Research: determinant obstruction for rank-homogeneous register paths (no kappa)

research/graded-path-obstruction/.gitattributes

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+* text eol=lf
+*.gz binary

research/graded-path-obstruction/EXPECTED.json

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+{
+  "status": "PASS_GRADED_PATH_OBSTRUCTION",
+  "counts": {
+    "excluded": 64,
+    "dense_middle_paths": 1377,
+    "determinant_polynomial_controls": 73,
+    "not_excluded_by_this_criterion": 9,
+    "positive_nine_call_controls": 3,
+    "false_homogeneity_rejected": 1,
+    "inhomogeneous_cancellation_control": 1,
+    "corrupt_rank_bill_rejected": 1
+  },
+  "by_origin": {
+    "four_block_followup": {
+      "excluded": 8
+    },
+    "collision_generators": {
+      "excluded": 23,
+      "not_excluded_by_this_criterion": 9
+    },
+    "pr680_regression": {
+      "excluded": 33
+    }
+  },
+  "words_payload_sha256": "a6acbdcbf7e8b28a6b0e0e334b16a3fdc1e06731fef6c6421ff71801f844a511",
+  "certificates_sha256": "6c6d968b9e1f733f02e1b485356a0fb81c6c38246f3c8e0cc55c36dffc450c5c"
+}

research/graded-path-obstruction/MANIFEST.json

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+{
+  "files": {
+    ".gitattributes": "abeba78466be388a1c5f3cb4ad595b915a0aa9402d1d87e37dfe42762e5b91c5",
+    "EXPECTED.json": "48825c6d56001d65f5d84eae1225c5b4be567524c140ff309890337487eb3bd3",
+    "NOTICE.md": "4a5c9fd916f12337f38e1d2ef4f4db32539cc07418b3f59fb81c4193725d18c2",
+    "ORIGINS.json": "ac91e9b89eee000ce8fd112e97c9f2b5f753426180369a886ff7bcbabc5f8f0a",
+    "PROOF.md": "696e58423e320033f09bda17f81689273bb8947fe9e48f8379ae1b397e407b4a",
+    "README.md": "4c3a8ff6a27edde28ae5ccb19398d49c21a9add8036dd41c2233f4148e27e92b",
+    "VALIDATION.json": "ef454895445f2b85bf1a07db74a09fd29af0ffd8300e70ddc44cf54d5b1cc650",
+    "dense.py": "760df8621caaa81ab841df39667f910b45b40d9167c02747db346e80649632af",
+    "model.py": "5d8c14e1aa62185fcb75725f0c337f00423642cebf121545d38a024567fc6dbf",
+    "screen.py": "1e53ee94f2049491dfdea0078cdb02696544668d22fb84db9185e41405dc6ff4",
+    "verify.py": "0a45ffdbdaa401e599cfd7d88e1206910ba952b08dda4fcf00dfabdffa237247"
+  },
+  "words_payload_sha256": "a6acbdcbf7e8b28a6b0e0e334b16a3fdc1e06731fef6c6421ff71801f844a511"
+}

research/graded-path-obstruction/NOTICE.md

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+# Attribution, pins and overlap
+
+Prepared by [littleBro](https://github.com/littleBro) with substantial OpenAI
+Codex assistance in the research, proof, implementation and validation.
+New material is under the repository's Apache-2.0 license. No contributor
+endorsement, independent human review, or literature-priority claim is made.
+
+The open complex m=3, W=3, s=8 register problem and general framework are
+credited to eumemic's [PR #288](https://github.com/CrocSwap/integer-mult-bounds/pull/288),
+head `475a088b5862f75f0b10596cfd6842d19531a5ec`, and its retained predecessors,
+including Swapnil Jain's P1/P2 and the upstream Clifford-frame work.
+
+The same-author [PR #680](https://github.com/CrocSwap/integer-mult-bounds/pull/680),
+head `c3a5b6b7afec35d259c29bc2d8bb8516e975ad31`, supplies the unchanged
+`model.py` and `dense.py`, the formal phase-quotient convention, and 33
+regression words. That PR's notices and credited framework are retained by
+reference here. This package supplies all runtime dependencies locally and
+does not require PR #680 to be merged.
+
+The eight four-block words and 32 evolved words come from same-author local
+research. Their literal contents are supplied in full. Local snapshot hashes
+in ORIGINS.json establish extraction provenance only; they are not external
+dependencies or claims of public review.
+
+Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`. The selected result and
+`upstream/` are unchanged. The selected supplier was not cold-replayed for
+this standalone mathematical addition.
+
+Overlap review revisited #288 and #680, current PR metadata through #691,
+and repository PR searches for homogeneous, determinant and grading. #673
+and #689 concern dirty echo composition, a different object. The reviewed
+material did not identify this fixed-path homogeneity criterion or these
+eight four-block closures. This is not a review of every PR diff or proof
+of novelty. The proof and its scope remain open to maintainer review.

