Certify PR288's restricted sign-gauge obstruction with four constraints (no new κ)

.github/workflows/verify.yml

@@ -9,6 +9,25 @@ permissions:
   contents: read
 
 jobs:
+  pr288-sign-cycle:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 5
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.11', '3.13', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Check the restricted sign-cycle certificate and corruption controls
+        run: make pr288-sign-cycle-verify
+      - name: Require unchanged tracked artifacts
+        run: git diff --exit-code
+
   verify:
     runs-on: ubuntu-24.04
     timeout-minutes: 35

Makefile

@@ -13,6 +13,7 @@ community-followup-check:
 # CI runs these in separate checkouts. Keep local verification sequential:
 # different groups regenerate certificates that another group may read.
 verify:
+	$(MAKE) pr288-sign-cycle-verify
 	$(MAKE) verify-community
 	$(MAKE) verify-producers
 	$(MAKE) verify-partial-gauge
@@ -31,6 +32,10 @@ verify:
 	$(MAKE) verify-pair
 	$(MAKE) verify-tests
 
+.PHONY: pr288-sign-cycle-verify
+pr288-sign-cycle-verify:
+	python3 -B research/pr288-sign-cycle-certificate/run_checks.py
+
 .PHONY: selected-record-check entrance-bank-verify verify-entrance-banks
 selected-record-check:
 	python3 -B scripts/verify_selected_result.py

research/pr288-sign-cycle-certificate/NOTICE

@@ -0,0 +1,18 @@
+PR288 sign-gauge obstruction certificate
+
+Mark Braverman and Zhongtian He supplied the published graph and numerical
+instance, arXiv:2610.10108v1. eumemic reported the restricted sign-gauge
+obstruction in CrocSwap/integer-mult-bounds PR288 and prepared its original
+checker with substantial Anthropic Claude assistance.
+
+The explicit certificate, separate checker, producer, tests and proof
+exposition were prepared with substantial OpenAI ChatGPT assistance,
+including separate assistant reviews. The local integration and checks
+used OpenAI Codex. This is not independent human peer review or Lean
+formal verification. Douglas Colkitt and the CrocSwap/OpenAI/community
+contributors supplied the host framework and review conventions.
+
+New material is offered under the repository's Apache License, Version 2.0.
+Published numerical facts retain attribution. No companion source code
+or full paper is distributed or relicensed here. Existing repository
+notices remain applicable and unchanged.

