Mister-Meeseeks · Oct 23, 2026 · Oct 23, 2026 · Oct 23, 2026 · Oct 23, 2026
diff --git a/research/spd-dirty-circuit-witness/NOTICE.md b/research/spd-dirty-circuit-witness/NOTICE.md
@@ -0,0 +1,23 @@
+# Attribution and frozen source
+
+New proof, degree adapter, literal replay and checks: Ali Romman, with substantial
+OpenAI Codex assistance. Apache-2.0.
+
+The two files in vendor and wrong-order.json are unchanged copies from
+[PR #5235](https://github.com/CrocSwap/integer-mult-bounds/pull/5235), head
+`48976daa4dce59b4867ef046da72a370e389925d`. The source is prepared for zxv with
+substantial OpenAI Codex assistance, under Apache-2.0. This PR is pending and
+is not part of our main base. SOURCES.json records each original path, Git blob
+and SHA256, and binds the retained copies to an ancestor in this publication
+history. The original proof and README are also retained in vendor. The replay
+checks the copied runtime bytes; metadata checks bind all retained sources. References inside the unchanged
+vendor files refer to the predecessor's proof, not to a claim of our authorship.
+
+Retain its weighted-automata/PIT credit to Stefan Kiefer and Ran Raz / Amir
+Shpilka. The echo lineage credits androolloyd, Chafik Boukhalfa / chafreaky
+[PR #628](https://github.com/CrocSwap/integer-mult-bounds/pull/628), and Ali Romman
+[PR #621](https://github.com/CrocSwap/integer-mult-bounds/pull/621).
+The source also retains Cinna Davis, ToikY, Sagar Thakrar, roger-sato, Charles
+Bethin, Ross Richardson and the wider dirty-phase/tensor lineage. Their results
+are not new results of this package. No identities are inferred from a PR's
+submitter account. The source README and proof are pinned provenance.
diff --git a/research/spd-dirty-circuit-witness/PROOF.md b/research/spd-dirty-circuit-witness/PROOF.md
@@ -0,0 +1,108 @@
+# Positive-definite separation of complete dirty circuits
+
+## Statement
+
+Use the complete linear register grammar from zxv's [PR #5235](https://github.com/CrocSwap/integer-mult-bounds/pull/5235),
+head `48976daa4dce59b4867ef046da72a370e389925d`. All input and returned roles
+are independent. Coefficients are central elements of Q or Q(i). Symbols have
+no inverse, commutation, Clifford or other relations. Compare two finite
+programs, or compare one program with an explicit polynomial endpoint.
+
+Let F be their matrix of noncommutative polynomial differences. If F is
+nonzero, let D be the largest word degree with a nonzero coefficient in any
+entry. For every rational 0 < e <= 1/4, there are real symmetric rational
+matrices A_a of dimension D+1 such that:
+
+- every spectrum is contained in [1-2e,1];
+- each matrix is positive definite, has operator norm at most 1, and has
+  spectral condition number at most 1/(1-2e) <= 2;
+- the literal complete endpoint difference is nonzero for one input role
+  equal to a coordinate vector and all other roles equal to zero.
+
+Thus identity on this class, in every payload dimension, is equivalent to
+free-polynomial identity. Taking e arbitrarily small also gives witnesses
+arbitrarily close to I in operator norm. This statement gives no lower bound
+on numerical detectability. The nonzero discrepancy can be exponentially small.
+
+## Construction and proof
+
+Choose an entry (o,i) and a degree-D word w=a_1...a_D with coefficient c != 0.
+Number the path vertices 0,...,D. Let S_a contain both symmetric entries
+(j-1,j) and (j,j-1) exactly when a_j=a. All other entries are zero. Put
+
+    A_a = (1-e)I + (e/2) S_a.
+
+Each S_a is real symmetric and has absolute row sum at most 2. For every real
+vector x, the inequality 2|x_j x_k| <= x_j^2+x_k^2 gives
+|x^T S_a x| <= 2||x||_2^2. The spectral interval for A_a is therefore
+[1-2e,1]. The same statement applies on complex payload space.