research/graded-path-obstruction/ORIGINS.json

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+{
+  "base": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "predecessor_pr": 680,
+  "predecessor_head": "c3a5b6b7afec35d259c29bc2d8bb8516e975ad31",
+  "vendored_files": {
+    "model.py": "5d8c14e1aa62185fcb75725f0c337f00423642cebf121545d38a024567fc6dbf",
+    "dense.py": "760df8621caaa81ab841df39667f910b45b40d9167c02747db346e80649632af"
+  },
+  "same_author_local_input_sha256": {
+    "watched-search.json.gz": "2e1cb2797e46144fc33481a06576f638399521aeed785f10d644b2e36b807f0c",
+    "evolution.json.gz": "ff105da6453acf3974243e650e8202fdaf054e7878bc6dddf351e9a656e6e804"
+  },
+  "generator_seed": 6801302,
+  "note": "Local hashes are provenance only; every literal word required for verification is included."
+}

research/graded-path-obstruction/PROOF.md

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+# A determinant obstruction for rank-homogeneous path collisions
+
+This is a written proof in a formal group algebra, followed by finite exact
+hypothesis checks. It is not a proof-assistant verification or a new integer
+multiplication bound. No literature-priority claim is made.
+
+## 1. General statement
+
+Let G be a group and let r:G -> Z_{≥0} satisfy
+
+    r(gh) <= r(g) + r(h),     r(1) = 0.
+
+Fix W registers and L >= 1 arbitrary invertible scalar matrices
+B_1,...,B_L over a field K. On the two boundaries put
+
+    U = diag([u_1],...,[u_W]),   V = diag([v_1],...,[v_W]).
+
+For each interior layer 1 <= k < L put
+
+    D_k = diag([g_{k,1}],...,[g_{k,W}]).
+
+All u,v,g are fixed group elements; the B matrices can have any coefficients
+and zero pattern compatible with invertibility. Consider
+
+    S = V B_L D_{L-1} B_{L-1} ... B_2 D_1 B_1 U
+
+over K[G]. An interior path a=(a_1,...,a_{L-1}) has ordered product and degree
+
+    h(a) = g_{L-1,a_{L-1}} ... g_{1,a_1},
+    d(a) = sum_k r(g_{k,a_k}).
+
+For L=1 the only interior path is empty, with product 1 and degree 0.
+
+**Hypothesis (finite path homogeneity).** Whenever h(a)=h(b), d(a)=d(b).
+This concerns this finite collection of paths. It does not assert that rank
+is a grading or a homomorphism on G.
+
+**Endpoint assumption.** In every entry of S, every group element with a
+nonzero coefficient has rank at least m. Different entries may use different
+group elements; no shared final frame is required by this assumption.
+
+Then
+
+    W*m <= sum_i r(u_i) + sum_i r(v_i) + sum_{k,i} r(g_{k,i}).       (1)
+
+Thus any word whose right side is less than W*m and satisfies path
+homogeneity is impossible for **all** invertible choices of the B matrices.
+There is no restriction on L, on the scalar zero patterns, or on the size of
+the scalar coefficients.
+
+## 2. Proof
+
+Replace each interior D_k by the commuting scalar diagonal matrix
+
+    T_k(t) = diag(t^{r(g_{k,1})},...,t^{r(g_{k,W})}),
+
+and form the ordinary polynomial matrix
+
+    T(t) = B_L T_{L-1}(t) B_{L-1} ... B_2 T_1(t) B_1.
+
+Its determinant is exactly
+
+    det T(t) = c * t^D,
+    c = product_k det B_k != 0,
+    D = sum_{k,i} r(g_{k,i}).                                      (2)
+
+For fixed output i and input j, group the paths in the entry T_ij(t) by
+their ordered product h. Homogeneity assigns each such product a single
+degree d_h, so
+
+    T_ij(t) = sum_h c_{ij,h} * t^{d_h}.
+
+The same coefficient c_{ij,h} is the coefficient of [v_i h u_j] in S_ij.
+Left and right multiplication by fixed group elements are injective, so
+distinct h cannot merge in this latter coefficient comparison.
+
+If c_{ij,h} != 0, the endpoint assumption and subadditivity give
+
+    m <= r(v_i h u_j)
+      <= r(v_i) + r(h) + r(u_j)
+      <= r(v_i) + d_h + r(u_j).
+
+Consequently every nonzero term of T_ij(t) has degree at least
+m-r(v_i)-r(u_j). Cancellation between different h at the same degree can
