[petalas]

Summary

Prove a scoped obstruction for the fixed H6 source complement over D=Z[1/2]: a dyadic correction B with BK=KB and I-K+B in GL(D) has exact minimum rational rank 816. In fact, ker B is contained in ker K. Every polynomial correction is covered, and arbitrary mixing inside repeated eigenspaces is allowed.

Unrestricted dyadic repair: 560, cited from Ross Richardson #3655
Commuting dyadic repair:    816, this proof
Rank-560 repair:            must break commutation with K

B=K attains 816 by cancelling the kernel response. This is an algebraic sharpness witness, not a useful source encoder. The contribution is a scoped written proof with finite exact corroboration, not a new kappa or a physical supplier. The invariant saturated kernel of B must carry a dyadic automorphism, which excludes all three nonunit characteristic factors of I-K.

Only the six-file research/h6-commuting-repair/ package is added. This contribution was produced by Nick Petalas. Ross Richardson #3655 supplies the H6 spectrum, symmetry and unrestricted 560-row completion as cited premises; those results are not replayed or audited here. Credit Dominik Scholz #606, DreamingOfClouds' H6/Walsh proposal, and the retained #1830 spectral, compensation and encoder lineage. PID module facts and characteristic polynomials are classical.

Evidence

Before, the cited 560-row completion left a paid physical synthesis open. After, the new proof shows that no commuting or polynomial correction can realize it at rank 560. Such a synthesis must break commutation or change the endpoint contract.

• Exact enumeration covers 390,625 ordered small matrix pairs, including 12,465 commuting pairs and 6,552 with a dyadic-unit repaired determinant. Every accepted pair satisfies the general rank obstruction; 1,856 attain their bound.
• 36 dyadic conjugations check literal abstract attaining endpoints. Three altered factor certificates, four hypothesis controls and ten invalid certificates pass. Noncommuting and odd-denominator examples show why the hypotheses matter.
• Local warning-strict CPython 3.12.15 normal and optimized commands pass, timed individually at 0.080 and 0.085 seconds. The registered replay passes in 0.412 seconds. These are timeout checks, not performance improvement claims.
• Source pins, exact six-file committed bytes, executable modes and declared source history pass at 6df64badfa3b1c2680c8306b863c4a4ca187a8d5. Official committed export integrity and a read-only, Git-free optimized replay with an empty executable PATH pass. Exported bytes and sealed modes remain unchanged. The verifier emits a generated SHA256-bound receipt without writing files.
• JSON parsing, Python compilation and AST inspection pass; all functions are below 60 lines. No external linter was available. Registered runs enable warnings as errors.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/h6-commuting-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/h6-commuting-repair/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/h6-commuting-repair
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/h6-commuting-repair --check-source-history --check-modes

Another CPython 3.12 executable may replace the local path. Direct output ends PASS_H6_COMMUTING_REPAIR rank=816 no_new_kappa; the launcher ends PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed. The universal H6 statement rests on PROOF.md and the cited spectral premise, not a finite full-H6 enumeration.

Canonical base is 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. The decoder card is explicitly changed to this fixed-source algebraic side obstruction. The selected metadata baseline passes. Live PRs, Discussions, main package statements and merged H6 work were checked before the advance CLAIM and refreshed before committing. The broad overlap snapshot covered 393 open PRs and 5,638 Discussion records; local sparse omissions were supplemented with main-tree and merged-PR searches. This is not an exhaustive novelty certificate.

Limitations and unrun obligations

The matrix dimension and dyadic coefficient ring are fixed. Noncommuting corrections, odd-denominator rational coefficients and stable extensions are outside this theorem. Correction rank is not a paid native-call count. The rank-816 witness removes the kernel response rather than implementing the intended source map.

No dense H6 matrix reconstruction, predecessor spectrum/Smith/repair replay, paid native word, source-frame synthesis, dirty-helper restoration, complete cost/precision/tape/stock calculation, supplier admission or selected cold replay was run. No registered scientific command is left for CI alone. Independent human and formal proof-kernel review are unclaimed. The selected frontier is unchanged.

Open next questions

• Construct explicit noncommuting coefficients for the cited 560-row completion and price their actual source-frame access, scalar work and dirt restoration. Those coefficients and a physical word remain missing.
• Test an enlarged dirty-helper interface only with a declared arbitrary-input endpoint and complete paid/target invoice. Stable extensions can evade this fixed-dimensional obstruction but carry new restoration and hosting obligations.

Related open work

The completion refresh inspected this account's twelve other open PRs created since the advance CLAIM: #6016 through #6022 and #6024 through #6028. None computes the fixed-H6 commuting-repair quantity or reports a competing rank-816/rank-560 value. No additional scientific computations were run to compare or validate those works.

• #6019 proves chronological helper-spill and cross-bank repair ranks. Its matrix, coefficient constraint and endpoint differ from fixed-H6 commuting repair; no common numerical repair rank is computed or compared.
• #6027 decides paired 2-integral lattices with rank-one projector anchors. It supplies no H6 repair or decoder rank, so no numerical comparison is made.

Merged #3655 is the cited unrestricted predecessor. Merged #3649 and #5233 change fitting-kernel families; this result fixes K and studies source-map repairs.

CI notes

At head 6df64badfa3b1c2680c8306b863c4a4ca187a8d5, both hosted checks passed: plan and replay (research/h6-commuting-repair), in run 40592506520. The completed check list was read once. Run metadata confirms that exact head, completed status and successful conclusion. The local scientific, source-history/committed-byte/mode and read-only export checks above also pass.

This is hosted execution of the registered finite checks and metadata gate. It is not independent human or formal proof-kernel acceptance of the unbounded written proof or its inherited spectral premises.

Merge danger

Door: two-way. One isolated research package can be reverted independently.

Blast radius: research. Shared code, frozen predecessors and selected records are unchanged.