[petalas]

Summary

Scoped proof over characteristic zero. For the complementary four-subset kernel of Cinna Davis / 1337isnot1337's #2225, prove the full spectrum and that I-K is invertible for every odd n >= 5.

K(S,T) = binomial(|S intersection T|/2,2)
n = 9: rank(K) = 36; rank(I-K) = 126
whole side response through h linear intermediates, no direct bypass: h >= binomial(n,4)

This closes one algebraic part of #2225's open side-map obligation. The no-bypass floor is sharp as a linear factorization statement; its attaining factorization is not assigned a physical cost. The source's fitting/support theorem is cited, not audited. Ali Romman's merged #5458 studies a cube restriction and separate binary endpoint hulls; this result concerns the complete constant-weight shell. Dominik Scholz's #606 already observes full-rank compensation in another geometry. The new contribution is this four-subset spectrum and its consequence, not the general rank-obstruction method.

Only research/four-subset-side-spectrum/ is added. This contribution was produced by Nick Petalas. Credit Cinna Davis for the kernel, dyadic incidence construction and side-map question, with the retained Zhihao Chen subset/compiler ancestry. No supplier, new kappa or selected-frontier change.

Evidence

Before, #2225 leaves I-K unimplemented and supplies no full-shell side spectrum. After, the written incidence decomposition proves the spectrum and excludes eigenvalue one for every allowed n. The standalone verifier builds the new side matrices from intersections without importing the predecessor's implementation.

Local CPython 3.12.15 results:

• All 17,126 literal matrix entries at n=5,7,9, ten exact eigenspace dimensions, three full side ranks and three complete determinants pass.
• Literal counting checks 1,737 incidence-Gram entries. Scalar formula checks cover 499 odd n through 1001. Three malformed spectra reject, and an actual smaller compression has deficient rank. These bounded checks corroborate the written all-n proof.
• Warning-strict normal and optimized direct commands pass in 0.80 and 0.77 seconds locally, below their 60-second limits. Their generated receipts agree byte for byte and bind all six package files by SHA256. These are replay durations, not a performance improvement claim.
• Registered replay, metadata/pins, in-memory compilation and whitespace checks pass. Committed-byte/source-history checks pass for all six files at 4219727be784f329a5bf2949cb4c29f0a04456e8.
• Official committed export integrity and input coverage pass. Both direct modes pass in the read-only Git-free export with PATH=/nonexistent; their receipts equal local receipts and all seven exported files retain bytes and modes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/four-subset-side-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/four-subset-side-spectrum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/four-subset-side-spectrum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/four-subset-side-spectrum --check-source-history

Another CPython 3.12 path can be substituted. Expected markers are PASS_FOUR_SUBSET_SIDE_SPECTRUM, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

This covers k=4 in characteristic zero. Direct source access plus a compressed -K correction escapes the no-bypass hypothesis. No general arithmetic-circuit, native-call or recursive-rank lower bound follows. The rational kernel rank does not implement a smaller dyadic feature basis. A physical side word, dirty restoration, both native orientations, hosting, precision and a complete invoice remain open. The five selected framework assumptions remain. Independent human/formal proof-kernel review, a selected cold replay, Lean and Docker are unrun.

Open next questions

• Construct a legal identity route plus compressed -K correction at n=9, starting with explicit frames and arbitrary dirty restoration, then price the complete word.
• Determine when one is an eigenvalue for other even k by deriving its incidence action. This PR proves only k=4.

Related open work

No open PR studying this complementary four-subset side spectrum or its no-bypass intermediate dimension was found in the complete cached 393-PR snapshot and refreshed live open lists. Merged #2225 is the direct predecessor; merged #5458 concerns the cube restriction. Sparse local notes and unpublished work remain coverage limits. No extra computations were run to audit other contributions.

The completion refresh read this account's fifteen other open PRs created since the CLAIM at 2026-10-27T21:56:38Z: #6013 and #6015 through #6028. None computes this four-subset side spectrum. One adjacent side-complement result is distinguished:

• Prove H6 commuting dyadic repairs need rank 816 (no new κ) #6023 studies the minimum commuting dyadic correction rank for a fixed H6 complement I-K. This PR determines the full side-map rank of the complementary four-subset shell. The objects and rank objectives differ; no common numerical quantity is compared.

CI notes

Both hosted checks passed at head 4219727be784f329a5bf2949cb4c29f0a04456e8: plan and replay (research/four-subset-side-spectrum), in run 40591648558. The completed checks were read once; the run conclusion is success and its head SHA matches the PR. Hosted replay certifies the registered bounded checks. The every-n statement remains a written proof, with the limitations above. All registered commands and publication/export checks also passed locally.

Merge danger

Door: two-way. Blast radius: isolated research package. Shared code, frozen evidence and the selected record are unchanged.

Coordination: #1166 (comment)