[ssafwann]

Change and mathematical scope

PR #4671 characterizes sharp identity-pivot Schur deletion but leaves its complete-header frequency open. This adds a general exact counting law for any fixed commuting binary idempotent pair, specified by its four joint-sector dimensions. The sample space is uniform complete GL_n(F2) headers; the event asks for common I_d leading blocks and Schur-commutator rank 2d after that one deletion.

The proof counts sector factorizations conditional on the shared compressed product H, then supplies the exact GL_(n-d) completion multiplier. It gives both the unconditional density and the density conditioned on identity pivots. The shared-H sum preserves the dependence among the six rank conditions.

Sector dimensions (00,10,01,11)	d	Sharp complete headers	Fraction of all headers	Fraction given identity pivots
(1,1,1,2)	1	60,480	3/496	3/172
(0,2,2,1)	1	181,440	9/496	9/100
(2,2,2,2)	2	52,251,375,697,920	27/2,763,520	27/20,128

The first two pairs have the same individual projector ranks but different counts. For the second profile, multiplying the six conditioned marginals gives 81/625, whereas their actual joint fraction is 9/100.

The README contains the complete factor/count proof and a credited derivation of the six-map criterion. Later full-guard chains, restricted production covers and physical supplier admission remain open; no new κ. Only the five-file research/schur-header-count/ package is added. The selected frontier and frozen sources are unchanged.

Task and overlap

New branch from canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7, selected conditional κ=0.00302015. Publication head: af9cf80.

The explicit #4671 handoff was claimed in Discussion #4653. formal-interface-lemma is marked changed to an algebraic counting side result. The family decision keeps the completed dimension question closed and narrows this round to one specified header event. Physical selection/emission, reads, restoration, stock, dispatch and the full consumer bill remain unknown, not zero.

Final overlap covered 445 open PR bodies (2026-10-22T15:29:10.770941+00:00), 3,589 Discussion records (2026-10-22T15:31:35.834501+00:00), and all 5,860 tracked in-scope notes on main, with no coverage warnings. It found the credited handoff and this claim, with no duplicate count located. The search excludes arbitrary diffs, closed remote PRs and unpublished work.

#4598 and #4599 concern corank-one full-guard events, with no shared numerical quantity here. All five inspected predecessor/related heads were unchanged on the final check. Exact source paths, Git blobs, SHA256 hashes and coverage limits are in TASK.json. Pending sources are citations; none is a replay-time dependency.

Validation

Python 3.12.11, macOS arm64; standard library only:

python3.12 -B scripts/research_package.py --package research/schur-header-count
python3.12 -B scripts/check_research_commit.py --package research/schur-header-count --check-source-history

• Registered ordinary/optimized (-O) replays pass, using explicit exceptions; the combined local replay took 9.92 seconds.
• All 270,759 small matrix-factor pairs independently reproduce both factor-count formulas for every target matrix in the stated domains.
• Literal conjugation/Schur checks cover 60,648 small complete-header/context cases and both 496-element scalar split domains.
• All completions of the directly accepted scalar split pairs are explicitly formed, inverted and checked: 241,920 positive complete-header cases, with distinctness verified for each fixed pair. This is a complete positive census through the proved bijection, not a raw scan of all GL5 headers.
• Seven rejection controls distinguish dependent events, the conditional denominator, the completion multiplier, the unguarded sector, missing surjectivity and invalid dimensions.
• The d=2,n=8 count is evaluated by the proved formula; its 5.3e18-element header space is not enumerated.
• All five committed files and declared source-history pins pass. Official export/integrity checks pass. The read-only, Git-free export passes -I -W error -O -B with PATH=/nonexistent, producing identical scientific stdout; all six exported files, including the receipt, remain unchanged. The affected-package planner selects only this package, and the working tree is clean.

Universal claims rest on the written proof, with finite corroboration. No independent human or proof-kernel acceptance is claimed. Lean, Docker and the selected scientific cold replay were not run. The Registered research packages run on exact head af9cf805c501e4e9e947c98c0b97b2281b6b6e97 reports action_required; fork workflow approval is pending. No hosted replay pass is claimed.

Credit

ssafwann's #4671 supplies the six-map criterion and counting handoff; #4657 supplies its identity-pivot precursor. Nick Petalas's #4597 supplies the ordered factorization, and #4278/#4259/#4246/#4228 retain the scalar Schur and complete-header completion lineage. Ryanson Jonathan (#4609/#4617 and the full-guard census) and lee-phys (#4634) retain their binary sharpness/dimension contributions and disclosed Claude Code/OpenAI Codex assistance. Cinna Davis supplies the earlier BIT setting; source527, commdata2338 and Jose Manuel retain the normalizer lineage credit.

Prepared for ssafwann with substantial OpenAI Codex (gpt-6-astra) assistance, by one agent. Apache-2.0.

Open next questions

To obtain a full-guard recursion, first determine the residual pair's simultaneous conjugacy state distribution after one deletion. The pair generally ceases to commute, so repeating this formula is unjustified. Start with one n=5 sector profile and retain the multiplicity of headers reaching each residual state, as suggested by #4597's orbit question. Any actual producer also needs its specified header cover and complete cost allowance. No follow-up claim is held.