in range(v):
+        assert inside(cur[v + t], fr[c["final"][v + t]]) and gx.reduce(ports[t], cur[v + t]) != 0
+    assert mass == R * h + 2 * v * (h - 1), mass
+    gx.scal(c)                                                   # scalar half: gates and coefficients unchanged
+
+    # exhaustive optimum over all chains of cut frames inside W^perp (targets only climb between cuts)
+    subs = [()] + [span([x]) for x in rad_nonzero] + [span(rad_nonzero)]
+    subs = [s for i, s in enumerate(subs) if s not in subs[:i]]
+    def leq(a, b):
+        return inside(a, b)
+    best, best_chain = None, None
+    old_format = None
+    for n in range(1, len(subs) + 1):
+        for chain in itertools.combinations(subs, n):
+            if not all(leq(a, b) or leq(b, a) for a in chain for b in chain):
+                continue
+            total = sum(min(d(f, g) for g in chain) for _, f in tot.values())
+            if chain == ((),):
+                old_format = total
+            if best is None or total < best:
+                best, best_chain = total, chain
+    assert old_format == 18 and best == 16 == cst, (old_format, best, cst)
+
+    # control: a common frame not inside W^perp would put a target outside q^perp
+    not_rad = [x for x in range(1, 1 << h) if x not in rad_nonzero]
+    ctrl = all(any(dot(x, q) for q in ports) for x in not_rad)
+    assert ctrl, "control failed"
+    # Thm C bound for this instance (degenerate span): lower bound of the pairing ranks
+    lb = sum(len(f) - 1 for _, f in tot.values())
+    # on the nondegenerate families the gain is zero: W^perp = 0 forces g = 0 (window_rank = h in certs.py)
+    return dict(h=h, v=v, R=R, old_cst=old_cst, new_cst=cst, optimum_over_cut_chains=best, old_format_optimum=old_format,
+                cut1=sorted(cut1), cut2=sorted(cut2), line=bin(L), radical_nonzero=[bin(x) for x in rad_nonzero],
+                port_span_dim=len(W), moves_old=R * h + 2 * v * (h - 1) + old_cst, moves_new=mass + cst,
+                thmC_per_total_bound=lb, D_old=4 * v - 5 * old_cst, D_new=4 * v - 5 * cst,
+                cut_outside_radical_rejected=ctrl)
diff --git a/research/format-obstructions/expected.json b/research/format-obstructions/expected.json
new file mode 100644
index 0000000000..44faf6094e
--- /dev/null
+++ b/research/format-obstructions/expected.json
@@ -0,0 +1,545 @@
+{
+ "certificates": {
+  "p10": {
+   "N_rank": [
+    19,
+    19
+   ],
+   "R": 6515,
+   "check1": "PASS",
+   "copy_lb": 360,
+   "copy_paid": 360,
+   "coread_all": true,
+   "cst": 360,
+   "file": "gcert1-p10-anneal-mw-flow.json.gz",
+   "h": 20,
+   "incap_deg": [
+    620,
+    620
+   ],
+   "incap_perp": true,
+   "self_lossless": false,
+   "thmA_gate_checks": 1081204,
+   "thmA_violations": 0,
+   "v": 960,
+   "window_rank": 20
+  },
+  "p11": {
+   "N_rank": [
+    21,
+    21
+   ],
+   "R": 9412,
+   "check1": "PASS",
+   "copy_lb": 440,
+   "copy_paid": 440,
+   "coread_all": true,
+   "cst": 440,
+   "file": "gcert1-p11-pr193.json.gz",
+   "h": 22,
+   "incap_deg": [
+    884,
+    884
+   ],
+   "incap_perp": true,
+   "self_lossless": false,
+   "thmA_gate_checks": 2063837,
+   "thmA_violations": 0,
+   "v": 1320,
+   "window_rank": 22
+  },
+  "p3": {
+   "N_rank": [
+    3,
+    3
+   ],
+   "R": 16,
+   "check1": "PASS",
+   "controls": {
+    "corrupted_port_check1_rejects": "E1",
+    "corrupted_port_replay_violations": 14,
+    "shifted_labels_nonorthogonal_pairs": 12
+   },
+   "copy_lb": 12,
+   "copy_paid": 18,
+   "coread_all": true,
+   "cst": 18,
+   "file": "gcert1-p3-cone-R16.json",
+   "h": 6,
+   "incap_deg": [
+    4,
+    4
+   ],
+   "incap_perp": true,
+   "self_lossless": false,
+   "thmA_gate_checks": 110,
+   "thmA_violations": 0,
+   "v": 8,
+   "window_rank": 4
+  }
+ },
+ "e8_labels": {
+  "calibrations": {
+   "k2_positive_control": {
+    "clauses": 605809,
+    "r18_direct_sum_accepted": true,
+    "vars": 173825
+   },
+   "r8": {
+    "sat": false
+   },
+   "r9": {
+    "rank": 9,
+    "sat": true
+   }
+  },
+  "cliques": {
+   "degree": 56,
+   "edges": 3360,
+   "maximal_cliques_by_size": {
+    "3": 1120,
+    "7": 34560,
+    "8": 8640
+   },
+   "omega": 8,
+   "vertices": 120
+  },
+  "k2_planes": {
+   "r16": {
+    "patterns": 255,
+    "r": 16,
+    "sat": 0
+   },
+   "r17": {
+    "patterns": 256,
+    "r": 17,
+    "sat": 0
+   }
+  },
+  "uniqueness_h9": {
+   "clauses": 166685,
+   "gram_is_sussman": true,
+   "h8_control_unsat": true,
+   "incap_degree": 56,
+   "labels_distinct": true,
+   "neighbourhood_ranks": [
+    8
+   ],
+   "solutions": 1,
+   "vars": 42645
+  }
+ },
+ "families": {
+  "e8_bitdual": {
+   "betas": [
+    8,
+    8,
+    8,
+    8,
+    8,
+    8,
+    8,
+    8,
+    8
+   ],
+   "copy_lb": 72,
+   "decoder_equals_UUt": true,
+   "h": 8,
+   "neighbourhood": [
+    [
+     63,
+     7
+    ]
+   ],
+   "totals": 9,
+   "v": 120
+  },
+  "e8_complex": {
+   "N_rank": 8,
+   "h": 9,
