[petalas]

Summary

Scoped proof with exact finite corroboration: no real or complex Gram line system attains the even-window Lagrangian cap at any h=2k >= 6.

- v <= 2^k T
+ v <= 2^k T - 1, for k >= 3

This answers Matarener's PR5282, open question 1: saturation v=8T at h=6 and v=16T at h=8 is impossible. The contract is rank_F2 A=2k and 1 in col A for the nonorthogonality adjacency. The proof also covers the stronger affine symplectic label contract with additional orthogonalities.

The new equality argument forces E_p+E_(p+w)=2^(2-k) I for orthogonal projections. Positivity contradicts this scalar being smaller than one. The prior cap and label reduction belong to DreamingOfClouds's PR2244; PR5282 supplies the explicit question and historical screen. Neither predecessor's construction nor invoice is audited or replayed. The new obstruction, written proof and standalone exact verifier were produced by Nick Petalas.

Package: research/even-window-saturation-gap. Card complex-new-helper-inventory, scope changed to the explicit line-system equality question. No new kappa or selected-frontier change.

Evidence

Before: the source question left the two saturated even windows open. After: PROOF.md excludes saturation uniformly for every k>=3.

The verifier independently enumerates affine generators by RREF and checks every complementary-label elimination certificate. At k=2,3,4 it enumerates respectively 12, 120 and 2160 affine generators, with 4, 16 and 64 pair certificates, forcing scalars 1, 1/2 and 1/4. The k=2 complementary-projection control survives. Thirty exact scalar instances and nine malformed-certificate/dimension cases pass their required checks. These finite checks corroborate the new elimination step; the all-k claim rests on the written proof.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/even-window-saturation-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/even-window-saturation-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/even-window-saturation-gap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/even-window-saturation-gap --check-source-history

All pass locally. Direct normal and optimized checks took 0.480 and 0.481 seconds, below their 120-second registered limits, and their generated source-bound receipts agree. The package runner reports PASS_REGISTERED_RESEARCH_PACKAGE; publication checks report PASS_RESEARCH_COMMIT_BYTES for all eight package/input files and confirm the declared base is an ancestor of the publication commit. Syntax inspection and warnings-as-errors pass. CPython 3.12 and the standard library suffice; another machine may substitute its Python 3.12 path.

Limitations

The integer gap of one does not establish an optimal density gap independent of T, or exclude ratios approaching 2^k. Arbitrary pairings and non-Gram decoders are outside scope. No native word, dirty restoration, complete current signed invoice, compiler admission or complete multiplication theorem is supplied. The selected frontier and five framework assumptions remain unchanged. Independent human and formal proof-kernel review are unclaimed; Lean, Docker and historical supplier suites were not run.

Open next questions

• Determine the optimal v/T supremum at h=6 or h=8. Start with positive semidefinite deficits in the affine basis equations; a quantitative deficit estimate is missing.
• Price a concrete nonsaturated even-window system on a current complex caller. A literal decoder, native realization, dirty restoration and a complete signed invoice are required before a gain can be claimed.

Related open work

The account's open PRs created since the advance CLAIM were refreshed after CI. No additional PR on even-window line-system saturation was found. Related decoder and Gram context is:

• PR6267 studies owned-lineage floors on the fixed P104 mask; this result excludes equality for new even-window Gram inventories.
• PR6275 determines the P104 rank-only owned-delivery support minimum; it concerns a fixed-mask matrix relaxation rather than even-window Gram saturation.
• PR6278 proves a P104 common-metric defect-energy identity; it concerns quantitative orthogonality under deformation rather than affine-label saturation.

No shared numerical quantity is computed by this result, so no numerical agreement or disagreement is asserted. The question source PR5282 and line-count predecessor PR2244 are merged archival work.

CI notes

Both hosted checks passed at 4280ef185156ac701d6b84c21ec57a182066611d: plan and replay (research/even-window-saturation-gap). Run 41066904694 completed successfully at the matching head. The completed checks were read once, and run metadata separately confirmed the head and conclusion. Both registered commands also passed locally. CI certifies the registered finite replay and package controls; it is not independent human or proof-kernel review of the all-k written proof.

Merge Danger

Door: two-way. Blast radius: isolated. Only the new six-file research package is added; shared code and the selected frontier are unchanged.