+
+In a product of k <= D such matrices, a path from vertex 0 to vertex D must
+make D steps. Each factor can stay at its vertex or move one edge. If k<D,
+the (0,D) entry is zero. If k=D, each factor must move forward exactly once.
+It can do so only when its letter is the corresponding letter of w. Hence
+
+    F_(o,i)(A)_(0,D) = c (e/2)^D != 0.
+
+There are no terms of degree greater than D. Set input i to the last basis
+vector and all other inputs to zero. The first coordinate of returned role o
+is exactly this discrepancy. D=0 uses a one-dimensional space; every A_a is
+(1-e)I and the discrepancy is c. This proves the claim for both coefficient
+fields. The generators themselves are real in either case.
+
+The maximum degree is essential for this construction. For F(X)=X-3I/4,
+the shortest nonzero word is the constant word. Its one-dimensional shifted
+path is X=3I/4 and gives zero. The degree-one construction gives discrepancy
+1/8 at e=1/4. The verifier checks this distinction.
+
+## Extract the degree without expanding words
+
+Retain the predecessor's exact acyclic coefficient machine. Let N be its
+state count, U_o the initial row for output o, T_a its transitions, and V its
+terminal map to input roles. A word coefficient is U_o T_w V. Every transition
+strictly decreases the state index. Thus all products of N transitions vanish.
+
+For each d, form a basis B_d of the span of all rows U_o T_w with |w|=d.
+Use the original reachable rows as the basis vectors and retain their output
+and word. Begin with the initial rows at d=0. Apply every T_a to B_d and use
+exact elimination to select B_(d+1). Induction proves that these rows span
+exactly the length-d reachable space. A nonzero degree-d coefficient exists
+if and only if some selected row has a nonzero terminal projection. Keep a
+nonzero projection at the last such degree. This returns an actual word and
+coefficient, not a linear combination of words.
+
+The implementation does not merge bases from different degrees. The source
+compiler has N=2r+q_left+q_right states; an explicit target contributes the
+sum of its displayed term lengths in place of q_right. We use at most N
+layers, at most N basis rows per layer, and at most |alphabet| N candidates
+per later layer. Straight exact elimination needs O(|alphabet| N^4) field
+operations. This is an arithmetic bound, not a bit-complexity or wall-time
+claim. D is at most the maximum unary dependency depth and at most N-1.
+
+If all projections vanish, the argument proves equivalence. The exported
+negative-witness checker deliberately accepts only nonzero literal witnesses;
+use the predecessor's separate certificate checker for positive certificates.
+The new checker does not rely on the producer's degree computation: it builds
+the prescribed symmetric path action and executes the original program and
+its reference or target separately. It checks the predicted nonzero output.
+
+## Credit, evidence and limits
+
+The path separation argument is an elementary constructive specialization of
+free-algebra evaluation. No global literature priority or new general PIT
+algorithm is claimed. The source compiler and exact scalar code are retained
+byte-for-byte from zxv. Its weighted-automata foundation credits Stefan Kiefer,
+[Notes on Equivalence and Minimization of Weighted Automata](https://arxiv.org/abs/2009.01217),
+and Ran Raz and Amir Shpilka,
+[Deterministic Polynomial Identity Testing in Non Commutative Models](https://www.cs.tau.ac.il/~shpilka/publications/RazShpilka_PIT.pdf).
+The new contribution is the quantified SPD/conditioning interface, exact-degree
+adapter, and independently transcribed literal diagnostic. See NOTICE.md for
+retained project credit and SOURCES.json for exact pending-predecessor pins.
+
+The source-bound wrong-order echo and random finite checks corroborate the
+written proof. They do not replace its quantified argument. Python, exact
+scalar arithmetic, source transcription and the mathematical proof remain in
+the trust base. No independent human or proof-kernel acceptance is claimed.