research/pr288-sign-cycle-certificate/PROOF.md

@@ -0,0 +1,180 @@
+# Two sessions certify the PR288 sign-gauge obstruction
+
+## Status and attribution
+
+This is a new explicit certificate and independent reconstruction of the
+obstruction already reported by eumemic in PR288, §1.4(d). The obstruction
+theorem itself is not claimed as new. Braverman and He supplied the graph and
+terminal pairs in arXiv:2610.10108v1, §4 and Appendix A. Exact source pins and attribution are recorded in `SOURCES.md`.
+
+The proof below is an elementary written argument. The finite path premises
+are checked by `check_certificate.py`. The argument has been examined by a
+separate assistant agent in the same AI-assisted workflow; it has not been
+formally checked in Lean or independently reviewed by a human.
+
+## 1. Exact statement
+
+Let G be the 182-vertex undirected physical graph defined below. There is no
+assignment of fixed additive matrices A_v in M_5(Q), one for each vertex, with:
+
+1. rank(A_v - A_u) <= 1 on every physical edge {u,v};
+2. one fixed invertible C_0 in GL_5(Q);
+3. for each session (p,l), A_l - A_p = sigma_(p,l) C_0, where each session
+   independently chooses sigma_(p,l) in {+1,-1}.
+
+It suffices to impose item 3 on just the two sessions (0,155) and (3,93).
+This includes the original sign-gauge model with C_0 invertible lower
+triangular. It does not cover independently chosen nonproportional Levi
+targets, higher-rank physical edges, or non-additive/nonabelian frames.
+
+For convenience every session is oriented point-to-line. The paper orients
+each session from its earlier to its later vertex in pi. Reversing one
+session replaces its free sign variable sigma by -sigma, a bijective change
+of variables. Thus this orientation choice changes no existence question.
+It does not assume any symmetry of the unknown matrices A_v.
+
+## 2. Exact finite instance
+
+Construct F_9 = F_3[t]/(t^2+1). Write eta_r = (r mod 3) + floor(r/3)t for
+0 <= r <= 8. The ordered projective representatives are
+
+    v_0 = (0,0,1)
+    v_(1+r) = (0,1,eta_r)
+    v_(10+9r+s) = (1,eta_r,eta_s).
+
+Vertex i represents point v_i; vertex 91+j represents line v_j. An incidence
+edge joins i to 91+j exactly when v_i dot v_j = 0 in F_9. Number the 910
+candidate edges lexicographically by (i,j), starting at zero. If the ten
+incident line indices for point i are j_(i,0) < ... < j_(i,9), then
+e_(10i+r) = {i,91+j_(i,r)}.
+
+Remove the 175 edges R listed in `instance.json`. The remaining 735 edges
+are physical. The 157 session indices are B = R minus (C union J), where C
+and J are the disjoint nine-element sets in the same file. The complete
+lists match the paper and its pinned companion data. PR288's raw JSON has a
+176th R entry, the paper page number 13; the original PR checker explicitly
+discards it. Our normalized data records the 175 actual edge indices.
+The immutable source links and numerical normalization are documented in
+`SOURCES.md`; the checker binds the normalized file to its reviewed SHA-256.
+
+F_9 defines only this combinatorial graph. The matrix ranks and traces in
+the theorem are over Q. They are not ranks or traces over the bit payload
+field F_2.
+
+## 3. A rank-tight path forces unit traces
+
+**Lemma.** Over any field, if m matrices Q_1,...,Q_m of rank at most one sum
+to I_m, each Q_j has trace 1. In fact Q_i Q_j = delta_(i,j) Q_i.
+
+**Proof.** Factor Q_j = u_j v_j^T, allowing zero vectors initially. Let U
+have columns u_j, and let V have rows v_j^T. Then UV = I_m. Since U and V
+are square, they are invertible and VU = I_m. Therefore
+v_i^T u_j = delta_(i,j). In particular trace(Q_j) = v_j^T u_j = 1, and
+the asserted multiplication identity follows. This also proves that no
+Q_j is zero. The term "orthogonal idempotents" here would mean this
+algebraic multiplication identity, not orthogonal or Hermitian projections.
+
+Now take any path p=x_0,x_1,...,x_5=l of exactly five physical edges for a
+session whose endpoint difference is sigma C_0. Put
+
+    Q_j = sigma C_0^(-1) (A_(x_j) - A_(x_(j-1))).
+
+These five matrices have rank at most one and sum to I_5, since sigma^2=1.
+The lemma implies
+
+    trace(C_0^(-1) (A_(x_j) - A_(x_(j-1)))) = sigma.
+
+No shortest-path algorithm is used in this implication. A path of length
+five and a rank-five target suffice. Rank subadditivity would itself
+prohibit a shorter path under the assumed embedding. Paths need not
+follow the coding paper's causal orientation.
+
+Orient each physical edge from point to line and define
+
+    tau_e = trace(C_0^(-1) (A_line - A_point)).
+
+If a chosen session path traverses e in direction epsilon in {+1,-1}
+relative to this orientation, then
+
+    tau_e = epsilon sigma, equivalently sigma tau_e = epsilon.
+
+Thus the same fixed edge variable must satisfy every path constraint in
+which that edge is used. This is a necessary condition for an embedding;
+consistency would not establish sufficiency.
+
+## 4. Three explicit paths
+
+The certificate uses session edge indices 7 and 30 and physical edge
+indices 114 and 113:
+
+| Index | Kind | Endpoints in point-to-line order |
+| --- | --- | --- |
+| 7 | Session | (0,155) |
+| 30 | Session | (3,93) |
+| 114 | Physical edge | (11,130) |
+| 113 | Physical edge | (11,121) |
+
+The three distinct path witnesses are:
+
+| Session | Five-edge path | Candidate edge indices along path |
+| --- | --- | --- |
+| 7 | 0, 92, 11, 130, 66, 155 | 0, 110, 114, 664, 667 |
+| 7 | 0, 92, 11, 121, 68, 155 | 0, 110, 113, 683, 687 |
+| 30 | 3, 121, 11, 130, 78, 93 | 33, 113, 114, 784, 780 |
+
+Every listed edge is an incidence edge outside R. Both session indices
+belong to B. The independent checker reconstructs these facts directly
+from F_9 and the source-bound input. The union uses only eleven physical
+edges and eleven vertices.
+
+The session-7 paths traverse edges 114 and 113 in point-to-line direction.
+The session-30 path traverses edge 114 in point-to-line direction but
+edge 113 in the reverse direction. Hence
+
+    tau_114 = sigma_7,
+    tau_114 = sigma_30,
+    tau_113 = -sigma_30,
+    tau_113 = sigma_7.
+
+The first pair gives sigma_7 = sigma_30; the second pair gives
+sigma_7 = -sigma_30. This is impossible over Q for signs in {+1,-1}.
+This proves the theorem.
+
+Equivalently, the session/edge cycle
+
+    session 7, edge 114, session 30, edge 113, session 7
+
+has constraint signs +1,+1,-1,+1, whose product is -1. Every sign variable
+occurs twice in the product of the four equations, forcing the product to
+be +1. The exact certificate stores four incidence records, with the
+session-30 path reused for two records.
+
+## 5. Coverage and trusted components
+
+This is a finite contradiction witness, not a search-based exclusion.
+The proof universally covers all rational vertex matrices, all invertible
+common C_0, and all 2^157 choices of session signs under the stated
+premises: any proposed embedding must satisfy these four equations.
+There is no need to enumerate the sign assignments, possible matrices,
+all paths, or all geodesic incidences.
+
+The discovery producer may be wrong without compromising soundness of an
+accepted certificate: the checker imports no producer or upstream code,
+rebuilds the graph by a different method, checks the fixed instance hash,
+validates every path and selected incidence, recomputes every sign, checks
+cycle closure and checks the negative product. Its implementation, the
+Python interpreter, the written lemma and the provenance binding remain
+part of the trusted argument. No solver status is treated as proof.
+
+## 6. What this does not establish
+
+- No reproduction or closure of PR288's m=3 SAT, MITM or Groebner reports.
+- No general obstruction to distinct Levi targets or nonabelian frames.
+- No conclusion when paths have more than five edges or edges may have
+  rank greater than one; the trace lemma needs exactly m rank-one terms.
+- No disproof or re-verification of the Braverman-He network-coding theorem.
+- No new multiplication exponent, practical benchmark, or complete formal
+  verification of integer multiplication.
+- No worldwide novelty claim. The already reported obstruction retains
+  its attribution; the explicit certificate, separate checker and tests
+  are the contribution prepared here.