+   "incap_deg": 56,
+   "v": 120
+  },
+  "omega": {
+   "bit_4": {
+    "h": 8,
+    "kind": "bit",
+    "omega": 4,
+    "p": 4
+   },
+   "bit_5": {
+    "h": 10,
+    "kind": "bit",
+    "omega": 5,
+    "p": 5
+   },
+   "bit_6": {
+    "h": 12,
+    "kind": "bit",
+    "omega": 6,
+    "p": 6
+   },
+   "bit_7": {
+    "h": 14,
+    "kind": "bit",
+    "omega": 7,
+    "p": 7
+   },
+   "cx_4": {
+    "h": 8,
+    "kind": "cx",
+    "omega": 8,
+    "p": 4
+   },
+   "cx_5": {
+    "h": 10,
+    "kind": "cx",
+    "omega": 8,
+    "p": 5
+   },
+   "cx_6": {
+    "h": 12,
+    "kind": "cx",
+    "omega": 12,
+    "p": 6
+   },
+   "cx_7": {
+    "h": 14,
+    "kind": "cx",
+    "omega": 12,
+    "p": 7
+   }
+  },
+  "p10_clique": {
+   "all_pairs_lossless_in_program": true,
+   "label_dim_sum_le": 20,
+   "orthonormal": true,
+   "size": 20,
+   "window_dim": 20
+  },
+  "paired_bit": {
+   "10": {
+    "N_rank": 19,
+    "h": 20,
+    "incap_deg": 339,
+    "p": 10,
+    "v": 960
+   },
+   "11": {
+    "N_rank": 21,
+    "h": 22,
+    "incap_deg": 435,
+    "p": 11,
+    "v": 1320
+   },
+   "4": {
+    "N_rank": 7,
+    "h": 8,
+    "incap_deg": 15,
+    "p": 4,
+    "v": 32
+   },
+   "5": {
+    "N_rank": 9,
+    "h": 10,
+    "incap_deg": 39,
+    "p": 5,
+    "v": 80
+   }
+  }
+ },
+ "mutable_425": {
+  "J_is_identity": true,
+  "K_nonzeros": 14246,
+  "S_K_nonzeros": 14582,
+  "S_b": 0.0009102159764952497,
+  "S_b_vs_b_percent": -13.24,
+  "S_exact_after_compensation": true,
+  "S_helpers": 904,
+  "S_helpers_formula": "R + b_x + b_y",
+  "S_local_hist": {
+   "1": 2472,
+   "2": 896,
+   "3": 168,
+   "4": 336,
+   "5": 392,
+   "8": 248,
+   "9": 8
+  },
+  "S_mass": 10128,
+  "S_uncompensated_rejected": true,
+  "S_z_restored": true,
+  "V_offdiag": 112,
+  "all_targets_at_frame0_at_cut": true,
+  "b": 0.001049104978040586,
+  "b_pre425": 0.000999388595603943,
+  "cst_bound": 72,
+  "cst_paid": 72,
+  "detector_on_converted_word": {},
+  "detector_on_word_as_written": {
+   "source modified": 1640,
+   "target read": 112
+  },
+  "h": 9,
+  "helpers": 680,
+  "local_hist": {
+   "1": 2248,
+   "2": 896,
+   "3": 168,
+   "4": 336,
+   "5": 392,
+   "8": 24,
+   "9": 8
+  },
+  "m": 73,
+  "m_min_nine_stages": 73,
+  "mass": 8112,
+  "modified_sources": 112,
+  "omit_K_control_rejected": true,
+  "ports_realise_E8_gram": true,
+  "preconditioned_map_exact": true,
+  "read_targets": 112,
+  "registers": 920,
+  "sources_gathered_per_centre": [
+   57,
+   57,
+   57,
+   57,
+   57,
+   57,
+   57,
+   57
+  ],
+  "stats": {
+   "cut": 232,
+   "scale": 644,
+   "src_modified": 1640,
+   "target_read": 112
+  },
+  "target_neighbourhood_ranks": [
+   8
+  ],
+  "thmA_violations": 0,
+  "v": 120,
+  "z_restored": true
+ },
+ "pivots": {
+  "copy_count_p10": {
+   "forced_extra_copy_per_overlap": 10,
+   "frame_dims": [
+    18
+   ],
+   "max_noncarrier_totals": 1,
+   "max_totals_containing_b": 9,
+   "totals": 20
+  },
+  "overlap_scan_p10": {
+   "candidates": 1160,
+   "h": 20,
+   "min_carriers": 5984,
+   "min_contaminated_registers": 8435,
+   "min_contaminated_targets": 960,
+   "min_port_rank_of_contaminated": 20,
+   "registers": 8435,
+   "slots": 6515,
+   "slots_with_gates": 6515
+  },
+  "thm_P": {
+   "p10": {
+    "K_solution_dim": 0,
+    "K_zero_gives_m_ge": "s*h, under (P1)-(P2)",
+    "cross_cube_slots": 5880,
+    "file": "gcert1-p10-anneal-mw-flow.json.gz",
+    "h": 20,
+    "h_squared": 400,
+    "rank_all_carried_pairs": 400,
+    "rank_orthogonal_pairs_only": 399,
+    "slot_written_targets_port_rank": 20,
+    "slots": 6515,
+    "writer_slots": 3708
+   },
+   "p11": {
+    "K_solution_dim": 0,
+    "K_zero_gives_m_ge": "s*h, under (P1)-(P2)",
+    "cross_cube_slots": 9180,
+    "file": "gcert1-p11-pr193.json.gz",
+    "h": 22,
+    "h_squared": 484,
+    "rank_all_carried_pairs": 484,
+    "rank_orthogonal_pairs_only": 483,
+    "slot_written_targets_port_rank": 22,
+    "slots": 9412,
+    "writer_slots": 4125
+   }
+  }
+ },
+ "pricing": {
+  "D": 24480,
+  "W": 131760,
+  "m": 100,
+  "per_stage_rank1_copy_percent": "-0.2457",
+  "per_stage_rank1_register_descent_percent": "-0.4911",
+  "pinned_root": "7.702769095630e-04",
+  "pivot_free_m96": {
+   "blocks 2->1": {
+    "with_40_forced_copy_percent": "+4.30",
+    "zero_cost_percent": "+6.39"
+   },
+   "largest blocks shortened": {
+    "with_40_forced_copy_percent": "+1.53",
+    "zero_cost_percent": "+3.56"
+   },
+   "mid blocks shortened": {
+    "with_40_forced_copy_percent": "+3.68",
+    "zero_cost_percent": "+5.75"
+   },
+   "rank-1 blocks removed": {
+    "with_40_forced_copy_percent": "+5.99",
+    "zero_cost_percent": "+8.11"
+   }
+  },
+  "root": "7.702769095630e-04"
+ },
+ "thmB_table": {
+  "bit_p10": {
+   "s3": {
+    "layout_s_times_h": 60,
+    "m_min": 58
+   },
+   "s5": {
+    "layout_s_times_h": 100,
+    "m_min": 96
+   },
+   "s7": {
+    "layout_s_times_h": 140,