+No native-relation realization, nonlinear or rounded leaf, physical operation
+count, complete multiplication theorem, supplier admission or new kappa is
+proved. The selected frontier is unchanged.
diff --git a/research/spd-dirty-circuit-witness/README.md b/research/spd-dirty-circuit-witness/README.md
@@ -0,0 +1,65 @@
+# Well-conditioned positive-definite circuit counterexamples
+
+Every failed complete dirty linear identity over Q or Q(i), with independent
+operator symbols, has a counterexample using real symmetric positive-definite
+contractions. The construction has dimension D+1, where D is the largest
+nonzero coefficient degree. Every spectrum lies in [1/2,1], so every generator
+has condition number at most 2. A rational parameter also places all generators
+arbitrarily close to I. [PROOF.md](PROOF.md) gives the quantified statement.
+
+This strengthens the negative diagnostic in zxv's
+[PR #5235](https://github.com/CrocSwap/integer-mult-bounds/pull/5235).
+Conditioning or positivity alone cannot repair a false independent-symbol
+identity. Native inverse, commutation or Clifford relations can still exclude
+the counterexample. The complete kappa calculation is unchanged.
+
+[witness.py](witness.py) uses exact-length reachable spaces to find a top-degree
+word without expanded words. [literal.py](literal.py) separately executes every
+register operation under the prescribed matrices. It retains all dirty input
+and returned roles, including self-source queries. It checks negative witnesses
+only. Positive certificates retain the predecessor's separate checker.
+
+## Evidence and replay
+
+Requires CPython 3.12 and the standard library:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/spd-dirty-circuit-witness
+python3.12 -I -B research/spd-dirty-circuit-witness/witness.py research/spd-dirty-circuit-witness/wrong-order.json
+```
+
+Expected registered output: `PASS_SPD_DIRTY_WITNESS` in normal and optimized
+modes. [RESULT.json](RESULT.json) records exact counts and source hashes.
+The replay includes 240 expanded-word differential circuits over Q and Q(i),
+literal matrix actions, 96 spectral-enclosure fixtures, fifteen control groups,
+and the source's complete wrong-order echo. The (X+Y)^64 stress circuit has
+2^64 top words; the adapter retains small exact-length spaces and returns a
+65-dimensional rational witness. Finite checks support the written proof;
+no independent human or formal proof-kernel acceptance is claimed.
+
+The package starts at main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority
+revision 7. Two vendor files and the example are frozen copies from pending
+head 48976daa4dce59b4867ef046da72a370e389925d. [SOURCES.json](SOURCES.json)
+records original upstream locations and pins the exact retained copies in this
+publication history. The original proof and attribution are retained with the
+compiler. No pending upstream Git object is required by a fresh checkout.
+The first hosted replay failed before mathematical execution because that
+pending Git object was absent. The source-history repair preserves all bytes
+and all mathematical checks. No dependency is promoted to main.
+The full-history wrapper was not run in the shallow checkout. No selected cold
+replay, native/Lean/Docker run, physical implementation or kappa change is claimed.
+
+## Open next questions
+
+Can a specified native child family realize one such witness while retaining
+its extra relations? First write its exact relations and test the labelled path
+against each one. This matters because the current theorem permits independent
+symbols; it does not prove failure inside a restricted native family. No such
+relation-preserving substitution is supplied here. The inverse-aware frontend
+claimed separately in Discussion #5236 remains outside this work.
+
+## Credit
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.
+See [NOTICE.md](NOTICE.md) for the unchanged zxv compiler and scalar code,
+classical weighted-automata/PIT credit, and retained echo/tensor lineage.
