[petalas]

Summary

Prove the sharp local spectral envelope at the larger-support boundary of JoyboyBrian's PR #5507. For every characteristic-two field and idempotent D of rank k in dimension r:

A+B=D, z = algebraic zero multiplicity
max(z(A)+z(B)) = r       if k=r
                2r-1    if 0<k<r is odd
                2r      if 0<=k<r is even

Explicit binary matrices attain every case. At odd r>=3 and k=r-1, every maximizer has two full-length nilpotent Jordan chains. Already r=5,k=4 attains ten, while extrapolating the ordinary 2r-k bound would give six. The excess over that expression is unbounded as support size grows.

Claim type: scoped algebraic proof with exact finite corroboration. The contribution was produced by Nick Petalas. Credit JoyboyBrian #5507 for the triple exception and global compensation, Ali Romman #5456 for the early-helper model, and the retained #5452/#5388/#589/#555 BIT/native/owner/compiler lineage. General nilpotent-sum and zero-diagonal splitting methods are classical, with primary mathematical attribution in PROOF.md. No general linear-algebra priority is claimed. Predecessor mathematical verifiers are neither imported nor replayed.

Evidence

Before: #5507's local lemma covers dimensions at most three; its larger-support boundary has no sharp local envelope there. After: the written proof supplies the exact all-dimension law, coordinate witnesses and corank-one chain constraint.

Local CPython 3.12.15 normal and optimized checks pass:

• 594 constructed rank/dimension cases through dimension 33.
• All 329,780 binary ordered pairs with coordinate D through dimension four.
• 66,066 independent characteristic-polynomial checks against matrix powers/elimination.
• Ten invalid-domain and forged-witness rejection controls.

Both registered commands ran locally and each direct run took 0.78 seconds. Metadata/source pins, all six committed files and full source-history ancestry pass. The official export integrity check passes. Both read-only, Git-free export replays with empty PATH pass in 0.78 and 0.79 seconds; their outputs equal both local reports byte-for-byte. A generated external receipt binds all six package files by SHA256. Local environment: macOS 27.0.1 arm64, CPython 3.12.15, standard library only. These timings describe bounded replay duration, not a performance improvement.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/shared-block-spectral-envelope/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/shared-block-spectral-envelope/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/shared-block-spectral-envelope
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/shared-block-spectral-envelope --check-source-history

Any CPython 3.12 installation can replace the local interpreter path. Expected markers are PASS_SHARED_BLOCK_SPECTRAL_ENVELOPE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Base: canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8, refreshed unchanged before publication. The bit-helper-contract card changes to this local spectral side result. Advance CLAIM. Exact predecessor bytes are provenance pins, with no external runtime inputs. Live PRs, handoffs, local reservations, merged-object/formula searches and the overlap cache were checked. Older arbitrary diffs, exhaustive historical Discussion coverage and unpublished work remain outside that search.

Limitations and unrun obligations

Only characteristic two is covered. The finite binary census ends at dimension four; universal quantifiers rely on the written proof. Local matrices do not supply chronological support histories or legal native programs. Larger-support global compensation, native-chain constraints, independent symbols, ownership, physical coefficients, dirty restoration, precision, tape, compiler and complete supplier/outer accounting remain open. No kappa or selected-frontier change is claimed. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, selected scientific replay or predecessor replay was run. Every registered package command ran locally; none is deferred to CI.

Open next questions

• Determine whether a declared larger-support caller can compensate this exact local spectral surplus. Start with its chronological blocks, native chain and complete ownership/target/paid ledger. No completed caller is supplied here.
• Determine the sharp envelope with an explicit nilpotent-chain cap. At r=5,k=4, chains of length at most four cannot attain ten by the proved rigidity; the next-best constrained value is left open.

Related open work

At publication, live object/source searches found no open PR studying this shared-block zero-multiplicity envelope. The completion refresh lists this account's open PRs created since the advance CLAIM at 2026-10-27 13:32:51 UTC and returns zero; no additional related open work arises from those records.

#5507 and #5456 are merged predecessors. #5527 studies a shared-D controller rank/invoice obstruction under a different alphabet; its result is cited as adjacent work, with no extra replay or numerical comparison. Spectral-completion and Clifford-Gram results concern different operators and quantities.

CI notes

Both hosted checks passed at head 0f4419bc55adf4fe9b2469f98f7e8f95a21b6b6d in Registered research packages run 40514233480: plan in 4m40s and replay (research/shared-block-spectral-envelope) / replay (research/shared-block-spectral-envelope) in 1m27s. The run is completed with conclusion success, and its head SHA matches this contribution. Hosted replay executes both registered normal and optimized checks. All local checks listed above also passed; no registered check was deferred to CI.

GitHub records an archive merge at 2026-10-27 13:43:54 UTC through bulk queue commit aacce1a2c411a2f083b5fd5f1cdcf87661267ca3. Under the research archive policy, this preserves the source and attribution without establishing mathematical approval or a selected-frontier change. Passing hosted replay certifies the registered checks at the stated head; independent human and formal proof-kernel review remain unclaimed. No new commit was pushed after the archive merge.

Merge danger

Door: two-way. Blast radius: isolated. Six files are added in one new research package; no shared code, frozen evidence, predecessor credit or selected-frontier records change.