[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. This settles the displacement-rank-three part of Ali Romman's #6525 Open next question 1, over every binary symplectic ambient space and with prescribed fixed spectators.

For d=h+I, K=ker d, D=d(U), dim D=3, alternating residual form b of rank two, R=rad b and U_R=d^-1(R):

U_R contained in K+D      -> exact unsigned minimum 5
U_R not contained in K+D  -> exact unsigned minimum 4

The new result is the closed-case lower bound and the necessary commuting structure of every four-turn extension. A four-dimensional quadratic obstruction forces the hypothetical four-turn displacement space to be totally isotropic. The extension would then fix every closed radical lift, a contradiction.

Credit Ali Romman #6525 for the existing [4,5] interval, escaping-lift construction and known five-turn fixture. Its upper bounds are cited premises, not audited deliverables. Credit Krzysztof Wojt #5097, Nick Petalas #5085/#6494, Brian / JoyboyBrian #5051, David Ganz / Dugongue #4841 and the native/compiler lineage retained there. The lower-bound proof is self-contained; no predecessor executable or scientific receipt is replayed. Wall forms and binary quadratic forms are classical mathematics; no literature-priority claim is made.

Evidence

Before, #6525 leaves a one-turn interval for general closed rank-three contracts. After, PROOF.md excludes four turns for all of them, and the credited five-turn upper bound is sharp.

The generated source-hash-bound RESULT.json records:

• 1,024 binary quadratic forms and 15 hyperplanes per form; all 210 eligible nonzero-polar pairs have anisotropic span dimension three.
• 64 independent four-axis Gram topologies and 960 hyperplane restrictions. The only alternating rank-two canonical pair has commuting axes and 16 escaping radical lifts.
• 16,384 literal input vectors, 65,536 Wall basis equalities, 180 nonalternating closed controls, 256 omitted-turn controls and two exact-reason rejects.

These are finite coefficient-model counts. They do not enumerate all partial isometries or replace the arbitrary-dimension written proof.

From the repository root, using CPython 3.12.15:

~/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/rank-three-closed-completion/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/rank-three-closed-completion/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/rank-three-closed-completion
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/rank-three-closed-completion --check-source-history

All four commands passed locally at 876e984b1297918ef5ff9c4badf3c8ecc0afd484. Both direct checks printed PASS_RANK_THREE_CLOSED_COMPLETION and took 0.117 seconds each. The registered runner executed both VERIFY.json commands and printed PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed in 0.526 seconds total. Each registered check is bounded at 120 seconds. All seven committed files and declared base source ancestry passed PASS_RESEARCH_COMMIT_BYTES. Source compilation and whitespace checks also passed. The official committed-package export and declared-input coverage checks passed; both normal and optimized standalone replays passed from a read-only, Git-free export with empty PATH and unchanged file hashes. The selected-binding positive control passed, without a selected scientific replay.

Limitations and unrun obligations

The theorem is binary, has displacement rank three and prices unsigned transvections only. Attainment uses #6525's credited constructive upper bounds. Higher-rank closed contracts, signed Gaussian realization, absolute phase, physical execution/cost, owner cleanup, precision, stock, tape, compiler and outer admission are not supplied. The five selected assumptions remain. No new kappa or selected-frontier change is claimed. Lean, Docker and the selected cold replay were not run. Independent human and formal proof-kernel acceptance are unclaimed.

Open next questions

1. Classify alternating closed contracts at displacement rank four and above. Start with minimal Wall extensions containing a totally singular codimension-two subspace, retaining prescribed radical-lift action. The present quadratic obstruction no longer applies directly; Construct sharp rank-two residual completions and a one-turn bound #6525 already exhibits a closed r=4 attainer at the lower endpoint.
2. Bind a qualifying partial contract to an actual signed native contact and price its complete implementation. A short unsigned word alone gives no recursive saving.

Related open work

No other open PR studying this partial symplectic rank-three completion quantity was found in the live title/body and Discussion overlap search. The direct predecessor #6525 is merged. Other decoder, Schur residual-rank and scalar restoration questions use different objects and are not numerical comparisons here.

The final own-open-PR check since the CLAIM at 2026-11-11T13:19:11Z found #7261, #7263, #7264, #7265 and #7266 in addition to this PR. Their objects are bounded-reserve contraction, rational Stein factorization, BIT terminal rank capacity, Boolean single-insertion cost and the P120 quadratic pattern space. None studies this partial symplectic completion minimum; no numerical comparison is claimed.

Advance claim. Canonical main e545235, priority revision 8, selected #7000. Main, open PRs and claims were refreshed before committing; no matching concurrent target was found. The search does not cover every nested diff or unpublished result.

CI notes

CI certified nothing for this package at unchanged head 876e984b1297918ef5ff9c4badf3c8ecc0afd484. In run 43029249083, plan job 130551727341, the Onboarding helper tests step ran 760 tests with one failure and 24 skips. The exact failing test was test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record. Its assertion failed because the literal `make frontier7000-verify` was absent from docs/ci-verification.md. The package replay job was skipped because plan failed.

This mismatch is inherited from main commit e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both exact main files were inspected. This PR changes only its seven new package files; neither navigation file nor the failing test is changed.

The failed job logs were also inspected for every run below. Each reports the same sole failing test, missing make frontier7000-verify assertion, 760 tests, one failure and 24 skips:

• 42905094314
• 42903313790
• 43023514635
• 43024464841
• 43024693032

All registered scientific checks passed locally at this exact head as recorded above. No fix, retrigger commit or job rerun was made. Hosted package replay remains unrun.

Merge danger

Door: two-way. Blast radius: one new research package containing seven files. No shared code, selected record, frozen predecessor source or frontier is changed. New work is Apache-2.0.