+    "m_min": 134
+   },
+   "s9": {
+    "layout_s_times_h": 180,
+    "m_min": 172
+   }
+  },
+  "bit_p11": {
+   "s3": {
+    "layout_s_times_h": 66,
+    "m_min": 64
+   },
+   "s5": {
+    "layout_s_times_h": 110,
+    "m_min": 106
+   },
+   "s7": {
+    "layout_s_times_h": 154,
+    "m_min": 148
+   },
+   "s9": {
+    "layout_s_times_h": 198,
+    "m_min": 190
+   }
+  },
+  "complex_p10": {
+   "s3": {
+    "layout_s_times_h": 60,
+    "m_min": 58
+   },
+   "s5": {
+    "layout_s_times_h": 100,
+    "m_min": 96
+   },
+   "s7": {
+    "layout_s_times_h": 140,
+    "m_min": 134
+   },
+   "s9": {
+    "layout_s_times_h": 180,
+    "m_min": 172
+   }
+  },
+  "complex_p11": {
+   "s3": {
+    "layout_s_times_h": 66,
+    "m_min": 64
+   },
+   "s5": {
+    "layout_s_times_h": 110,
+    "m_min": 106
+   },
+   "s7": {
+    "layout_s_times_h": 154,
+    "m_min": 148
+   },
+   "s9": {
+    "layout_s_times_h": 198,
+    "m_min": 190
+   }
+  },
+  "e8_bit_dual": {
+   "s3": {
+    "layout_s_times_h": 24,
+    "m_min": 22
+   },
+   "s5": {
+    "layout_s_times_h": 40,
+    "m_min": 36
+   },
+   "s7": {
+    "layout_s_times_h": 56,
+    "m_min": 50
+   },
+   "s9": {
+    "layout_s_times_h": 72,
+    "m_min": 64
+   }
+  },
+  "e8_complex": {
+   "s3": {
+    "layout_s_times_h": 27,
+    "m_min": 25
+   },
+   "s5": {
+    "layout_s_times_h": 45,
+    "m_min": 41
+   },
+   "s7": {
+    "layout_s_times_h": 63,
+    "m_min": 57
+   },
+   "s9": {
+    "layout_s_times_h": 81,
+    "m_min": 73
+   }
+  }
+ },
+ "toy_p3": {
+  "D_new": -48,
+  "D_old": -58,
+  "R": 16,
+  "cut1": [
+   2,
+   3,
+   4,
+   5
+  ],
+  "cut2": [
+   0,
+   1
+  ],
+  "cut_outside_radical_rejected": true,
+  "h": 6,
+  "line": "0b111100",
+  "moves_new": 192,
+  "moves_old": 194,
+  "new_cst": 16,
+  "old_cst": 18,
+  "old_format_optimum": 18,
+  "optimum_over_cut_chains": 16,
+  "port_span_dim": 4,
+  "radical_nonzero": [
+   "0b1111",
+   "0b110011",
+   "0b111100"
+  ],
+  "thmC_per_total_bound": 12,
+  "v": 8
+ }
+}
diff --git a/research/format-obstructions/gx/gx.py b/research/format-obstructions/gx/gx.py
new file mode 100644
index 0000000000..6d14c0a976
--- /dev/null
+++ b/research/format-obstructions/gx/gx.py
@@ -0,0 +1,296 @@
+#!/usr/bin/python3 -I
+"""gx.py.  OWN code (imports nothing of port-producer, nothing outside).
+(1) convert(c0): gcert/0 (port-producer's event list) -> gcert/1, the LEAN-FACING NORMAL FORM:
+    only the main phase (no read0 / undo / uninj: the invocation theorem derives the dirt gate and the undo
+    gate from the product), every gate an elementary fan-out or fan-in add, the 4x4 K block written as three
+    such adds plus a sign carried by one register, the star scatter as a table.
+(2) check1(c1): the local decidable conditions of EXTENDED-CERT-SPEC.md on a gcert/1 file:
+    E0 sizes, E1 labels (nested subspaces by containment, equal frame at a gate), E2 scatter cut,
+    E3 shape (kinds of gates; X1; S1), E4 bracket rule S2, E5 scalar (x-content replay: hid and X3),
+    E6 price data (block histograms, N), E7 exterior gauges only (Q2).  Raises AssertionError("<rule>: ...").
+"""
+from fractions import Fraction as Fr
+from collections import Counter
+from math import lcm
+
+
+def need(c, msg):
+    if not c:
+        raise AssertionError(msg)
+
+
+def reduce(u, basis):
+    """basis: reduced echelon, pivot = leading bit, descending.  Returns u modulo the span."""
+    for b in basis:
+        if u >> (b.bit_length() - 1) & 1:
+            u ^= b
+    return u
+
+
+def echelon_ok(f):
+    piv = [b.bit_length() - 1 for b in f]
+    if any(b == 0 for b in f) or piv != sorted(piv, reverse=True) or len(set(piv)) != len(piv):
+        return False
+    return all(not (b >> p & 1) for i, b in enumerate(f) for j, p in enumerate(piv) if i != j)
+
+
+def inside(a, b):
+    return all(reduce(u, b) == 0 for u in a)
+
+
+def par(x):
+    return bin(x).count("1") & 1
+
+
+KJ = [[[1, 2] if i == j else [-1, 2] for j in range(4)] for i in range(4)]
+
+
+def convert(c0):
+    need(c0["format"] == "gcert/0", "input is not gcert/0")
+    v, h, R = c0["v"], c0["h"], c0["R"]
+    ev = c0["events"]
+    cen = [n for n, e in enumerate(ev) if e[0] == "centre"]
+    need(cen == list(range(cen[0], cen[0] + len(cen))), "centres are not contiguous: no single scatter cut")
+    A, B, ret = [], [], []
+    neg = set()            # x registers currently holding MINUS their value (between the two K blocks)
+    openq = {}
+    for n, e in enumerate(ev):
+        k = e[0]
+        out = A if n < cen[0] else B
+        if k in ("read0", "undo", "uninj"):
+            continue
+        if k == "inj":
+            need(e[2] not in neg, "inj reads a negated x register")
+            out.append(["out", e[3], e[2], [[e[1], 1, 1]], []])
+        elif k == "op":
+            if e[3] == -1:
+                out.append(["neg", e[5], e[1]])
+            else:
+                need(e[3] == 1, "op ca")
