[petalas]

Summary

Scoped counting proof with finite corroboration. For any ordered idempotents S,T over F2 on a positive-dimensional space, invertibility of ST-TS forces dimension 2r, r>=2, and gives

passing first-pivot splits = 2^(2r-3)(2^(2r)-1)
passing complete headers  = |GL_(2r)(F2)| / 4

Both literal first pivots must be nonzero. The proof counts every first-vector case and then restores the complete-header fiber. It does not assert a full-guard chain, a general characteristic-two law, supplier admission or a new kappa. Only the five-file research/generic-idempotent-first-guard package is added.

This contribution was produced by Nick Petalas. Credit MatShep #4762 and codyw912 #4735 for the open transition-count direction, factor invariant and split/completion framework; Mihir Srivastava #4689 for the earlier residual-state question; ssafwann #4678/#4671; Nick Petalas's Schur/header lineage; Ryanson Jonathan, lee-phys, Cinna Davis and the retained normalizer contributors. The elementary block normalization is background, not claimed as a new general classification. No predecessor executable is imported or replayed.

The F2-scoped CLAIM replaces the earlier prediction, whose excessive characteristic-two wording is preserved in the explicit correction. The (q-3)E correction persists over F4; this PR claims only F2. Canonical base remains 624de9d, priority revision 7. Publication head is fc36a3b.

Evidence

Before, the cited invariant packages leave transition frequencies open. After, the written proof supplies an exact dimension-uniform first-pivot density for the invertible-commutator stratum.

• All 16 binary 2-by-2 and 512 binary 3-by-3 compressions are enumerated. Their 2 and 48 admissible cases respectively have 30 and 504 passing splits.
• A separate scan of all 65,536 binary 4-by-4 partner matrices finds 802 idempotents and 12 partners with invertible commutator for fixed S=diag(I2,0). All 12 have 30 passing splits.
• Literal inversion and conjugation of every GL4 header gives exactly 5,040 passing headers out of 20,160. This independently checks the 30*168 completion factor. An explicit first-passing header fails a second leading minor.
• Seven scope/semantic controls pass. Checks use explicit exceptions and remain enabled under optimization. Generated stdout receipts bind all five source files; none is handwritten or committed.
• Both registered warning-strict commands and the package runner pass locally on CPython 3.12.15 / macOS arm64. Final direct durations were 0.192 s normal and 0.192 s optimized, below each 60-second cap. These are check durations. The measured registered replay took 0.640 s.
• Committed-byte/source-history preflight passes for all five files at fc36a3b. Official committed-export integrity passes. An optimized Git-free replay with empty PATH and read-only exported files passes in 0.190 s, matches the local source-bound output and leaves all six exported files unchanged.
• Syntax and function-size inspection pass; the longest function has 23 physical lines. Every domain is explicitly bounded. Managed-runtime allocation is documented; no external static analyzer was installed.

From the repository root:

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

Any CPython 3.12 interpreter may replace the local path. Expected markers are PASS_GENERIC_IDEMPOTENT_FIRST_GUARD, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct verifier writes no files and runs from a standalone package without Git. Source-history preflight requires full Git history.

Limitations and unrun obligations

The all-dimension F2 claim rests on the written proof, with finite corroboration. It requires idempotents and an invertible commutator on a positive-dimensional space. Other fields, singular commutators, restricted/weighted header covers and later guards are outside scope. A successful first pivot does not justify another quarter-law step.

Header selection/emission, actual producer compatibility, reads, dirty return, precision, stock and the complete physical consumer bill remain unknown. No physical supplier or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Every registered local check ran; Lean, Docker, native builds and the selected scientific cold replay were not run.

Related open work

• Prove an all-field central-spectrum law for Schur deletion (no κ) #4801 proves a central-spectrum identity after scalar Schur deletion. This result counts first pivots for a specified residual stratum; it does not compute that identity or its deletion-profile counts.
• Count shared Schur-product rank pairs without matrix enumeration (no κ) #4803 counts compressed-product rank pairs and sharp one-block Schur headers. Our already-enumerated compression populations, 2 at r=2 and 48 at r=3, agree with its displayed joint-nullity formula at (0,0). The first-pivot quarter law is a different header event.
• Count positive correlation for overlapping ternary full guards (no κ) #4806 counts full guards for a commuting ternary rank-two pair. This theorem is binary, starts with noncommuting idempotents and counts only the first pivots.
• Prove positive binary correlation for a common-covector guard pair (no κ) #4807 counts full guards for a common-covector corank-one pair. This theorem requires an invertible commutator and counts a different request orbit and event.
• Prove the exact scalar-block Schur dimension formula (no κ) #4785 studies minimum dimensions for common scalar-block Schur pivots and prescribed products. The present package supplies a header frequency, not a minimum-dimension construction.
• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 studies minimum dimensions for unequal scalar Schur pivots. That dimension question and this first-pivot count are distinct.
• Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777 proves a full-guard inequality for three orthogonal binary corank-one requests. This theorem concerns two idempotents with invertible commutator and a single guard order.
• Prove all-field negative correlation for one-way full guards (no κ) #4778 studies full-guard correlation for one-way coupled kernels. The coupling and guard event differ from this theorem.

Before DONE, this account's open PRs were refreshed. Among those created since the F2-scoped CLAIM at 2026-10-22 20:39:45 UTC, only this PR was returned. No additional related open work or numerical comparison was needed.

The compression populations above are the only shared numerical quantity already computed by this package. Their comparison evaluates the credited displayed formula; no predecessor verifier was replayed. No other numerical comparison was added. Their bodies and recent claims were checked for overlap. Refreshed helper coverage included 340 open PR bodies and 3,772 Discussion records with complete remote pagination; direct main/merged searches and a fresh precommit open-PR/claim check supplemented it. Sparse local notes, arbitrary diffs and unpublished work remain coverage limitations.

Open next questions

1. Determine the successor-state distribution after one successful generic binary pivot. Start with r=2 and retain ordered residual types and split multiplicities; the present proof gives no transition distribution.
2. Count first pivots when the binary commutator has a kernel. Start with the singular factor data of Prove a complete factor invariant for ordered idempotents (no new kappa) #4762; the invertible off-diagonal reduction here does not classify it.
3. Find a named producer and fully paid header cover. Pin its actual field and allowed headers before pricing selection, emission, reads and dirty return; the complete physical bill is unresolved.

CI notes

Both hosted checks passed on contribution head fc36a3b8636a200de26f55aaa7393a05b8ee3f69 in Registered research packages run 39851676157: plan in 6m44s and the registered research/generic-idempotent-first-guard replay in 1m16s. Run metadata confirms this exact head and conclusion success. The registered replay executes both warning-strict normal and optimized verification commands. Every registered local command, committed-byte/source-history preflight, official export check and read-only Git-free replay also passed. Hosted CI certifies execution of these scoped checks; independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

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