P112: exclude cost 296 in the aligned six-mode factor family

[flyingviper]

Result and scope

This closes one concrete remaining route in the PR2757 rational-decoder investigation: for the specified aligned six-mode bilateral factor spaces and fixed endpoint K = R R^T, every finite rational decomposition costs at least 297 under the whole-group tariff. It therefore cannot meet the necessary integer gate C <= 296.

The result permits arbitrary rational coefficients and finite term counts within the stated spaces. It does not cover other fitting matrices, partitions or relative orientations, and does not claim that 297 is attained. The known complete decoder still costs 334. No native supplier, selected-record change or improved multiplication bound is claimed.

The proof combines the canonical high-price-excess bound, a mandatory negative residual loss, proportional coalescence and a quotient-deficit inequality. A support-preserving rational gauge reduces all hypothetical target representations to 4,928 families and 5,440 term-count cases. Explicit subspace/line-cover certificates exclude every case. The proof accounts for opposite factor ranks potentially dropping during coalescence.

Review and reproduction

Only research/p112-six-mode-exclusion/ is added. It contains the complete proof, the unchanged source-bound PR2757 cover, a deduplicated explicit cover certificate, source/task metadata, and two standalone Python 3.12 standard-library replays. Each reconstructs the finite geometry, validates actual threshold edges and computes exact ranks and bounds; neither trusts saved lower-bound numbers or reruns discovery searches.

python3 -I -O -B research/p112-six-mode-exclusion/verify.py
python3 -I -O -B research/p112-six-mode-exclusion/independent.py
python3 scripts/research_package.py --package research/p112-six-mode-exclusion

Validation: both mathematical replays, source/metadata checks, optimized-mode guards, meaningful malformed-certificate controls, and clean Git-free export verification pass. The selected cold replay and historical Lean suites were not run for this isolated scoped proof. Independent checks are AI-agent checks, not human scientific admission or a proof-assistant verification of the entire theorem.

Source binding and overlap

Base: canonical main d8e0f83f00191df62174b641e9fff055c357f641, refreshed before packaging; priority revision 6 and the selected result are unchanged. Scientific inputs retain the exact PR2757 candidate and selected replay archive/member hashes. The new portable package does not depend on unpublished scratch files.

The coverage-law-change task is narrowed to this source-bound decoder obstruction; its scope status is changed from the original coverage-law construction card. TASK.json records the concrete spaces, finite domain, budget connection and stop rule. The necessary optimistic positive-saving gate is C < 2672/9, hence integer C <= 296; this obstruction closes only the stated family's route through that gate.

The submission overlap snapshot covers the 100 most recently updated open PR bodies, with exact heads in TASK.json. PR3026 is a different five-triad low-tariff theorem. PR2985 and PR3029 concern root-line residue/deficiency budgets, and PR2832 different singleton-decoder contracts; none is imported as a proof premise. Discussions remained unavailable after an earlier HTTP406, so cached context was retained without retry and no complete novelty coverage is claimed.

Published head 71a0ec5012a04965826c62ad27532b5a4e014480 changes exactly the nine package files; every published file was fetched and byte-compared with the validated clean-export package.

Prepared by flyingviper with substantial OpenAI Codex assistance in the original research, proof/certificate packaging and separate-agent audits. Apache-2.0; predecessor geometry, mask and pricing attribution is retained in the README and source manifest.

Open next questions

Changed fitting matrices or relative source/reader geometries remain open: first specify the changed spaces and fully price their supports. The sharp minimum within this family is also unknown between 297 and 334. These are handoff questions; further research remains paused.