+            out.append(["out", e[5], e[2], [[e[1], e[4], 1]], []])
+        elif k == "centre":
+            ret.append([e[2], e[1], e[3]])
+        elif k == "gread":
+            hit = set(t for t, _, _ in e[3])
+            out.append(["out", e[2], e[1], [[v + t, a, b] for t, a, b in e[3]], [v + t for t in e[4] if t not in hit]])
+        elif k == "rread":
+            out.append(["out", e[2], e[1], [[v + t, a, b] for t, a, b in e[3]], []])
+        elif k == "ysh":
+            out.append(["out", e[4], e[2], [[e[1], e[3], 1]], []])
+        elif k == "yw":
+            out.append(["out", e[4], e[2], [[e[1], e[3][0], e[3][1]]], []])
+        elif k == "kd":
+            out.append(["out", e[3], e[2], [[e[1], -1 if e[2] in neg else 1, 1]], []])
+        elif k == "ktr":
+            q = tuple(e[1])
+            need(e[2] == KJ, "ktr matrix is not 1 - J/2")
+            x1, rest = q[0], list(q[1:])
+            if q not in openq:          # first block: x := M x, register x1 keeps the sign
+                out.append(["in", e[3], x1, [[r, 1, 1] for r in rest]])
+                out.append(["out", e[3], x1, [[r, -1, 2] for r in rest], []])
+                out.append(["in", e[3], x1, [[r, 1, 1] for r in rest]])
+                openq[q] = n
+                neg.add(x1)
+            else:                       # second block: the three adds undone in reverse order
+                out.append(["in", e[3], x1, [[r, -1, 1] for r in rest]])
+                out.append(["out", e[3], x1, [[r, 1, 2] for r in rest], []])
+                out.append(["in", e[3], x1, [[r, -1, 1] for r in rest]])
+                del openq[q]
+                neg.discard(x1)
+        else:
+            raise AssertionError("unknown event kind %r" % k)
+    need(not openq and not neg, "a K block is not closed")
+    ext = sorted([z[3], z[1]] for z in c0["gauge"] if z[5])
+    c1 = dict(format="gcert/1", derived_from=c0["name"] + " (gcert/0)", p=c0["p"], h=h, v=v, R=R, cst=c0["cst"],
+              N=c0["N"], ports=c0["ports"], frames=c0["frames"], start=c0["start"], final=c0["final"],
+              ext=ext, ret=sorted(ret), scat=dict(inside=c0["scat"]["inside"], outside=c0["scat"]["outside"]),
+              A=A, B=B, blocks=c0["blocks"])
+    return c1
+
+
+def scat_row(c, t):
+    """the scatter of target t: [(k, coefficient)], from the table or from the star rule"""
+    sc = c["scat"]
+    if "table" in sc:
+        return [(k, Fr(a, b)) for k, a, b in sc["table"][t]]
+    sin, sout = Fr(*sc["inside"]), Fr(*sc["outside"])
+    return [(k, sin if c["ports"][t] >> k & 1 else sout) for k in range(c["h"])]
+
+
+def gate_regs(g):
+    if g[0] == "neg":
+        return [g[2]], [], []
+    if g[0] == "out":
+        return [g[2]] + [t for t, _, _ in g[3]] + list(g[4]), [(t, g[2], Fr(a, b)) for t, a, b in g[3]], g[4]
+    if g[0] == "in":
+        return [g[2]] + [s for s, _, _ in g[3]], [(g[2], s, Fr(a, b)) for s, a, b in g[3]], []
+    raise AssertionError("E0: unknown gate kind %r" % (g[0],))
+
+
+def check1(c, scalar=True, stats=None):
+    need(c.get("format") == "gcert/1", "E0: format")
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    ports, fr = c["ports"], [tuple(f) for f in c["frames"]]
+    need(len(ports) == v == len(set(ports)) and all(par(q) and 0 < q < 1 << h for q in ports), "E0: ports")
+    need(fr[0] == () and fr[1] == tuple(1 << i for i in range(h - 1, -1, -1)), "E0: frames 0 and 1")
+    need(all(echelon_ok(f) for f in fr) and len(set(fr)) == len(fr), "E0: frame table not reduced echelon / repeats")
+    dim = [len(f) for f in fr]
+    start, final = c["start"], c["final"]
+    need(len(start) == len(final) == nreg, "E0: start / final")
+    ext = {r: f for r, f in c["ext"]}
+    for t in range(v):
+        need(fr[start[t]] == (ports[t],) and final[t] == 1, "E1: x start / final")
+        f = fr[final[v + t]]
+        need(start[v + t] == 0 and len(f) == h - 1 and all(par(u & ports[t]) == 0 for u in f), "E1: y start / final")
+    for r in range(2 * v, nreg):
+        need(start[r] == ext.get(r, 0) and final[r] == 1, "E1: slot start / final")
+    cls = lambda r: "x" if r < v else "y" if r < 2 * v else "s"
+    cur = list(start)
+    H = {"x": Counter(), "y": Counter(), "s": Counter(), "c": Counter()}
+    steps = set()
+
+    def climb(r, f, where):
+        o = cur[r]
+        if o != f:
+            if (o, f) not in steps:
+                need(dim[o] < dim[f] and inside(fr[o], fr[f]), "E1: step not nested (%s, register %d)" % (where, r))
+                steps.add((o, f))
+            H[cls(r)][dim[f] - dim[o]] += 1
+            cur[r] = f
+    okind = {("x", "s"), ("s", "s"), ("s", "y"), ("x", "y"), ("x", "x"), ("y", "y")}
+    ysum, piv, ytg = Counter(), set(), set()
+    inc = 0
+    for ph, gates in (("A", c["A"]), ("B", c["B"])):
+        if ph == "B":      # ---- E2: the scatter cut
+            need(sorted(k for k, _, _ in c["ret"]) == list(range(len(c["ret"]))), "E2: retained totals numbered 0..")