research/pr288-sign-cycle-certificate/README.md

@@ -0,0 +1,40 @@
+# Four constraints certify PR288's sign-gauge obstruction
+
+This package gives an explicit certificate and a separate exact checker for
+the obstruction already reported by eumemic in PR288 §1.4(d). It adds no
+multiplication bound. Braverman and He supplied the PG(2,9) graph and sessions.
+Read [PROOF.md](PROOF.md) for the complete statement and ordinary proof,
+[SOURCES.md](SOURCES.md) for immutable sources, and [NOTICE](NOTICE) for credits.
+
+The restricted model has fixed additive 5×5 rational vertex matrices,
+rank-at-most-one physical edge differences, and each session target equal
+to ± one common invertible matrix. Three five-edge paths force sessions 7
+and 30 to have both equal and opposite signs. Their four constraints have
+sign product −1. This excludes all assignments in that model; it covers
+neither unrelated Levi targets nor nonadditive frames or higher-rank edges.
+
+Run from the repository root with Python 3.11 or newer:
+
+```sh
+make pr288-sign-cycle-verify
+```
+
+Or run `python3 -B run_checks.py` in this directory. The standard-library
+runner checks the certificate, runs 24 tests normally and under `-O`,
+regenerates into a temporary directory, requires identical certificate
+bytes, and checks the new file. Tests include a geometrically valid
+balanced cycle, a false sign, removed path edges, wrong endpoints and
+source-pin rebinding. No dependency or network access is needed.
+
+The checker reconstructs the graph by projective-vector normalization and
+imports neither the producer nor upstream checker. The producer uses a
+separate direct construction and BFS; it is not trusted by the verifier.
+The matrix-to-sign implication remains an ordinary written proof. This is
+not human peer review, a Lean proof, or a full multiplication theorem audit.
+The unprovided PR288 SAT/MITM/Gröbner searches are outside this package.
+
+This small integration omits the original handoff's large PR snapshots,
+copied upstream documents and historical execution receipts. It retains
+the complete witness and mathematical premises. Validation of this local
+integration is reported separately; no broad gate success is asserted here.
+New work is Apache-2.0 with substantial OpenAI ChatGPT/Codex assistance.

