Prove exact height-one integral shear biclique optimum six (no κ)

research/height-one-shear-bicliques/PROOF.md

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+# Exact balanced optimum at coefficient height one
+
+All ranks in this note are over the rationals. Let
+
+```text
+D = {M in Mat_2(Z) : every entry belongs to {-1,0,1}}.
+```
+
+A balanced rank-distance biclique of order r consists of two r-element
+subsets E and O of D. Distinct matrices in the same part have difference
+rank two; every cross-part difference has rank one.
+
+**Result.** The maximum r is exactly six. The lower bound has a written
+construction. The upper bound is an exact finite enumeration with two
+implementations and the coverage argument below. No formal-kernel acceptance
+of that finite computation is claimed.
+
+## Six-by-six construction
+
+Write q(a,b)=a^2+ab+b^2 and take
+
+```text
+S = {(-1,0), (-1,1), (0,-1), (0,1), (1,-1), (1,0)}.
+E(a,b) = [[a,b],[-a-b,a]]
+O(a,b) = [[a,b],[b,-a-b]]       for (a,b) in S.
+```
+
+Every point in S satisfies q=1 and every displayed entry is in {-1,0,1}.
+The determinant identities are
+
+```text
+det(E(a,b)-E(u,v)) =  q(a-u,b-v)
+det(O(a,b)-O(u,v)) = -q(a-u,b-v)
+det(E(a,b)-O(u,v)) =  q(a,b)-q(u,v) = 0.
+```
+
+The identity `4q(a,b)=(2a+b)^2+3b^2` proves strict positivity away from
+zero over Q. Thus distinct same-part differences have rank two. The parts
+are disjoint because E has determinant one and O has determinant minus one.
+Consequently every cross difference is nonzero and singular, with rank one.
+
+For the actual shear frames
+
+```text
+Phi(M) = [[I_2,M],[0,I_2]],
+```
+
+we have `Phi(M)Phi(N)=Phi(M+N)` and
+`rank(Phi(M)-Phi(N))=rank(M-N)`. Hence these twelve commuting integral
+four-by-four frames have the rank-distance metric K_(6,6).
+
+## Upper bound by a complete finite certificate
+
+For every unordered triple {A,B,C} of distinct matrices in D whose three
+pairwise differences have rank two, count
+
+```text
+N(A,B,C) = #{X in D : rank(X-A)=rank(X-B)=rank(X-C)=1}.
+```
+
+The enumeration gives the following complete distribution.
+
+| N(A,B,C) | Number of triples |
+|---:|---:|
+| 0 | 1,480 |
+| 1 | 4,368 |
+| 2 | 9,624 |
+| 3 | 5,760 |
+| 4 | 2,096 |
+| 6 | 160 |
+
+There are 23,488 such triples. In particular each has at most six common
+rank-one contacts. A purported K_(7,7) contains three matrices in its first
+part, while all seven matrices of its second part would be contacts of that
+triple. This contradicts the certified ceiling six. Any larger balanced
+biclique contains a K_(7,7), so it is excluded as well.
+
+This upper certificate does not assume that the contacts themselves have
+pairwise rank-two differences. Its relaxation is sufficient for the ceiling.
+
+## Why both enumerations cover the domain
+
+`verify.py` creates exactly the 3^4=81 lexicographically ordered matrices.
+For every ordered matrix pair, it computes rank from its determinant and
+whether the difference is zero. It visits each of the binomial(81,3)=85,320
+unordered triples once. Rank-two adjacency selects the admissible triples;
+intersection of three 81-bit rank-one neighbor sets counts their contacts.
+It checks the complete histogram and emits a hash of every retained triple
+and its full contact bitset, in order.
+
+`ceiling.py` imports no scientific code from the first verifier. It derives
+all pair ranks by exact Fraction Gaussian elimination. Two matrices with
+the same first row cannot have rank-two difference. An admissible triple
+therefore uses three distinct first rows among the nine possible rows.
+For each of the binomial(9,3) choices, it tries all 9^3 second-row choices,
+giving 61,236 candidates. Sorting first rows gives every admissible triple
+exactly once, even though this ordering differs from the first enumeration.
+For each accepted triple it scans all 81 matrices and tests the three
+contact ranks individually. It recovers the same complete histogram.
+It also checks all 144 ordered differences of the twelve four-by-four
+shear frames by Fraction elimination, covering attainment directly.
+
+The first verifier rejects a missing witness matrix, duplicates, a height
+violation, Boolean or noninteger entries, a malformed shape, overlapping
+parts, a same-part rank-one change, and missing/repeated domain vertices.
+The second verifier checks elimination on zero, singular, nonsingular,
+pivot-swap and rational fixtures. All acceptance checks remain enabled under
+optimized Python.
+
+## Scope
+
+This is an exact finite result for D, with a rational construction proof.
+It does not bound larger coefficient heights, rational entries of bounded
+denominator, different matrix orders, or other fields. The elementary
+determinant and rank arguments are classical. No literature-priority claim
+is made for them. The new claim is the specified height-one optimum and
+its twelve-matrix attaining configuration.