+            if "table" in c["scat"]:
+                need(len(c["scat"]["table"]) == v and all(0 <= k < len(c["ret"]) for row in c["scat"]["table"] for k, _, _ in row),
+                     "E2: scatter table")
+            else:
+                need(len(c["ret"]) == h, "E2: one retained total per coordinate (star rule)")
+            for k, s, f in c["ret"]:
+                need(cls(s) == "s" and cur[s] == f, "E2: retained slot %d is not at its frame" % s)
+                H["c"][dim[f]] += 1
+            need(all(cur[v + t] == 0 for t in range(v)), "E2: a y role left frame 0 before the scatter")
+        for i, g in enumerate(gates):
+            regs, pairs, _ = gate_regs(g)
+            need(all(0 <= r < nreg for r in regs) and len(set(regs)) == len(regs), "E3: registers of gate %s%d" % (ph, i))
+            inc += len(regs)
+            for r in regs:
+                climb(r, g[1], "%s%d" % (ph, i))
+            for t, s, co in pairs:
+                kd = (cls(s), cls(t))
+                need(kd in okind, "E3: gate kind %s -> %s (gate %s%d)" % (kd + (ph, i)))
+                need(ph == "B" or kd in {("x", "s"), ("s", "s")}, "E3: bank gate before the scatter (gate A%d)" % i)
+                if kd == ("y", "y"):
+                    ysum[s, t] += co
+                    piv.add(s)
+                    ytg.add(t)
+            need(ph == "B" or all(cls(r) != "y" for r in regs), "E2: y role named in phase A")
+    for r in range(nreg):
+        climb(r, final[r], "final")
+    need(not (piv & ytg), "E4: a pivot is the target of a gate reading y")
+    need(all(x == 0 for x in ysum.values()), "E4: y-reads of a (pivot, target) pair do not cancel")
+    Hs = {k: {str(r): n for r, n in sorted(w.items())} for k, w in H.items()}
+    need(Hs == c["blocks"], "E6: block histograms")
+    N = sum(r * n for w in H.values() for r, n in w.items())
+    tails = sum(dim[f] for f in ext.values())
+    need(N == c["N"] and N + tails == R * h + 2 * v * (h - 1) + c["cst"], "E6: N")
+    if stats is not None:
+        stats.update(steps=len(steps), incidences=inc, blocks=sum(sum(w.values()) for w in H.values()),
+                     step_work=sum(dim[a] * dim[b] for a, b in steps), ysh_pairs=len(ysum))
+    if scalar:
+        scal(c, stats)
+        q2(c)
+    return True
+
+
+def scal(c, stats=None):
+    """E5: exact replay of the x-content of every register through A, scatter, B."""
+    h, v, R = c["h"], c["v"], c["R"]
+    nreg = 2 * v + R
+    reg = [({r: Fr(1)} if r < v else {}) for r in range(nreg)]
+    den = {"x": 1, "y": 1, "s": 1}
+    mx = {"x": Fr(0), "y": Fr(0), "s": Fr(0)}
+    cls = lambda r: "x" if r < v else "y" if r < 2 * v else "s"
+
+    def axpy(t, src, co):
+        d = reg[t]
+        for k, u in src.items():
+            w = d.get(k, 0) + co * u
+            if w:
+                d[k] = w
+                kk = cls(t)
+                if w.denominator != 1:
+                    den[kk] = lcm(den[kk], w.denominator)
+                if abs(w) > mx[kk]:
+                    mx[kk] = abs(w)
+            else:
+                d.pop(k, None)
+
+    def run(gates):
+        for g in gates:
+            if g[0] == "neg":
+                reg[g[2]] = {k: -u for k, u in reg[g[2]].items()}
+            elif g[0] == "out":
+                src = dict(reg[g[2]])
+                for t, a, b in g[3]:
+                    axpy(t, src, Fr(a, b))
+            else:
+                for s, a, b in g[3]:
+                    axpy(g[2], reg[s], Fr(a, b))
+    run(c["A"])
+    tot = {k: dict(reg[s]) for k, s, _ in c["ret"]}
+    for t in range(v):
+        for k, co in scat_row(c, t):
+            axpy(v + t, tot[k], co)
+    run(c["B"])
+    need(all(reg[t] == {t: 1} for t in range(v)), "E5: an x role is not restored (X3)")
+    need(all(reg[v + t] == {t: 1} for t in range(v)), "E5: a target does not receive exactly its source (hid)")
+    if stats is not None:
+        stats.update(den=dict(den), maxabs={k: str(x) for k, x in mx.items()},
+                     gates_A=len(c["A"]), gates_B=len(c["B"]))
+
+
+def q2(c):
+    """E7 (only for files with exterior gauges, rule (2)): for every slot q that starts off the zero frame, the
+    column of q in the y rows of the product is zero (its dirt is removed in line, the derived dirt gate at
+    frame 0 does not touch it).  Replay with the gauged slots as variables, x = 0."""
+    h, v, R = c["h"], c["v"], c["R"]
+    ext = [r for r, _ in c["ext"]]
+    if not ext:
+        return True
+    reg = [dict() for _ in range(2 * v + R)]
+    for r in ext:
+        reg[r] = {r: Fr(1)}
+
+    def axpy(t, src, co):
+        d = reg[t]
+        for k, u in src.items():
+            w = d.get(k, 0) + co * u
+            if w:
+                d[k] = w
+            else:
+                d.pop(k, None)
+
+    def run(gates):
+        for g in gates:
+            if g[0] == "neg":
+                reg[g[2]] = {k: -u for k, u in reg[g[2]].items()}
+            elif g[0] == "out":
+                src = dict(reg[g[2]])
+                for t, a, b in g[3]:
+                    axpy(t, src, Fr(a, b))
+            else:
+                for s, a, b in g[3]:
+                    axpy(g[2], reg[s], Fr(a, b))
+    run(c["A"])
+    tot = {k: dict(reg[s]) for k, s, _ in c["ret"]}
+    for t in range(v):
+        for k, co in scat_row(c, t):
+            if tot[k]:
+                axpy(v + t, tot[k], co)
+    run(c["B"])
+    need(all(not reg[v + t] for t in range(v)), "E7: the dirt of an exterior-gauged slot reaches a y role (Q2)")
+    return True
diff --git a/research/format-obstructions/inputs/gcert1-p10-anneal-mw-flow.json.gz b/research/format-obstructions/inputs/gcert1-p10-anneal-mw-flow.json.gz
new file mode 100644
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diff --git a/research/format-obstructions/inputs/gcert1-p11-pr193.json.gz b/research/format-obstructions/inputs/gcert1-p11-pr193.json.gz
new file mode 100644
index 0000000000000000000000000000000000000000..a98026a72f574fce17edd94bcaff10e15a314736
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diff --git a/research/format-obstructions/inputs/gcert1-p3-cone-R16.json b/research/format-obstructions/inputs/gcert1-p3-cone-R16.json
new file mode 100644
index 0000000000..8908f77525
--- /dev/null
+++ b/research/format-obstructions/inputs/gcert1-p3-cone-R16.json
@@ -0,0 +1 @@
+{"format": "gcert/1", "derived_from": "p3 cone-A + B0", "p": 0, "h": 6, "v": 8, "R": 16, "ports": [21, 37, 25, 41, 22, 38, 26, 42], "frames": [[], [32, 16, 8, 4, 2, 1], [21], [37], [25], [41], [22], [38], [26], [42], [37, 21], [25, 3], [41, 3], [38, 12], [21, 12], [37, 12], [22, 12], [21, 12, 3], [37, 21, 12], [38, 22], [37, 12, 3], [38, 22, 12], [37, 21, 12, 3], [41, 25, 3], [37, 21, 3], [21, 15], [42, 25], [37, 22], [51, 15], [38, 15], [38, 21, 15], [37, 22, 15], [35, 16, 9, 6], [33, 18, 11, 4], [33, 18, 8, 7], [34, 17, 11, 4], [33, 17, 8, 4, 3], [33, 16, 8, 5, 2], [34, 16, 10, 4, 1], [34, 16, 8, 6, 1], [32, 17, 8, 5, 2], [32, 17, 9, 4, 2], [33, 16, 9, 4, 2], [32, 18, 8, 6, 1], [32, 18, 10, 4, 1]], "start": [2, 3, 4, 5, 6, 7, 8, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], "final": [1, 1, 1, 1, 1, 1, 1, 1, 40, 37, 41, 42, 43, 39, 44, 38, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], "ext": [], "ret": [[0, 16, 18], [1, 17, 21], [2, 18, 24], [3, 19, 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\ No newline at end of file
diff --git a/research/format-obstructions/inputs/pr425/FULL-WORD.json.gz b/research/format-obstructions/inputs/pr425/FULL-WORD.json.gz
new file mode 100644
index 0000000000000000000000000000000000000000..28cb945cfff79b1439aa717e8e8368017e981444
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diff --git a/research/format-obstructions/inputs/pricing/cost.py b/research/format-obstructions/inputs/pricing/cost.py
new file mode 100644
index 0000000000..677584c1c6
--- /dev/null
+++ b/research/format-obstructions/inputs/pricing/cost.py
@@ -0,0 +1,62 @@
+"""Fast exact cost evaluator for a five-stage banked bit core-block word.
