Strengthen odd-unit rank bounds with balanced symplectic flags

[jmbarrancoml]

I strengthen PR #3512's odd-dimensional fitting-rank theorem by balancing two symplectic flags. In dimension five, I improve rank N >= ceil(v/9) to rank N >= ceil(17v/105). For bit rank at most five, I reduce the necessary port limit from 45 to 30. Under the retained model, copy charges of at least five now require C/v >= 17/21, above the source's .7275 scouting screen.

For odd d>=3, put t=2^(d-2) and rho=((d-3)t+1)/((2t-1)(t-1)). If p port occurrences have an all-ones source or sink label, I prove rank N >= p+ceil(rho(v-p)) over any coefficient field, with arbitrary multiplicities and nonzero diagonal values. I also retain the stronger maximum of two orbit-weighted inequalities. At dimension seven I obtain rho=43/651. For dimension-five identity sums of rank-15 and rank-1 idempotents over characteristic zero, I derive the necessary count 44 J_1 >= 75 J_15.

I also prove that no convex average of triangular principal-minor size bounds improves the uniform regular-type coefficient rho. I bound arbitrary triangular selectors by successive sink spans and show that either flag attains the maximum. I leave optimal fitting rank open.

I start this follow-up from canonical main 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Mister-Meeseeks merged PR #3512 during this extension; its archival intake remains partially reviewed. I retain its original source pins and frozen archive. I compared open work and Discussion claims through 2026-10-18 05:16 UTC, including complete cached coverage of 250 open PRs and 2,375 Discussion records. I found no matching claim within that coverage; I did not inspect arbitrary PR diffs or unpublished LP drivers.

I verified head 6cfa4333a2e593ea092fe4bda686bb172b5b6d22 with CPython 3.12.13 and the standard library:

• All six registered commands passed, including separate ambient and abstract symplectic implementations, 14 semantic/source controls, and retained baseline checks.
• Both flag engines agree on six selectors at dimensions three, five and seven. I checked 68,216 triangular seed positions and exact special-port elimination over Q/F2/F3/F5/F7.
• An explicit-annihilator dynamic program checked 32,768 sink subsets at dimension five, covering all 15! sink orders, and reconstructed a maximum 17-type regular selector. I also checked the dimension-three endpoint.
• Committed-byte/source-history validation passed for 17 package files. All six commands passed again from a Git-free export; I then compared all 18 exported regular files byte-for-byte with the archive.
• I confirmed successful planning and package replay for this final head in GitHub CI run 39100394180.

You can replay the package with python3.12 -B scripts/research_package.py --package research/odd-unit-triangular-rank. I record the proof, exact hypotheses, finite evidence and open questions in that package. I claim neither an attaining unit nor a complete native supplier invoice or improved integer exponent. Independent scientific review remains open; I leave the selected frontier unchanged.

I retain Philippe Van Velzen's acyclic-minor method and source examples, DreamingOfClouds' Lagrangian averaging in PR #2244 and pairing-rank lineage, and the predecessor notices. My contribution concerns the directed odd-label mask, the two flag counts and weights, and their unit consequences. Jose Manuel (jmbarrancoml), with substantial OpenAI Codex assistance in proof, code, checks and prose. Canonical investigation.