+only increase that lower bound; an identically zero entry causes no problem.
+
+Every term in the ordinary Leibniz determinant expansion therefore has
+degree at least
+
+    sum_i (m-r(v_i)-r(u_{sigma(i)}))
+      = W*m - sum_i r(v_i) - sum_j r(u_j).
+
+This remains valid if some entry bounds are negative. Since (2) is nonzero,
+its degree D is at least this quantity. Rearranging proves (1).
+
+The argument never maps the whole noncommutative group algebra into K[t].
+Only the coefficients of this fixed path expansion are compared.
+
+## 3. Why the collision hypothesis matters
+
+Without homogeneity, a group coefficient can cancel between different
+powers of t. The weighted scalar coefficient then need not vanish.
+
+For example, take an element C of positive rank, and in two interior layers
+use diag(1,C) and diag(1,C^{-1}), with identity mixing between them. The two
+constant-role paths have product 1 but degrees 0 and 2 when both ranks are 1.
+With A=[[1,1],[1,-1]] at entry and A^{-1} at exit, the unweighted group word
+is the identity. The weighted matrix is
+
+    A^{-1} diag(1,t) diag(1,t) A
+      = (1/2) [[1+t^2, 1-t^2], [1-t^2, 1+t^2]].
+
+The off-diagonal group coefficients vanish, but the displayed polynomials
+do not. The verifier includes this control and rejects a forged homogeneous
+classification of an exact Clifford inverse-pair word.
+
+Failure of homogeneity is only a necessary escape from this obstruction.
+It is not a construction or evidence that invertible coefficients exist.
+
+## 4. Clifford specialization and scope
+
+The finite checker uses the same three-qubit projective Clifford group
+algebra as PR #680. For a Clifford symplectic matrix [[A,B],[C,D]], define
+r(g)=rank_F2(B). Composition gives B_{gh}=A_g B_h+B_g D_h, hence the required
+subadditivity. Every listed free monomial operation has rank zero; each
+paid C_v=((1+i)/2)I+((1-i)/2)X_v has rank one. The rank of a complete unary
+layer is therefore no larger than its literal paid-call count.
+
+The usual endpoint condition, one full-rank left monomial coset for each
+input stream, implies the weaker endpoint assumption above with m=3.
+For W=3 and eight paid calls the right side of (1) is at most eight, which
+contradicts nine whenever path homogeneity holds.
+
+Projective equality here retains Pauli conjugation signs and quotients only
+scalar phase. The supplied rational-Gaussian generators have central phases
+in {1,-1,i,-i}; the intended tensor realization removes them when the tensor
+multiplicity is divisible by four, as in the predecessor model. The result
+is stated in that formal group algebra. It does not infer formal linear
+independence from the concrete 8-by-8 representation. That representation
+is used only to independently check group equalities and ranks.
+
+## 5. What the supplied words establish
+
+All 73 literal words have eight paid calls. The criterion excludes 64:
+
+- Eight four-block descendants previously left unresolved by bounded
+  support/polynomial searches.
+- Twenty-three of 32 words from two generations of a collision generator.
+- All 33 three-block words already excluded by PR #680, as regressions.
+
+Nine evolved words fail the homogeneity hypothesis and remain unclassified
+by this theorem. The cases are not an exhaustive enumeration of any
+four-block family or of the full register model. The fixed unary words and
+mixing locations matter. Inserting or changing unary operations can change
+the hypothesis.
+
+The proof supplies a search filter: a successful word with paid cost below
+W*m must admit coincident interior products of different accumulated ranks.
+Merely counting repeated products, including repetitions of the same unary
+choice on different registers, is insufficient. The new criterion is
+independent of the support, torus and polynomial solvers used in PR #680.

