Exclude rank 15 in the fixed P120 quadratic space (no κ)

research/p120-char3-rank31-window/NOTICE

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+A characteristic-three rank-31 pattern on P120
+Copyright 2026 Nick Petalas. Licensed under Apache-2.0.
+
+This contribution was produced by Nick Petalas.
+
+The new construction builds directly on unpaidmoon's rank-32 P120 pattern
+and open rank-31 question, PR #7087 at
+043b334e7249e8b79e437a0765773a7e2fc8b8e8, and its quadratic predecessor #6953.
+The unrestricted rank-15 clique construction is unpaidmoon's PR #7014.
+The port-window model and port family are DreamingOfClouds's #6933.
+unpaidmoon's #6946 supplies the earlier odd-characteristic upper bounds.
+The fractional Haemers framework is due to Blasiak and Haemers, with the
+cited formulation by Bukh and Cox. Classical finite-field linear algebra
+retains its prior credit. No predecessor code is included in this package.

research/p120-char3-rank31-window/PROOF.md

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+# A fixed quadratic space misses the rank-15 P120 pattern
+
+Claim type: finite certificate with a scoped algebraic proof. The main result
+is a restriction on one column space, not an unrestricted minimum-rank theorem.
+
+Let V be the ascending list of integers v with 0 <= v < 512 and binary weight
+3 or 7. It has 120 elements. Write x_i(v) for bit i of v, with bits numbered
+0 through 8. Two vertices are orthogonal when their binary dot product is zero.
+A legal one-block pattern has nonzero diagonal entries and zeros at every
+ordered orthogonal pair. Symmetry is not required.
+
+## The fixed space
+
+For every field K of characteristic three, let U_K be the space of functions
+on V spanned over K by these 36 functions:
+
+```text
+x_i + x_8 - 1,       0 <= i < 8,
+x_i x_j + 1 - x_8,  0 <= i < j < 8.
+```
+
+Use this displayed order, with pairs ordered lexicographically. Let F be their
+120 by 36 evaluation matrix. This is the fixed quadratic space specified by
+the nine coefficient relations in unpaidmoon's #7087, expressed without the
+redundant coefficient coordinates. Indeed, if the free coefficients are a_i
+and b_ij for indices below 8, then
+
+```text
+a_8 = sum_i a_i - sum_(i<j) b_ij,
+c = -a_8,
+b_i8 = 0.
+```
+
+Thus c+a_8=0 and sum_i a_i+2a_8+2 sum_(i<j) b_ij=0 over characteristic three,
+exactly the declared relations. The omitted b_78 coefficient is zero here.
+The uniqueness minors below also prove that the 36 evaluation functions are
+independent. The result uses this explicitly defined space, irrespective of
+other representations or conventions.
+
+## The obstruction
+
+**Theorem.** Over every field K of characteristic three, every legal P120
+pattern whose columns lie in U_K has rank at least 16. In particular, no such
+pattern has rank 15.
+
+**Proof.** Scale each column by the inverse of its diagonal entry. This preserves
+rank, membership in the linear space U_K, and forced zeros, and makes the
+diagonal one.
+
+For a vertex v, coefficients z in K^36 of a normalized legal column must satisfy
+
+```text
+F[u,:] z = 0   for every u orthogonal to v,
+F[v,:] z = 1.
+```
+
+`FACTOR.json` lists 16 vertices v and, for each, a coefficient vector z_v
+satisfying these equations. For each v it also lists 36 allowed equation rows.
+The 36 by 36 matrix of those rows has nonzero determinant in F3. Hence the
+coefficient solution is unique over F3 and over every K of characteristic three.
+The latter assertion follows because the same determinant is a nonzero prime-
+subfield element of K. Therefore every normalized legal matrix with columns
+in U_K contains all 16 displayed columns F z_v at the indicated vertices.
+
+The artifact supplies 16 row indices for these columns. Their 16 by 16 minor
+has determinant one in F3 and therefore in K. The mandatory columns are
+independent, so every such matrix has rank at least 16. QED.
+
+Only these 16 columns are needed. No classification of all rigid columns or
+all legal patterns is claimed. The exact checks cover 912 legal-column
+equations, 16 uniqueness determinants and the independence determinant.
+
+## A rank-31 construction in the same space
+
+The artifact also supplies A in F3^(120 x 31) and B in F3^(31 x 120). The checker
+computes M=AB and verifies its 120 unit diagonal entries and all 6,720 ordered
+orthogonal entries. It also supplies C in F3^(36 x 31); all 3,720 entries of
+A=FC are checked. Consequently every column of M lies in the same fixed space.
+
+The checker evaluates nonzero 31 by 31 minors of A and B and their corresponding
+product minor in M. Their determinants are 1, 2 and 2. Hence M has rank at least
+31, while the factorization gives rank at most 31. It has rank exactly 31, and
+scalar extension preserves the certificate over every characteristic-three K.
+
+This point was obtained by intersecting the legal affine column fibers in the
+span of #7087's rank-32 construction with a common hyperplane. A fiber c+N meets
+ker(L) precisely when L is nonzero on N or L(c)=0. This follows by cancelling
+L(c) with a direction in N, or by observing that the functional is constant on
+the fiber. Only the resulting point is claimed; unsuccessful search candidates
+prove no lower bound. The delivered factors are the complete construction,
+and checking them requires no predecessor code, Git or network.
+
+## Consequence for the next construction
+
+Accepting the existing clique construction in unpaidmoon's #7014, P120 has a
+rank-15 legal pattern over every field. Its result is cited, not replayed here.
+The new theorem shows that this rank cannot be attained with every column in
+U_K. Thus a rank-15 construction must leave this particular quadratic space.
+The bounds established here inside U_K are 16 <= minimum rank <= 31. Neither
+endpoint is claimed optimal in that restricted problem.
+
+The rank-31 point is not an improvement to the unrestricted window bound.
+The initial broad upper-bound target was superseded by #7014; the published
+scope is this fixed-space separation and its new rank-31 companion point.
+The scope correction is recorded in the linked Discussion update.
+
+## Remaining obligations
+
+No optimal restricted rank, physical decoder, native word or recursive supplier
+is constructed. Delivery, restoration, precision, stock, movement and a complete
+caller are unpriced. No characteristic-zero identity follows from these finite-
+characteristic factors. No new kappa or selected-frontier change is claimed.
+Independent human and formal proof-kernel review are unclaimed.

