[petalas]

Summary

Scoped algebraic proof with finite corroboration. For commuting idempotents P,Q over any field with common leading scalar entry 1, their Schur complements S,T obey

h = (PQ)[1,1], Z = (S-T)^2, C = ST-TS, theta = h(2-h)
(Z-theta I)C = 0
Z(Z-I)(Z-theta I) = 0

The proof confines im C to span(e-c,Tc) and computes the square of S-T there. When 0,1,theta are distinct, Z is diagonalizable and its theta-eigenspace is exactly im C, of dimension at most two. An attained ternary deletion has Z nonzero and Z^2=0, so the repeated zero factor is necessary.

This answers one bounded part of codyw912's #4735 open question on intrinsic higher-rank residual invariants. It stops at this identity. It supplies neither a sufficient conjugacy invariant nor a new full-guard count, supplier, kappa or selected-frontier change. Only the five-file research/schur-central-spectrum package is added.

This contribution was produced by Nick Petalas. Credit codyw912 #4735, Mihir Srivastava #4689, Ryanson Jonathan #4609, ssafwann #4678/#4671, Nick Petalas #4597/#4278, lee-phys, Cinna Davis and the retained BIT/header lineage. Their distinct predecessor results and exact source hashes are recorded in README.md and SOURCES.json. No predecessor executable is imported or audited.

Advance CLAIM. Canonical base is 624de9d, priority revision 7; formal-interface-lemma is explicitly changed to an algebraic side question.

Evidence

Before, #4735 leaves a sufficient higher-rank invariant open, and #4609 gives centrality for arbitrary idempotent pairs. After, the written block proof identifies h(2-h) as the scalar action on the commutator image for this deletion contract.

• All 80,656 split pairs are enumerated across three specified profiles. The 172/306/4,260 admitted pairs over F2/F3/F5 pass the new identities. The 172 binary admissions are credited corroboration of Classify seven residual Schur-pair states and full-guard counts (no new κ) #4689's first working example, not a new count.
• Twelve completing headers per profile directly compare the lifted matrices with literal Schur deletion. Another 28 literal constructions cover every h over F2,F3,F5,F7,F11. The separate ternary control constructs its full header and checks the repeated-root witness.
• Eight rejection controls cover required hypotheses and false replacements of the central identity. Explicit exceptions remain active under -O.
• Both registered warning-strict commands pass locally on CPython 3.12.15 / macOS arm64. Direct durations are 0.588 seconds normal and 0.592 seconds optimized, below each 60-second cap. These are check durations, not speed comparisons. The package runner emits PASS_REGISTERED_RESEARCH_PACKAGE with outcome=completed.
• Committed-byte/source-history preflight passes for 98bc071. Official export integrity and an optimized Git-free, empty-PATH, read-only export replay pass. All six exported files remain unchanged and the complete output, including generated hashes of the five package files, agrees.
• A separate read-only proof and verifier review found no mathematical or verification blocker. Independent human and formal proof-kernel review are unclaimed.

From the repository root:

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

Expected markers are PASS_SCHUR_CENTRAL_SPECTRUM, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Any CPython 3.12 interpreter can replace the local path. Source-history checking needs full Git history; the standalone verifier also runs from an exported package without Git.

Limitations and unrun obligations

The all-field claim rests on the written proof, with finite corroboration. It assumes commuting field idempotents and one common scalar identity pivot. It is a necessary residual-state constraint, not a complete classification. Larger block pivots, subsequent residual transitions, complete-header frequencies and coefficient rings remain open. Header selection/emission, physical reads, restoration, stock, precision and the complete consumer bill remain unpriced. No physical supplier admission or complete multiplication theorem follows.

Every registered command ran locally. Lean, Docker, the selected scientific cold replay and physical caller admission were not run. Bounded managed-runtime allocation and dispatch deviations are documented; the longest function is 35 lines, and no recursion or external executable is used by the verifier.

Related open work

• Prove the exact scalar-block Schur dimension formula (no κ) #4785 studies dimensions for common scalar block pivots and arbitrary product H. At F5, lambda=1,d=1,h=3, its formula gives minimum n=4. The present exhaustive n=4 profile has 256 admitted splits at h=3 with rank(C)=2 and theta=2, agreeing numerically on attained dimension. We prove a central polynomial, not a new dimension minimum.
• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 studies distinct nonzero scalar pivots p!=q. This theorem requires p=q=1, so there is no shared parameter point or overlapping numerical minimum to compare.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 20:13:10 UTC were refreshed. Nine were returned, including this contribution; the three new related bodies were read.

• Count shared Schur-product rank pairs without matrix enumeration (no κ) #4803 counts shared Schur-product rank pairs and one-block sharp binary headers. For the common (1,1,1,2), d=1 example, its sharp/identity split counts are (3,172). The present saved exact histogram has three rank-two commutator splits among 172 identity-pivot splits, agreeing numerically. The new central polynomial and its counting law are distinct results.
• Count positive correlation for overlapping ternary full guards (no κ) #4806 counts full guards for the commuting ternary rank-two pair with sector profile (1,1,1,1). Its separately reported 306 splits with both first pivots equal to one agree exactly with this package's ternary admission count. Its continuation probability is a different quantity and is not claimed here.
• Prove positive binary correlation for a common-covector guard pair (no κ) #4807 counts full guards for a noncommuting common-covector pair. This theorem requires commuting input idempotents and supplies a central polynomial, so there is no overlapping numerical probability or header count to compare.

The other new account PRs, #4804/#4805/#4808/#4809/#4810, concern right-vector factor costs and coefficient-algebra target counts. They assert no overlapping central-spectrum or Schur-header number. No disagreement was found in the shared numbers above. The related-work comparisons use this package's saved exact output; they are not predecessor verifier audits. Main remained unchanged at submission refresh. TASK.json records complete remote cache pagination for 325 open PRs and 3,742 Discussion records, plus the sparse local-note limit and direct main/merged searches. Arbitrary diffs and unpublished work are outside coverage.

Open next questions

1. Combine this central constraint with image/kernel data to obtain a sufficient residual invariant. Start with the ternary nilpotent central blocks; sufficiency is not established here.
2. Derive an ordered replacement for h(2-h) at a common block pivot I_d. Matrix coefficients need not commute, so scalar substitution is unjustified.

CI notes

Both hosted checks passed on contribution head 98bc071317b17fd8bb4f21044328914c7561d3cb in Registered research packages run 39847984317: plan in 3m18s and the registered research/schur-central-spectrum replay in 1m17s. The registered replay executes both warning-strict normal and optimized commands. All registered local checks, committed-byte/source-history preflight, export integrity and read-only Git-free export replay also passed. Hosted CI certifies execution of the stated bounded checks; independent human and formal proof-kernel acceptance of the all-field theorem remain unclaimed.

Merge danger

Door: two-way. Blast radius: package. This isolated addition changes no shared implementation, frozen source, predecessor attribution or selected-frontier file.