research/graded-path-obstruction/README.md

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+# Rank-homogeneous paths: a determinant obstruction
+
+**Scoped theorem and exact finite verification. No new kappa or primitive.**
+
+A scalar determinant tracks accumulated unary rank whenever all paths with
+the same interior group product have the same rank sum. The resulting
+inequality rules out a saving, for any number of scalar mixing blocks and
+arbitrary invertible complex coefficients, subject to that explicit path
+hypothesis. See [the complete proof](PROOF.md).
+
+This closes eight previously unresolved four-block Clifford words and 23
+additional evolved words at m=3, W=3, s=8. It also recovers the 33 exclusions
+in [PR #680](https://github.com/CrocSwap/integer-mult-bounds/pull/680) by a
+different argument. Nine supplied words fail the new hypothesis and are
+kept as unknown, not reported as solutions or impossibility results.
+
+## Replay
+
+Python 3.10+ and the standard library are sufficient. From the repository:
+
+```sh
+python3 -I -B research/graded-path-obstruction/verify.py
+```
+
+Expected: `PASS_GRADED_PATH_OBSTRUCTION`, with 64 exclusions and nine words
+not excluded by this criterion. The checker uses no network, Git checkout
+dependencies, external solvers, or sibling research directories. It writes
+no files. Assertions are required; optimized execution is explicitly rejected.
+
+The replay checks the finite hypotheses using both signed Clifford tableaus
+and exact 8-by-8 Gaussian matrices, including 1,377 interior paths. It also
+checks 73 polynomial determinant identities, three exact nine-call positive
+controls, a heterogeneous inverse-pair control, and deliberate certificate
+corruptions. It compares the complete result with [EXPECTED.json](EXPECTED.json).
+Input and source integrity checks are described in [VALIDATION.json](VALIDATION.json).
+
+The general conclusion rests on the written determinant argument, not on
+extrapolation from these finite controls. The obstruction is to formal group
+algebra words with the stated fixed unary layers. The full W=3, s=8 problem
+from [PR #288](https://github.com/CrocSwap/integer-mult-bounds/pull/288) remains
+open. There is no bit-side construction, supplier invoice, all-size compiler
+proof, or change to the selected frontier here.
+
+## Search consequence
+
+The eight follow-up words originally survived very short polynomial searches.
+Diagonal gauge normalization then provided exact refutations of the seven
+stored arithmetic supports, but bounded searches over other supports still
+stalled. Inspection of their collision classes exposed the determinant
+criterion, which closes the whole fixed words without coefficient search.
+
+For future synthesis, require a collision between **different accumulated
+ranks** before spending time on scalar coefficients. This is a necessary
+filter, not a score predicting existence. The 32 evolved words here are a
+finite literal corpus, not a claim that their generator families are exhausted.
+
+`words.json.gz` contains all 73 literal words; matrix entries at `mix3` gates
+are identity placeholders marking locations where the theorem permits any
+GL(3,C) matrix. `screen.py` checks the hypotheses. `verify.py` reconstructs
+them using independent Gaussian matrices. `model.py` and `dense.py` are
+unchanged, attributed copies from the public PR #680 head.
+
+See [NOTICE.md](NOTICE.md) and [ORIGINS.json](ORIGINS.json) for source pins and
+attribution. No independent human review, Lean verification, repository-wide
+replay, or remote CI is represented as completed.

research/graded-path-obstruction/VALIDATION.json

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+{
+  "status": "Exact local replay passed before publication",
+  "platform": "Windows, Python 3.14.4, standard library",
+  "command": "python3 -I -B research/graded-path-obstruction/verify.py",
+  "expected_status": "PASS_GRADED_PATH_OBSTRUCTION",
+  "evidence": "Signed tableau and exact Gaussian-matrix classifications agree on 73 words and 1377 internal paths; 64 exclusions, nine criterion-unknown words. Polynomial determinant checks and positive/rejection controls are listed in EXPECTED.json.",
+  "scope": "Written general proof plus independent finite hypothesis checks; not a proof assistant or independent human review.",
+  "not_claimed": [
+    "new kappa",
+    "new positive register primitive",
+    "all-size multiplication theorem",
+    "full supplier or repository-wide replay",
+    "remote CI"
+  ],
+  "isolation_and_corruption_checks": {
+    "relocated_isolated": {
+      "exit_code": 0,
+      "seconds": 0.5829,
+      "expected_result": true
+    },
+    "recompressed_input": {
+      "exit_code": 0,
+      "seconds": 0.6254,
+      "expected_result": true
+    },
+    "corrupt_payload": {
+      "exit_code": 1,
+      "seconds": 0.2365,
+      "expected_result": false
+    },
+    "corrupt_expected": {
+      "exit_code": 1,
+      "seconds": 0.2479,
+      "expected_result": false
+    },
+    "optimized_rejected": {
+      "exit_code": 1,
+      "seconds": 0.0807,
+      "expected_result": false
+    }
+  },
+  "full_replay_seconds": 0.63,
+  "selected_record_check": "PASS selected record, exact kappa, original source inventory and proof supplement: 51110230636731/31250000000000000"
+}