research/p120-char3-rank31-window/README.md

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+# The fixed P120 quadratic space cannot attain rank 15
+
+This contribution was produced by Nick Petalas.
+
+Claim type: finite certificate with a scoped algebraic proof. Over every field
+of characteristic three, legal P120 patterns whose columns lie in #7087's fixed
+36-dimensional quadratic space have rank at least 16. Sixteen uniquely forced
+columns and a nonzero minor certify this. A new explicit rank-31 point belongs
+to the same space, lowering its cited constructive ceiling from 32 to 31.
+Thus the restricted minimum lies in [16,31]; its exact value remains open.
+
+unpaidmoon's [#7014](https://github.com/CrocSwap/integer-mult-bounds/pull/7014)
+already gives an unrestricted rank-15 clique construction. The new obstruction
+shows that this particular quadratic space cannot attain that rank. The rank-31
+point is not an improvement to the unrestricted window bound. There is no new
+kappa, physical decoder or selected-frontier change.
+
+[PROOF.md](PROOF.md) defines the space and proves the uniform field consequence.
+[FACTOR.json](FACTOR.json) contains the obstruction and the complete rank-31
+factors. The standalone checker runs no predecessor verifier.
+
+## Reproduce
+
+From the repository root, using CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev research/p120-char3-rank31-window/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev -O research/p120-char3-rank31-window/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p120-char3-rank31-window
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p120-char3-rank31-window --check-source-history
+```
+
+Another CPython 3.12 path may replace the local interpreter path. The scientific
+marker is `PASS_P120_CHAR3_RANK31 no_new_kappa`. The verifier prints a generated
+receipt bound to all eight package files and writes no files. The publication
+check needs full Git history; the scientific checker needs neither Git nor
+network access.
+
+The exact checks cover all 14,400 product entries, 120 diagonal entries, 6,720
+forced zeros, three rank-31 determinants, 3,720 fixed-space membership entries,
+912 forced-column equations, 16 uniqueness determinants and a nonzero 16 by 16
+minor. Eleven rejection controls cover malformed certificates and violations of
+the mask, rank, fixed-space membership, forced columns and independence. Three
+small determinant fixtures cover identity, row-swap and singular matrices.
+
+## Scope and limitations
+
+Card `complex-rank-shape-decoder`, priority revision 8, changes scope from a
+complete signed-moment improvement to a fixed-space obstruction and finite
+construction. The source-bound baseline command passed at the selected #7000
+checkpoint. The older catalogue context is not the selected frontier.
+
+The initial target was a new unrestricted rank-31 upper bound answering #7087.
+A broader PR check exposed #7014's stronger prior construction. The
+[scope correction](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19542514)
+records this before publication. The main new result is the restriction forcing
+rank at least 16, so the fixed space misses the known unrestricted rank 15.
+
+No optimal restricted rank, characterization of all rigid columns, physical
+native implementation or complete supplier invoice is claimed. The five selected
+framework assumptions remain unchanged. No predecessor scientific suite,
+selected scientific replay, Lean or Docker run was performed. Independent human
+and formal proof-kernel review are unclaimed.
+
+The verifier has fixed dimensions and bounded loops, with a 64 KiB certificate
+limit. Python allocates bounded temporary lists during this one-shot check;
+this managed-runtime deviation is limited to matrices at most 120 by 120 and
+has no retained outputs. Indirect calls are restricted to the named determinant
+and negative-control routines. Warning-strict normal and optimized runs check
+the same artifact. Shared code and diagnostics configuration are unchanged.
+
+## Open next questions
+
+1. What is the exact minimum rank inside this fixed quadratic space? The new
+   bounds are 16 and 31. A first step is a common hyperplane meeting every legal
+   affine fiber inside the delivered rank-31 span. A failed search would not
+   prove a lower bound for all subspaces.
+2. Which enlargement or change of the quadratic space admits a rank-15 point?
+   Start by representing the cited #7014 clique columns in a larger feature
+   space. Their membership and the cost of a useful decoder remain open here.
+3. Is any of this finite-field structure useful in characteristic zero? Start
+   with a separate exact mask factorization and its complete paid invoice.
+   Reduction modulo three alone does not supply that implementation.
+
+## Credit and sources
+
+unpaidmoon supplies the fixed quadratic space, rank-32 construction and open
+questions in [#7087 at 043b334e](https://github.com/CrocSwap/integer-mult-bounds/blob/043b334e7249e8b79e437a0765773a7e2fc8b8e8/research/p120-char3-rank32-window/README.md),
+the quadratic predecessor in [#6953](https://github.com/CrocSwap/integer-mult-bounds/pull/6953),
+the stronger unrestricted clique construction in [#7014](https://github.com/CrocSwap/integer-mult-bounds/pull/7014),
+and the earlier odd-characteristic bound in [#6946](https://github.com/CrocSwap/integer-mult-bounds/pull/6946).
+DreamingOfClouds supplies the port-window definition and family in
+[#6933](https://github.com/CrocSwap/integer-mult-bounds/pull/6933).
+Bukh, Cox, Blasiak and Haemers retain the fractional Haemers framework credit.
+Finite-field elimination, rank factorization and scalar extension are classical.
+The new fixed-space obstruction, rank-31 factors and standalone checker were
+produced by Nick Petalas. Apache-2.0; see [NOTICE](NOTICE).
+
+Cited pending predecessors are not replay dependencies. Their bytes are not
+needed to check this certificate. SOURCES.json pins catalogue and selected-record
+provenance. The advance [CLAIM](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19542430)
+and the scope correction state the change in the research question.

