[petalas]

Summary

A scoped finite-field counting proof adds exact orbit weights to MatShep's #4762 ordered-idempotent descriptor. For its factor-data stabilizer H, h=a-c, j=b-c and removable zero dimension z=n-a-b+c:

|centralizer(P,Q)| = |H| |GL_z(F_q)| q^(hj+hz+jz)
|pair orbit|       = |GL_n(F_q)| / |centralizer(P,Q)|

The written exact-sequence proof applies to every finite field and arbitrary ordered idempotent ranks, including noncommuting pairs and zero/full ranks. The new step counts the extension group and proves surjectivity onto H using the credited inclusion/observation lemma. This contribution was produced by Nick Petalas. MatShep #4762, codyw912 #4735 and Mihir Srivastava #4689 retain their descriptor, extension-lemma and residual-state credit. Their classifications and earlier header numbers are cited, not audited.

Only the five files in research/idempotent-stabilizer-multiplicity/ are added. No new kappa or selected-frontier change. Advance CLAIM. Canonical main remains 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7.

Evidence and reproduction

Before, #4762 identifies residual types and leaves transition multiplicities open. After, the new formula supplies exact orbit weights conditional on computing H; split incidence is still open.

• Direct simultaneous-centralizer counts agree with the formula for all 859 fixed-first-projector pairs in seven complete small domains over F2, F3 and F5.
• Five distinct omission controls reject deletion of q^hj, q^hz, q^jz, |GL_z| and |H| using concrete direct-count witnesses. Six sharp-pair fixtures and an oversized-enumeration rejection pass.
• Both warning-strict normal and optimized registered commands pass on CPython 3.12.15 / macOS arm64. Direct durations are 2.14 seconds each, below separate 120-second timeouts. Generated JSON records are identical and bind all five files by SHA256. These are durations, not speed comparisons.
• Registered package replay and five-file committed-byte/source-history preflight pass at 906be31cc1337f599dae85718fbe8187a97453f2. Committed export integrity and an optimized Git-free exported replay pass. Syntax and function-size inspection pass; the longest function has 27 physical lines. Bounded managed-runtime allocation is documented.

From the repository root:

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

Any CPython 3.12 interpreter can replace the local path. Expected markers are PASS_IDEMPOTENT_STABILIZER_MULTIPLICITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. No predecessor executable is imported; its README pins are historical provenance.

Limitations and unrun obligations

The general claim rests on the written algebraic proof; finite checks use prime fields only. Computing H can still require exponential enumeration in r and s. This supplies orbit weights, not a complete split-transition table, a header probability or a polynomial-time canonical form. Physical header selection/emission, reads, dirty restoration, recursive execution, precision/tape, stock and full consumer costs remain open.

Every registered check ran locally. No selected scientific cold replay, Lean or Docker run was needed or performed. Independent human and formal proof-kernel review are unclaimed.

Related open work

• Classify shared native maps for three subspaces (no new κ) #4919 classifies shared invertible maps for three ordered subspaces and counts their stabilizers. This package counts simultaneous centralizers of two idempotent operators; the objects and enumerated domains differ, with no shared numerical quantity computed.
• Prove the binary first-pivot quarter law (no new κ) #4817 counts successful first pivots for binary pairs with invertible commutator. This formula counts stabilizers and pair orbits; it does not compute pivot events or successor-state distributions.
• Prove an all-field central-spectrum law for Schur deletion (no κ) #4801 constrains the central spectrum after Schur deletion. Here arbitrary residual pairs are weighted by their factor-data stabilizers; no spectral identity or deletion-profile count is asserted.
• Count shared Schur-product rank pairs without matrix enumeration (no κ) #4803 counts shifted compressed-product nullities and one-block Schur headers. The present formula counts simultaneous-centralizer fibers; no overlapping header count is computed.
• Count positive correlation for overlapping ternary full guards (no κ) #4806 counts full guards for a particular commuting ternary rank-two pair. This theorem supplies general orbit weights without counting its guard event or ternary rank-two population.
• Prove positive binary correlation for a common-covector guard pair (no κ) #4807 counts full guards for common-covector pairs. This theorem counts centralizers, without evaluating that guard probability.
• Prove the exact scalar-block Schur dimension formula (no κ) #4785 studies minimum Schur dimensions for common scalar block pivots. Here the active dimension is inherited from Prove a complete factor invariant for ordered idempotents (no new kappa) #4762; no Schur dimension minimum is computed.
• Prove the exact unequal-pivot Schur dimension law (no new κ) #4774 studies minimum dimensions for unequal scalar pivots. This theorem adds multiplicities, without imposing pivots or solving its dimension problem.
• Prove the binary pair-versus-singleton full-guard inequality (no κ) #4777 studies a three-request binary full-guard inequality. This theorem concerns two-projector orbit weights and computes no three-request event.
• Prove all-field negative correlation for one-way full guards (no κ) #4778 studies full-guard correlation for one-way coupled kernels. This theorem counts centralizers rather than that correlation.

These quantities differ from the new centralizer orders, so no numerical agreement or disagreement with their guard or dimension results is claimed. Live claims/open PRs and distinctive main README/merged-PR searches were refreshed before committing. Sparse local notes, arbitrary diffs and unpublished work remain coverage limitations.

The completion refresh listed all 81 open PRs on this account and read the scopes of the nine other open PRs created since the CLAIM at 2026-10-22 23:38:23 UTC. The adjacent stabilizer-count result #4919 is listed above. The others study bipartite/full-mask query capacity, a tensor-error dimension threshold, fixed-basis routing, square-zero call compilation, a quadratic inverse bound, ring-valued fanout tariffs and integer-unitriangular tariffs. None asserts an overlapping ordered-idempotent centralizer value. No additional computations were run to audit their results.

Open next questions

1. Compute H from its finite-dimensional endomorphism algebra for one actual residual family, beginning with the radical and semisimple quotient. No such structural count is supplied here.
2. Use these weights in one caller-specific split-transition table, beginning with split incidence into descriptor orbits. All physical implementation costs remain unpriced.

CI notes

Both hosted checks passed at exact head 906be31cc1337f599dae85718fbe8187a97453f2 in run 39873842073: planning in 2m59s and the registered package replay in 1m17s. Run metadata confirms the matching head, completed status and successful conclusion. Checks were read once at completion. The registered replay executes both complete warning-strict normal and optimized commands.

All registered local checks, committed-byte/source-history preflight, official export integrity and Git-free exported replay also passed. Hosted execution certifies these bounded checks; independent human and formal proof-kernel review of the general theorem remain unclaimed. No selected cold replay, Lean or Docker run was performed.

Merge danger

Door: two-way. Blast radius: package. This isolated research addition changes no shared implementation or selected-frontier record.