diff --git a/research/spd-dirty-circuit-witness/RESULT.json b/research/spd-dirty-circuit-witness/RESULT.json
@@ -0,0 +1,72 @@
+{
+  "controls": 15,
+  "files": {
+    "NOTICE.md": "0f1d9ac19527fb5d3176083e7b4540e1248d9ec96c00073c9e0d1461fa5f8453",
+    "PROOF.md": "b11c7bd44ceb7783e1e58d1f32e6a5c979f8be02b7ed91f7203a86cfbf7a19d9",
+    "README.md": "1df8ea3df8a2765aed1be207525ced7e682458b46ab8ba6bc106ab7a02c7c9f1",
+    "SOURCES.json": "ad1e637bafbf61a6fe9cf9d80528a1be9a55ee220e0a0424cd931d2d17baf481",
+    "TASK.json": "318d8c21d6b7b88793bdb34446a492598cfcca8b99107ee76c77108adf80a385",
+    "VERIFY.json": "d26d07c535953e4f972577c1e3b9a7bf58539254cb10661a0160876f465c4b3e",
+    "literal.py": "89f1b1e039119627ac512be9407e456fdb513adbe5104027a6d65865e1f598b9",
+    "vendor/UPSTREAM-PROOF.md": "ef4cfcfd65b1dde1f11eedace0ed90a1b47a2cc100daaf778fbecbff3ec685a7",
+    "vendor/UPSTREAM-README.md": "fb8d94a58b6c0448aa893e65f263b9d71639b1aa79cd10f054d0dc740a570dca",
+    "vendor/fields.py": "0829ad6ed3775ba6e3d11cf04cfb3513ac763549b730aaed74958b1238e660d7",
+    "vendor/linear_words.py": "fe825cf7c886de61bb5ec5a02171c5a392a7bddc185280726f4c359b3f289584",
+    "verify.py": "eed7802377767720f7f6a677d0d07e1d7889b0e3e567ee6a13831f1500a1c83d",
+    "witness.py": "ce52e13ba5fc038d484ced01c615e49172e0cd8eb3dd44d477f0aedbd8b2ee39",
+    "wrong-order.json": "137def9876c1195456a312b8c232d656ff063cc5caae645be6afc3d06709742c"
+  },
+  "fixtures": {
+    "differential_circuits": 240,
+    "equivalent_circuits": 48,
+    "literal_replays": 384,
+    "negative_circuits": 192
+  },
+  "source_example": {
+    "certificate": {
+      "coefficient": "1",
+      "dimension": 4,
+      "epsilon": "1/4",
+      "equivalent": false,
+      "input": 0,
+      "output": 1,
+      "problem_sha256": "d32cd250d576afa157b04f24b9583bec75b113fafddfc831a86079ce5e583ae9",
+      "schema": "spd-dirty-witness/v1",
+      "spectral_interval": [
+        "1/2",
+        "1"
+      ],
+      "statistics": {
+        "alphabet": [
+          "A0",
+          "A1",
+          "A2"
+        ],
+        "layer_dimensions": [
+          4,
+          2,
+          2,
+          2,
+          0
+        ],
+        "states": 19
+      },
+      "word": [
+        "A2",
+        "A1",
+        "A0"
+      ]
+    },
+    "discrepancy": "1/512"
+  },
+  "spectral_matrices": 96,
+  "stress": {
+    "degree": 64,
+    "dimension": 65,
+    "discrepancy": "1/6277101735386680763835789423207666416102355444464034512896",
+    "expanded_top_words": 18446744073709551616,
+    "maximum_layer_dimension": 2,
+    "states": 132,
+    "sum_layer_dimensions": 129
+  }
+}
diff --git a/research/spd-dirty-circuit-witness/SOURCES.json b/research/spd-dirty-circuit-witness/SOURCES.json
@@ -0,0 +1,78 @@
+{
+  "schema_version": 1,
+  "base_commit": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/spd-dirty-circuit-witness/vendor/fields.py",
+      "usage": "provenance",
+      "pin": "head:46bdee8dff4254afd2ed773e7f6c263156c6f84d:research/spd-dirty-circuit-witness/vendor/fields.py",
+      "git_blob": "f1b6dc83e26bd9fd9ec67ded3fedb330ef89598d",
+      "sha256": "0829ad6ed3775ba6e3d11cf04cfb3513ac763549b730aaed74958b1238e660d7",