+Word format (PR249 snapshot): records = int32 six-tuples
+  MOVE (0, role, old_frame, new_frame, rank, z)
+  ADD  (1, dst, src, coeff, frame, category)
+  COPY (2, center, temp, copy_frame, target_frame, rank)   [paid child of given rank]
+  ERASE(3, ...)
+"""
+import json, struct, collections
+import numpy as np
+from mpmath import mp, mpf, findroot
+mp.dps = 50
+
+def load_snapshot(d):
+    s = json.load(open(f'{d}/249-states.json'))
+    fr = json.load(open(f'{d}/frames.json'))
+    dim = {int(k): len(v['B']) for k, v in fr['frames'].items()}
+    rec = np.frombuffer(open(f'{d}/249-records.bin','rb').read(), dtype='<i4').reshape(-1, 6)
+    return s, fr, dim, rec
+
+def local_profile(rec, s, dim, h=24):
+    """Return (local child histogram incl. copies, sum of helper residuals, v)."""
+    v = s['v']; n = s['n']
+    H = collections.Counter()
+    moves = rec[rec[:, 0] == 0]
+    for r in moves:
+        if r[4] > 0: H[int(r[4])] += 1
+    copies = rec[rec[:, 0] == 2]
+    for r in copies:
+        if r[5] > 0: H[int(r[5])] += 1
+    init = s['initial']
+    resid = sum(h - dim[init[str(r)]] for r in range(2 * v, n))
+    return H, resid, v
+
+def five_stage(H, resid, v, h=24, stages=5):
+    m = stages * h
+    prof = collections.Counter()
+    for j, c in H.items(): prof[j] += stages * c
+    for j in (2*h - 2, h - 1, 2*h + 2, 4): prof[j] += 2 * v
+    W = mpf(4 * v) + mpf(resid) / h
+    mass = sum(j * c for j, c in prof.items())
+    return prof, W, m, W * m - mass
+
+def solve_moment(prof, W, m, bad=mpf('1e-16')):
+    calls = sum(prof.values())
+    items = [(mpf(j) / m, c) for j, c in prof.items()]
+    def mu(sv):
+        t = 1 - sv
+        return (sum(c * x ** t for x, c in items) + bad * 32 * m * m * calls * (mpf(1) / m) ** t) / W - 1
+    return findroot(mu, mpf('0.0007'))
+
+def evaluate(rec, s, dim, h=24):
+    H, resid, v = local_profile(rec, s, dim, h)
+    prof, W, m, deficit = five_stage(H, resid, v, h)
+    sv = solve_moment(prof, W, m)
+    return dict(saving=sv, kappa_approx=sv / (1 + sv), W=W, deficit=deficit, m=m,
+                calls=sum(prof.values()), local=dict(sorted(H.items())), resid=resid)
+
+if __name__ == '__main__':
+    import sys
+    s, fr, dim, rec = load_snapshot(sys.argv[1])
+    r = evaluate(rec, s, dim)
+    print({k: (float(v) if hasattr(v, 'real') and not isinstance(v, dict) else v) for k, v in r.items() if k != 'local'})
diff --git a/research/format-obstructions/inputs/pricing/hist_p10b.json b/research/format-obstructions/inputs/pricing/hist_p10b.json
new file mode 100644
index 0000000000..f1ec3796b4
--- /dev/null
+++ b/research/format-obstructions/inputs/pricing/hist_p10b.json
@@ -0,0 +1 @@
+{"m": 100, "W": 131760, "histogram": {"1": 1284000, "10": 14880, "11": 4800, "12": 60360, "13": 120, "14": 20640, "16": 57600, "17": 152160, "18": 58800, "19": 23040, "2": 548940, "3": 363900, "38": 23040, "4": 104580, "42": 23040, "5": 65520, "6": 60360, "7": 38400, "8": 9840, "9": 17580}, "deficit": 24480, "bit_root": "385138454781521/500000000000000000", "kappa": "0.000769684039383334"}
\ No newline at end of file
diff --git a/research/format-obstructions/requirements.txt b/research/format-obstructions/requirements.txt
new file mode 100644
index 0000000000..98c7bf4d74
--- /dev/null
+++ b/research/format-obstructions/requirements.txt
@@ -0,0 +1,4 @@
+python-sat==1.9.dev15
+networkx==3.7
+mpmath==1.3.0
+numpy==2.3.5
diff --git a/research/format-obstructions/verify.py b/research/format-obstructions/verify.py
new file mode 100644
index 0000000000..10c5e574a2
--- /dev/null
+++ b/research/format-obstructions/verify.py
@@ -0,0 +1,213 @@
+#!/usr/bin/env python3
+"""Format obstructions: run every exact check behind PROOF.md, offline, from pinned inputs.
+
+usage: python3 -B research/format-obstructions/verify.py [--output DIR] [--jobs N]
+
+Requires Python 3.11+ (tested: 3.12) and the pinned packages of requirements.txt (python-sat for the CaDiCaL checks,
+networkx, mpmath, numpy). Assertions must stay enabled (-O is refused). No network. The independent stages run in
+a process pool; the result facts are compared with expected.json (every number below is pinned there).