research/pr288-sign-cycle-certificate/SOURCES.md

@@ -0,0 +1,43 @@
+# Immutable sources and numerical normalization
+
+- Host integration base: CrocSwap/integer-mult-bounds
+  `12a4934d15006e8c10f9b2264f13f8f341be802b`.
+- Existing obstruction and original checker: eumemic,
+  [PR288 §1.4(d)](https://github.com/CrocSwap/integer-mult-bounds/pull/288),
+  source head `475a088b5862f75f0b10596cfd6842d19531a5ec` in
+  [eumemic/integer-mult-bounds](https://github.com/eumemic/integer-mult-bounds/tree/475a088b5862f75f0b10596cfd6842d19531a5ec/research/negative-results-paradigm-search).
+- Graph and numerical lists: Mark Braverman and Zhongtian He,
+  [arXiv:2610.10108v1 §4 and Appendix A](https://arxiv.org/html/2610.10108v1).
+- Corroborating numerical data:
+  [companion repository](https://github.com/zhongtianhe/multiple-unicast-counterexample/blob/3436359a85d13a1bf6e87c3a3dbdaa64788277f4/data/182-vertex-code.json)
+  at `3436359a85d13a1bf6e87c3a3dbdaa64788277f4`, data Git blob
+  `d84f5b3996d5e3b8bf886dfde4fde6bfc02981ca`.
+
+PR288's `data/bh_pg29_lists.json` has Git blob
+`056b3ccbb37584b5214c2db12fc65c2ab5b41b86` and SHA-256
+`0e950af1a3aa07b12377780c8a82f9de97bef6f7f048a82c60c3312b0f6c5c77`.
+Its R list has 176 entries with a final 13. The original checker explicitly
+omits that paper page number. `instance.json` records the 175 actual indices,
+the complete 182-entry π permutation, and disjoint nine-entry C and J sets.
+All four lists agree with the primary paper and pinned companion data in
+the independent preparation audit. This normalization is not a newly found
+failure of PR288.
+
+The normalized input SHA-256, including whitespace, is
+`06f62ef3c3b5448a67fd1b8ff1edf69012f6aa07c7dd60b1f3c83baa5343dabb`.
+The verifier hard-pins that value and requires the certificate's hash to
+match it. The distributed certificate SHA-256 is
+`12a627a89864b90ed611d156520b7c5213566f405554ea08fb1361a4bf54ae0f`.
+These pins identify reviewed inputs; checking them is not a fresh remote
+provenance audit or a proof of the paper's theorem.
+
+The graph field is F₉=F₃[t]/(t²+1). Matrix ranks and traces are over Q.
+Point-to-line session orientation replaces the paper's π orientation by a
+bijective relabeling of independently free signs. No symmetry of unknown
+vertex matrices is assumed.
+
+PR288 remains the predecessor and receives theorem credit. This package
+adds a compact four-constraint witness and independently written checking
+code. A bounded PR title/body and term search through #343 found no compact
+duplicate; it establishes neither worldwide novelty nor exclusive ownership.
+The companion's source code and the full paper are not redistributed.

research/pr288-sign-cycle-certificate/certificate.json

@@ -0,0 +1,69 @@
+{
+  "schema": "pr288-negative-cycle-v1",
+  "dimension": 5,
+  "instance_sha256": "06f62ef3c3b5448a67fd1b8ff1edf69012f6aa07c7dd60b1f3c83baa5343dabb",
+  "cycle_nodes": [
+    [
+      "session",
+      7
+    ],
+    [
+      "edge",
+      114
+    ],
+    [
+      "session",
+      30
+    ],
+    [
+      "edge",
+      113
+    ]
+  ],
+  "incidences": [
+    {
+      "path": [
+        0,
+        92,
+        11,
+        130,
+        66,
+        155
+      ],
+      "sign": 1
+    },
+    {
+      "path": [
+        3,
+        121,
+        11,
+        130,
+        78,
+        93
+      ],
+      "sign": 1
+    },
+    {
+      "path": [
+        3,
+        121,
+        11,
+        130,
+        78,
+        93
+      ],
+      "sign": -1
+    },
+    {
+      "path": [
+        0,
+        92,
+        11,
+        121,
+        68,
+        155
+      ],
+      "sign": 1
+    }
+  ]
+}