[petalas]

Summary

The height-one integral two-by-two matrix inventory has exact balanced rank-distance biclique optimum six. Twelve explicit commuting integral four-by-four shear frames attain K_(6,6), and a complete finite certificate excludes K_(7,7).

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]]
within-part difference rank = 2; cross-part difference rank = 1

Claim type: exact finite certificate, with a written construction and a proof reducing the upper bound to exhaustive computation. This contribution was produced by Nick Petalas. No new kappa or physical supplier is claimed.

This answers the first coefficient-height case of Open next question 2 in petalas #6174, published October 29. That predecessor supplies an unbounded construction with quadratic-height upper bound; it does not give this exact bounded optimum. Credit CubeJerry #907 for the earlier eight-frame example and its abstract distortion theorem, eumemic #288 for the rational shear model, and navinor #825 / jeremydiviney #836 for the separate obstruction scopes. The cited distortion formula gives 5/3 on the new twelve-frame configuration. No predecessor scientific verifier is replayed.

Overlap refreshed before committing: all 768 cached open PR bodies and 7,363 Discussion records had complete remote pagination; the sparse local scan omitted 9,962 note paths. Separate distinctive-term searches covered main READMEs, merged PRs, live open biclique work and current claims. No matching height-one result was found; this is bounded overlap evidence, not proof of novelty.

Base main: e54523517d0f03e58352e647c403a5af3e525a57. Advance claim. TASK.json pins the full revision-8 complex-rank-shape-decoder card and records the changed scope as a coefficient-inventory side result. The contextual assessment is historical #5599, while the selected record is #7000 at kappa 363005817/100000000000; that selected record is unchanged.

Evidence

Before: the cited coefficient-height question had no height-one ceiling. After: the explicit six-by-six construction and two exhaustive upper checks agree.

• The determinant checker visits all 85,320 unordered triples among all 81 matrices and finds exactly 23,488 pairwise rank-two triples. Every one has at most six common rank-one contacts. The complete contact-count histogram is {0:1480, 1:4368, 2:9624, 3:5760, 4:2096, 6:160}.
• The separate checker imports no scientific code from the first. Exact Fraction elimination and all 61,236 first-row candidates recover the same histogram. It also checks all 144 ordered differences of the actual four-by-four shear frames.
• All four warning-strict normal/optimized VERIFY.json commands pass locally, each below one second against a 60-second limit. Ten rejecting controls, five elimination fixtures and both unexpected-argument controls pass. Generated receipts bind the witness, complete ordered triple trace and all seven package hashes.
• Registered replay and source metadata pass. At head dadc7711829dfe8fbdcf663c466500395f5a0295, all seven committed files, executable modes and full declared source ancestry pass. The official export passes integrity checks. All four commands also pass from a read-only Git-free export with empty PATH; every exported byte and mode is unchanged.

Local environment: CPython 3.12.15, Darwin 27.0.0 arm64, standard library only. Exact reproduction from the repository root:

/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 may replace the recorded interpreter. Expected direct markers are PASS_HEIGHT_ONE_SHEAR_BICLIQUES and PASS_FIRST_ROW_SHEAR_CEILING. The registered replay runs both verifiers in normal and optimized mode.

Limitations and unrun obligations

Only integral two-by-two entries in {-1,0,1}, with rational ranks, are optimized. The result does not bound higher heights, bounded-denominator rational inventories, other dimensions or other fields. It gives no coding gain, legal native implementation, caller, dirty restoration, productive target bill, compiler closure or tape/precision implementation. The common-flag obstruction is not escaped by a non-L1 metric alone. All five selected framework assumptions remain. No selected cold replay, Lean, Docker, complete multiplication theorem, independent human review or formal proof-kernel acceptance is claimed.

Open next questions

• Determine the balanced optimum at height two. First reduce the contact equations for its 625 matrices; a larger finite search or a general coefficient-height theorem remains missing.
• Supply a legal completed interface using the twelve frames, with explicit targets, arbitrary dirty inputs and paid cleanup, identifying a hypothesis beyond the known common-flag obstruction. No such realization or favorable complete bill is supplied here.

Related open work

No open PR studying this bounded coefficient-height inventory was found in the live biclique/rank-metric searches and claim refresh. #6174 and #907 are merged predecessors. Other P104 subspace biclique packages study a different matrix inventory and do not compute a competing height-one optimum. The completion check read this account's seven other open PRs created since the CLAIM: #7469, #7471, #7472, #7473, #7474, #7476 and #7477. None studies this coefficient-height inventory or computes a competing optimum; no extra computations were run to check their results.

CI notes

CI did not certify this package. At unchanged head dadc7711829dfe8fbdcf663c466500395f5a0295, run 43289883200, plan job 131432985031, failed in Onboarding helper tests on exactly one of 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found

I read the completed checks once and inspected the actual job log. The package replay job was skipped because plan failed. No hosted mathematical or package verification is claimed.

I inspected main's own files at e54523517: certificates/selected-result.json requires make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. The test at line 58 requires the selected command in that section. Neither those files nor the test is changed by this package.

I independently read the failed-job logs of run 42905094314 / job 130140894651, run 42903313790 / job 130134720799 and run 43289715434 / job 131432409153. All three show the same sole failing test, assertion, 760-test total and 24 skips. This is the existing repository-level navigation mismatch. The package's local normal/optimized, registered replay, source-history and read-only export checks passed as recorded above. No fix, retrigger commit, push or workflow rerun was made; maintainers own the repair of main.

Merge danger

Door: two-way. Blast radius: one isolated seven-file research package. Shared code, predecessor files, frozen evidence and the selected frontier are unchanged.