+
+Stages (see README.md for what each one establishes):
+  certificates  gx.check1 (Sussman's reference checker, vendored unchanged) on three gcert/1 programs, then Thm A (taint
+                replay, every gate), Cor A1, Thm B data (neighbourhood rank), Thm C (copy lower bound vs. paid);
+                negative controls on the small program.
+  toy_p3        the one read-discipline gain (cst 18 -> 16) and its exhaustive optimum; control.
+  pivots        Theorem P (cross-Gram forced to zero), the copy count and the contamination scan of PROOF.md section 7.
+  mutable_425   PR #425's mutable-basis word: Thm A/B/C replays, exact scalar identity, Theorem S conversion; controls.
+  pricing       the pinned pricer on the p = 10 bit histogram (pivot-free variants, copy and descent units).
+  families      neighbourhood ranks and copy bounds for the port families, clique numbers, the explicit 20-clique.
+  e8_labels     SAT: uniqueness of the h = 9 labelling, k = 2 planes UNSAT at r = 16, 17; calibrations and controls.
+"""
+import argparse
+import hashlib
+import importlib
+import importlib.metadata as md
+import json
+import multiprocessing as mp
+import os
+import sys
+import tempfile
+import time
+import traceback
+from concurrent.futures import ProcessPoolExecutor, as_completed
+from pathlib import Path
+
+if sys.flags.optimize:
+    raise SystemExit("Assertions must remain enabled: run without -O")
+sys.dont_write_bytecode = True
+os.environ["PYTHONDONTWRITEBYTECODE"] = "1"
+ROOT = Path(__file__).resolve().parent
+sys.path.insert(0, str(ROOT / "checks"))
+sys.path.insert(0, str(ROOT / "gx"))
+
+PINNED_PACKAGES = {"python-sat": "1.9.dev15", "networkx": "3.7", "mpmath": "1.3.0", "numpy": "2.3.5"}
+P10 = str(ROOT / "inputs" / "gcert1-p10-anneal-mw-flow.json.gz")
+P11 = str(ROOT / "inputs" / "gcert1-p11-pr193.json.gz")
+
+
+def sha(path):
+    return hashlib.sha256(Path(path).read_bytes()).hexdigest()
+
+
+def package_files():
+    return sorted(p.relative_to(ROOT).as_posix() for p in ROOT.rglob("*") if p.is_file() and "__pycache__" not in p.parts)
+
+
+def integrity():
+    pins = json.loads((ROOT / "SOURCE.json").read_text())["files"]
+    for rel, want in pins.items():
+        assert sha(ROOT / rel) == want, "pinned input changed: " + rel
+    return len(pins)
+
+
+def tasks():
+    """(id, module, function, args). Long tasks first."""
+    t = [
+        ("cert_p11", "certs", "stage", ("p11",)),
+        ("cert_p10", "certs", "stage", ("p10",)),
+        ("overlap_p10", "pivots", "overlap_scan", (P10,)),
+        ("copy_count_p10", "pivots", "copy_count", (P10,)),
+        ("omega_cx_7", "families", "incap_omega", (7, "cx")),
+        ("thmP_p11", "pivots", "thm_P", (P11,)),
+        ("thmP_p10", "pivots", "thm_P", (P10,)),
+        ("cert_p3", "certs", "stage", ("p3",)),
+        ("toy_p3", "toy", "run", ()),
+        ("mutable_425", "mutable425", "run", ()),
+        ("pricing", "pricing", "run", ()),
+        ("e8_bitdual", "families", "e8_bitdual", ()),
+        ("p10_clique", "families", "p10_clique", (P10,)),
+        ("h9_unique", "sat_e8", "uniqueness_h9", ()),
+        ("calibrations", "sat_e8", "calibrations", ()),
+        ("e8_cliques", "sat_e8", "e8_clique_facts", ()),
+    ]
+    for p in (4, 5, 10, 11):
+        t.append((f"bit_p{p}", "families", "paired_bit", (p,)))
+    for p in (4, 5, 6, 7):
+        t.append((f"omega_bit_{p}", "families", "incap_omega", (p, "bit")))
+        if p != 7:
+            t.append((f"omega_cx_{p}", "families", "incap_omega", (p, "cx")))
+    sat = importlib.import_module("sat_e8")
+    npats = {}
+    for r in (16, 17):
+        chunks, n = sat.k2_tasks(r, 32)
+        npats[r] = n
+        for i, ch in enumerate(chunks):
+            t.append((f"k2_r{r}_{i}", "sat_e8", "k2_solve", (ch,)))
+    return t, npats
+
+
+def work(spec):
+    mod, fn, args = spec
+    sys.dont_write_bytecode = True
+    t0 = time.monotonic()
+    out = getattr(importlib.import_module(mod), fn)(*args)
+    return out, time.monotonic() - t0
+
+
+def strip_seconds(x):
+    if isinstance(x, dict):
+        return {k: strip_seconds(v) for k, v in x.items() if k != "seconds"}
+    if isinstance(x, list):
+        return [strip_seconds(v) for v in x]
+    return x
+
+
+def assemble(res, npats):
+    import families
+    import sat_e8
+    f = {}
+    f["certificates"] = {k: res[f"cert_{k}"] for k in ("p3", "p10", "p11")}
+    f["toy_p3"] = res["toy_p3"]
+    f["pivots"] = dict(thm_P={"p10": res["thmP_p10"], "p11": res["thmP_p11"]}, copy_count_p10=res["copy_count_p10"],
+                       overlap_scan_p10=res["overlap_p10"])
+    f["mutable_425"] = res["mutable_425"]
+    f["pricing"] = res["pricing"]
+    u_res, u = res["h9_unique"]
+    fam = dict(paired_bit={str(p): res[f"bit_p{p}"] for p in (4, 5, 10, 11)}, e8_bitdual=res["e8_bitdual"],
+               e8_complex=families.e8_complex(u), p10_clique=res["p10_clique"],
+               omega={f"{k}_{p}": res[f"omega_{k}_{p}"] for p in (4, 5, 6, 7) for k in ("bit", "cx")
+                      if f"omega_{k}_{p}" in res})
+    f["families"] = fam
+    k2 = {}
+    for r in (16, 17):
+        parts = [res[f"k2_r{r}_{i}"] for i in range(32)]
+        k2[f"r{r}"] = sat_e8.k2_collect(r, parts, npats[r])
+    f["e8_labels"] = dict(uniqueness_h9=u_res, calibrations=res["calibrations"], cliques=res["e8_cliques"],
+                          k2_planes=k2)
+    # Thm B table, derived from the measured neighbourhood ranks: m >= 1 + s * rank N versus the layout widths s * h
+    tb = {}
+    for name, h, rank in (("complex_p10", 20, f["certificates"]["p10"]["N_rank"]),
+                          ("complex_p11", 22, f["certificates"]["p11"]["N_rank"]),