research/height-one-shear-bicliques/README.md

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+# Height-one integral shear bicliques have exact balanced optimum six
+
+The largest balanced rank-distance biclique among the 81 two-by-two integer
+matrices with entries in {-1,0,1} is exactly **K_(6,6)**. Distinct matrices
+within each part have rank-two difference; every cross-part difference has
+rank one. The twelve associated integral four-by-four shear frames commute.
+
+Claim type: **exact finite certificate**, with a written construction and
+a proof reducing the ceiling to a complete finite enumeration.
+This contribution was produced by Nick Petalas. No new kappa is claimed.
+
+## Construction and ceiling
+
+For the six integer solutions used here of `a^2+ab+b^2=1`, take
+
+```text
+E(a,b) = [[a,b],[-a-b,a]],  O(a,b) = [[a,b],[b,-a-b]].
+```
+
+Their determinant identities prove attainment. For the ceiling, every
+pairwise rank-two triple in the complete 81-matrix inventory has at most
+six common rank-one contacts. Thus a K_(7,7) is impossible. Two separate
+enumerations agree on all 23,488 admissible triples and their complete
+contact-count histogram. [PROOF.md](PROOF.md) gives the identities, histogram
+and coverage arguments.
+
+Nick Petalas / petalas [#6174](https://github.com/CrocSwap/integer-mult-bounds/pull/6174)
+asks how coefficient height limits integral bicliques. Its all-size construction
+gives a quadratic-height upper bound and no exact height-one ceiling.
+This settles that first bounded inventory. Jerry Chen / CubeJerry
+[#907](https://github.com/CrocSwap/integer-mult-bounds/pull/907) supplies the
+earlier eight-frame example and the abstract distortion formula
+`c_1(K_(r,r))=2(r-1)/r`. Applying that cited theorem gives distortion 5/3
+for this new twelve-frame configuration; its cut proof is not replayed here.
+The rational shear model originates in eumemic
+[#288](https://github.com/CrocSwap/integer-mult-bounds/pull/288).
+The separate common-flag and signed-rank obstructions of navinor #825 and
+Jeremy Diviney #836 remain relevant. All predecessor attribution is retained
+in the unchanged source packages. Apache-2.0.
+
+Base main is `e54523517d0f03e58352e647c403a5af3e525a57`. The full original
+`complex-rank-shape-decoder` card is pinned in TASK.json. Its scope changes
+to a coefficient-inventory side result. Rank geometry and distortion alone
+do not supply a decoder, productive credit or a favorable complete invoice.
+The contextual picker reports its historical #5599 assessment differs from
+the selected #7000 record. Selected kappa `363005817/100000000000` is unchanged.
+The [advance claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19572817)
+reserves this exact height-one question.
+
+## Reproduce
+
+From the repository root, with CPython 3.12 and only its standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/height-one-shear-bicliques/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/height-one-shear-bicliques/ceiling.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/height-one-shear-bicliques
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/height-one-shear-bicliques --check-source-history --check-modes
+```
+
+A CPython 3.12 executable at another path can replace the local interpreter.
+VERIFY.json registers warning-strict normal and optimized runs of both
+verifiers, each with a 60-second timeout. Both commands are read-only and
+work without Git or predecessor files. The source pins are provenance only.
+No predecessor scientific executable is imported or replayed.
+
+The expected direct markers are `PASS_HEIGHT_ONE_SHEAR_BICLIQUES` and
+`PASS_FIRST_ROW_SHEAR_CEILING`. The primary JSON receipt includes the complete
+witness, histogram, ordered triple-trace hash and SHA256 of all seven package
+files. It is generated on every run. The separate checker uses Fraction
+elimination, a first-row enumeration and the actual four-by-four shear ranks.
+Ten rejection controls and five elimination fixtures test the stated boundary.
+
+## Limits
+
+Only the height-one integral two-by-two inventory is optimized. There is no
+physical native implementation, complete caller, dirty restoration, paid
+supplier invoice, compiler closure, tape/precision realization, complete
+multiplication theorem or discharge of any selected framework assumption.
+No selected cold replay, Lean or Docker run is relevant to this certificate.
+Independent human and formal proof-kernel review are unclaimed.
+
+## Open next questions
+
+1. What is the balanced optimum at height two? Start by reducing the contact
+   equations for its 625 matrices; a larger exact enumeration or a general
+   coefficient-height theorem is missing.
+2. Can these twelve frames participate in a favorable completed interface?
+   First specify legal gates, targets, arbitrary dirty inputs and paid cleanup,
+   and identify a hypothesis escaping the common-flag obstruction. None is
+   supplied here; the metric example alone does not imply a coding gain.
+
+## Verification-code bounds
+
+Both scripts use bounded loops and no recursion or subprocesses. The primary
+stores two 81-entry bitset graphs; the other stores an 81-by-81 rank table.
+No triple list is retained. Every function is below sixty physical lines.
+Errors propagate as exceptions, and checks do not depend on assertions.
+
+Power-of-Ten rule 3 has a managed-runtime deviation in both scripts. Python
+and Fraction allocate during enumeration. Input domains are fixed to 81
+four-entry integer matrices, matrices of order at most four for elimination,
+85,320 primary triple visits and 61,236 secondary candidates. Only the fixed
+graphs, one matrix/triple at a time and a histogram of at most 82 bins remain
+live. Normal and optimized replays verify those enforced domains. A port to
+fixed-capacity arithmetic would be needed for allocation-free execution.
+Python's dynamic dispatch is a runtime constraint; all scientific calls have
+fixed, direct targets and no callbacks, plugins or dynamic evaluation.