+      "upstream": {
+        "commit": "48976daa4dce59b4867ef046da72a370e389925d",
+        "path": "research/universal-dirty-certificates/fields.py",
+        "url": "https://github.com/zxv/integer-mult-bounds/blob/48976daa4dce59b4867ef046da72a370e389925d/research/universal-dirty-certificates/fields.py"
+      }
+    },
+    {
+      "path": "research/spd-dirty-circuit-witness/vendor/linear_words.py",
+      "usage": "provenance",
+      "pin": "head:46bdee8dff4254afd2ed773e7f6c263156c6f84d:research/spd-dirty-circuit-witness/vendor/linear_words.py",
+      "git_blob": "4a877e1ab9a87a0f01eb13e28ddf309c2060bd5e",
+      "sha256": "fe825cf7c886de61bb5ec5a02171c5a392a7bddc185280726f4c359b3f289584",
+      "upstream": {
+        "commit": "48976daa4dce59b4867ef046da72a370e389925d",
+        "path": "research/universal-dirty-certificates/linear_words.py",
+        "url": "https://github.com/zxv/integer-mult-bounds/blob/48976daa4dce59b4867ef046da72a370e389925d/research/universal-dirty-certificates/linear_words.py"
+      }
+    },
+    {
+      "path": "research/spd-dirty-circuit-witness/vendor/UPSTREAM-PROOF.md",
+      "usage": "provenance",
+      "pin": "head:46bdee8dff4254afd2ed773e7f6c263156c6f84d:research/spd-dirty-circuit-witness/vendor/UPSTREAM-PROOF.md",
+      "git_blob": "3a405e327a7fd5d59a570d7f9920f1ce7f229a76",
+      "sha256": "ef4cfcfd65b1dde1f11eedace0ed90a1b47a2cc100daaf778fbecbff3ec685a7",
+      "upstream": {
+        "commit": "48976daa4dce59b4867ef046da72a370e389925d",
+        "path": "research/universal-dirty-certificates/PROOF.md",
+        "url": "https://github.com/zxv/integer-mult-bounds/blob/48976daa4dce59b4867ef046da72a370e389925d/research/universal-dirty-certificates/PROOF.md"
+      }
+    },
+    {
+      "path": "research/spd-dirty-circuit-witness/vendor/UPSTREAM-README.md",
+      "usage": "provenance",
+      "pin": "head:46bdee8dff4254afd2ed773e7f6c263156c6f84d:research/spd-dirty-circuit-witness/vendor/UPSTREAM-README.md",
+      "git_blob": "be4e5981e57bdd2aa8d2ba8c36479594cf38b5d2",
+      "sha256": "fb8d94a58b6c0448aa893e65f263b9d71639b1aa79cd10f054d0dc740a570dca",
+      "upstream": {
+        "commit": "48976daa4dce59b4867ef046da72a370e389925d",
+        "path": "research/universal-dirty-certificates/README.md",
+        "url": "https://github.com/zxv/integer-mult-bounds/blob/48976daa4dce59b4867ef046da72a370e389925d/research/universal-dirty-certificates/README.md"
+      }
+    },
+    {
+      "path": "research/spd-dirty-circuit-witness/wrong-order.json",
+      "usage": "provenance",
+      "pin": "head:46bdee8dff4254afd2ed773e7f6c263156c6f84d:research/spd-dirty-circuit-witness/wrong-order.json",
+      "git_blob": "85482942f052694a3a6d15d364d18823b51e1b00",
+      "sha256": "137def9876c1195456a312b8c232d656ff063cc5caae645be6afc3d06709742c",
+      "upstream": {
+        "commit": "48976daa4dce59b4867ef046da72a370e389925d",
+        "path": "research/universal-dirty-certificates/examples/wrong-order.json",
+        "url": "https://github.com/zxv/integer-mult-bounds/blob/48976daa4dce59b4867ef046da72a370e389925d/research/universal-dirty-certificates/examples/wrong-order.json"
+      }
+    }
+  ]
+}