+                          ("bit_p10", 20, [fam["paired_bit"]["10"]["N_rank"]] * 2),
+                          ("bit_p11", 22, [fam["paired_bit"]["11"]["N_rank"]] * 2),
+                          ("e8_bit_dual", 8, [fam["e8_bitdual"]["neighbourhood"][0][1]] * 2),
+                          ("e8_complex", 9, [fam["e8_complex"]["N_rank"]] * 2)):
+        assert rank == [h - 1, h - 1], (name, rank)
+        tb[name] = {f"s{s}": dict(m_min=1 + s * (h - 1), layout_s_times_h=s * h) for s in (3, 5, 7, 9)}
+    f["thmB_table"] = tb
+    return f
+
+
+def main():
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument("--output", type=Path, help="new directory (outside the package) for facts.json / verification.json")
+    ap.add_argument("--jobs", type=int, default=max(1, min(12, (os.cpu_count() or 2))))
+    ap.add_argument("--write-expected", action="store_true", help="maintenance: rewrite expected.json from this run")
+    args = ap.parse_args()
+    assert sys.version_info >= (3, 11), "Python 3.11+ required"
+    for pkg, ver in PINNED_PACKAGES.items():
+        got = md.version(pkg)
+        assert got == ver, f"{pkg}=={ver} required, found {got} (python3 -m pip install -r requirements.txt)"
+    out = args.output or Path(tempfile.mkdtemp(prefix="format-obstructions-"))
+    if args.output:
+        assert not out.resolve().is_relative_to(ROOT) and not out.exists(), "output must be new and outside the package"
+        out.mkdir(parents=True)
+    before = package_files()
+    npins = integrity()
+    T0 = time.monotonic()
+    tk, npats = tasks()
+    res, secs = {}, {}
+    print(f"[{time.monotonic() - T0:6.1f}s] {npins} pinned inputs verified; {len(tk)} tasks on {args.jobs} workers", flush=True)
+    ctx = mp.get_context("spawn")
+    with ProcessPoolExecutor(max_workers=args.jobs, mp_context=ctx) as ex:
+        fut = {ex.submit(work, (m, fn, a)): tid for tid, m, fn, a in tk}
+        for f in as_completed(fut):
+            tid = fut[f]
+            try:
+                res[tid], secs[tid] = f.result()
+            except BaseException:
+                traceback.print_exc()
+                raise SystemExit(f"FAILED task {tid}")
+            if not tid.startswith("k2_") or len([k for k in res if k.startswith("k2_")]) % 16 == 0:
+                print(f"[{time.monotonic() - T0:6.1f}s] done {tid} ({secs[tid]:.1f}s)", flush=True)
+    facts = assemble(res, npats)
+    wall = time.monotonic() - T0
+    norm = json.loads(json.dumps(strip_seconds(facts), sort_keys=True))
+    exp_path = ROOT / "expected.json"
+    if args.write_expected:
+        exp_path.write_text(json.dumps(norm, indent=1, sort_keys=True) + "\n")
+        print("expected.json written")
+    else:
+        want = json.loads(exp_path.read_text())
+        assert norm == want, "facts differ from expected.json: " + ", ".join(
+            k for k in sorted(set(norm) | set(want)) if norm.get(k) != want.get(k))
+    after = package_files()
+    if args.write_expected:
+        after = [x for x in after if x != "expected.json"]
+    assert after == [x for x in before if not (args.write_expected and x == "expected.json")], \
+        "the package tree changed during the run"
+    stage_secs = {}
+    for tid, s in secs.items():
+        key = "k2_planes" if tid.startswith("k2_") else tid
+        stage_secs[key] = round(stage_secs.get(key, 0.0) + s, 1)
+    report = dict(status="PASS", python=sys.version.split()[0], packages=PINNED_PACKAGES, pinned_inputs=npins,
+                  tasks=len(tk), jobs=args.jobs, wall_seconds=round(wall, 1), cpu_seconds_sum=round(sum(secs.values()), 1),
+                  stage_cpu_seconds=dict(sorted(stage_secs.items(), key=lambda kv: -kv[1])))
+    (out / "facts.json").write_text(json.dumps(facts, indent=1, sort_keys=True) + "\n")
+    (out / "verification.json").write_text(json.dumps(report, indent=1) + "\n")
+    print(json.dumps(report), flush=True)
+    print(f"PASS: all format-obstruction checks, wall {wall:.1f}s (output in {out})", flush=True)
+
+
+if __name__ == "__main__":
+    main()

From 309482849b6ddf34db998f53612d34c19f6e3233 Mon Sep 17 00:00:00 2001
From: DreamingOfClouds <Ellerbroek.Mason@gmail.com>
Date: Sun, 11 Oct 2026 10:55:04 -0500
Subject: [PATCH 2/2] Re-pin research/joint-dual/SOURCE.json to main's current
 Makefile and README.md

main's repo-wide 'joint' verify job fails because the pinned root Makefile and
README.md hashes are stale; this updates exactly those two pins (no other change).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
---
 research/joint-dual/SOURCE.json | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/research/joint-dual/SOURCE.json b/research/joint-dual/SOURCE.json
index e3a8386b17..e1fbf9c5aa 100644
--- a/research/joint-dual/SOURCE.json
+++ b/research/joint-dual/SOURCE.json
@@ -3,9 +3,9 @@
   "base_pr57_commit": "cd350f76c9bc01489ec83568bded532cb69be938",
   "base_pr58_commit": "bc2f7ed4c20dc18898305ab17165c0c995cbb804",
   "files": {
-    "Makefile": "768131bd4ce18433f6abccdc9812a38c92e11220e3a1019c6d95334b42f529d1",
+    "Makefile": "cd0fb59dc078d1fdf54a163ea13bb3660d03c8fe69c1a7f99fc8e5f476a3d5c2",
     "NOTICE": "b49022cb4104d90b182e4c335f670faf45529025b15db77bf0291835083c28bd",
-    "README.md": "d104c31d5b7e0d3a8d011a0e3517ab615e5b499b3dc05684f7ad3cadcd28bdae",
+    "README.md": "b81f8035b859007add6693c6d860371c9100a6766e494045495a28a457a5fcb9",
     "certificates/joint-dual-compiler.json": "516c4d85158553dabd0a98216a064b2a7397957a9d5dd0df1d6e5aec53a4e805",
     "certificates/joint-dual-profiles-23.json": "159fa8fee175fc1d24f63e72d3c2f1dc9f4fcd78b938c5e7d2f735fd0fd37209",
     "certificates/joint-dual-profiles-25.json": "8d190ce42d097e828c5756918c52f2963df576cf4c6600b0fc7f2b7fdb6ad223",