research/height-one-shear-bicliques/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",
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+    },
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research/height-one-shear-bicliques/TASK.json

@@ -0,0 +1,89 @@
+{
+  "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": "Determine the exact largest balanced rank-distance biclique in the height-one integral two-by-two matrix inventory.",
+  "scope_change_reason": "This answers the first coefficient-height case of PR6174 Open next question 2. It is a frame-alphabet side result, not a decoder or complete signed-moment improvement.",
+  "bottleneck_link": "A changed frame alphabet must have explicit coefficient and rank costs. This settles its smallest nontrivial bounded inventory; non-L1 geometry alone does not escape the common-flag obstruction or earn productive credit.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first exact height-one optimum with an attaining configuration and independent finite ceiling; leave larger heights and physical realization open.",
+  "outcome": "completed",
+  "evidence": "PROOF.md gives the six-by-six construction and finite ceiling reduction. verify.py enumerates every matrix triple; ceiling.py separately uses rational elimination and first-row buckets.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "What is the balanced optimum at coefficient height two?",
+      "why_it_matters": "The height-one ceiling does not quantify how the inventory grows with coefficient storage.",
+      "first_step": "Use rank-one contact equations to reduce the 625-matrix inventory before a bounded exact search.",
+      "blocked_by": "No larger-height enumeration or general height theorem is supplied."
+    },
+    {
+      "question": "Can the twelve-frame configuration participate in a favorable completed interface?",
+      "why_it_matters": "The charged metric and its distortion do not imply a coding or multiplication gain.",
+      "first_step": "Specify legal gates, target jobs, arbitrary dirty inputs and all cleanup costs, identifying a hypothesis outside the existing common-flag obstruction.",
+      "blocked_by": "No physical caller, native realization or complete supplier invoice is supplied."
+    }
+  ],
+  "environment": {
+    "python": "3.12.15",
+    "implementation": "CPython",
+    "system": "Darwin",
+    "release": "27.0.0",
+    "machine": "arm64",
+    "dependencies": "standard library only"
+  }
+}