research/p120-char3-rank31-window/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "75d77b328ea02ded535e7f23aee5b1e55d0bb1a8",
+      "sha256": "154c3d30455d778fd3343b6b4851f0fa4665f0f7625b0adb3675e2bafbbf6bf1"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    }
+  ]
+}

research/p120-char3-rank31-window/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "e54523517d0f03e58352e647c403a5af3e525a57",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a legal decoder or coframe change improve the full signed moment, including changed helper targets?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Price one complete changed itinerary with both paid and productive histograms. Check #5693/#5698 for overlap and their exact hypotheses before another coframe search. Use the signed invoice preflight where its boundary contract agrees; an inconclusive concave comparison is not a power crossing. Stop if an optimistic complete budget fails at the chosen target or native delivery remains unspecified.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "coframe",
+      "signed moment",
+      "5693",
+      "5698"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Prove rank at least 16 for legal characteristic-three P120 patterns in the fixed 36-dimensional quadratic column space of PR #7087; supply a new rank-31 point in that space.",
+  "scope_change_reason": "The initial broad rank-31 target was already dominated by PR #7014 rank 15. A public scope correction precedes publication. The new contribution separates the fixed quadratic space from the stronger unrestricted construction, and supplies a new restricted rank-31 upper bound.",
+  "bottleneck_link": "Side result on decoder feature-space choice: rank 15 requires leaving this fixed quadratic space. No characteristic-zero decoder, paid invoice or kappa follows.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first verified original rank improvement; leave further reductions as open questions.",
+  "outcome": "partial",
+  "evidence": "PROOF.md; FACTOR.json; standalone verify.py checks 16 forced independent columns, uniqueness, fixed-space membership and a rank-31 attaining point.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "What is the exact minimum rank inside the fixed quadratic space?",
+      "why_it_matters": "The certificate narrows its range to 16 through 31 and excludes the unrestricted clique rank 15.",
+      "first_step": "Seek a hyperplane meeting every legal affine column fiber in the delivered rank-31 span.",
+      "blocked_by": "No sharper lower bound or smaller attaining pattern is provided."
+    },
+    {
+      "question": "What change of feature space admits a rank-15 pattern?",
+      "why_it_matters": "The known unrestricted clique construction cannot have all columns in this space.",
+      "first_step": "Express the cited clique columns in an enlarged feature space.",
+      "blocked_by": "The current certificate deliberately fixes the 36-dimensional space."
+    }
+  ]
+}

research/p120-char3-rank31-window/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "-W",
+        "error",
+        "-X",
+        "dev",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Fixed quadratic-space rank-16 obstruction, rank-31 construction and membership, all mask entries, nonzero minors and eleven rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "-W",
+        "error",
+        "-X",
+        "dev",
+        "-O",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Fixed quadratic-space rank-16 obstruction, rank-31 construction and membership, all mask entries, nonzero minors and eleven rejection controls."
+    }
+  ]
+}