P06 selection assessment

Selected title: Prime support and growth of Stirling repair factors.

This is one coherent, substantial specialist research paper. The main advance combines exact prime support, a factorial-to-LCM improvement in the uniform upper bound, explicit families attaining the local upper bound, and the asymptotic

Together these prove Miska–Ward Conjectures 1, 2, 3, and 4(d). The asymptotic determines the first-order logarithmic growth of the excess above the squarefree factor, beyond merely proving its divergence. The infinite conclusions have complete hand proofs; finite computations are diagnostic only. The elementary digit lemma or pole argument alone would have the scale of a small note, but the structural and asymptotic package supports retaining this consolidated paper.

The fresh audit reconstructs the local-congruence, companion-matrix, lifting, pole, digit, and asymptotic arguments. It checks the source conjectures and compares relevant 2024–2025 primary literature. No mathematical gap or already-known theorem subsuming the package was found in that bounded audit. This is an internal AI assessment, not external peer review or a guarantee of priority, journal acceptance, or broad breakthrough significance.

The exact general repair factor, the proposed finite cutoff, and Conjecture 4(a)–(c) remain explicitly open. The case of 4(c) is covered by the paper, but its general case is not. Zero residues in the digit formula do not prove higher divisibility, and pure prime-power witnesses do not remove the need to control mixed indices. No extra paper or significance claim is being created from a routine error-term refinement.

Changes in this selected copy

The source manuscript under research_program/papers/P06-stirling-repair/ was preserved byte-for-byte. This copy changes only two citation pinpoints to match the accepted Miska–Ward version: Theorem 7 becomes Theorem 2.2; Lemma 10 becomes the proof of Lemma 2.5. The mathematical statements and proofs are unchanged. Accepted source and publication record.

Original TeX SHA-256: 342863fbc8b75f0099606d96f1b913500f6e5fa39b59490df3025356e50f59fb.

Selected TeX SHA-256: dd0edaaea0b37293073801784badc058ae2cac8257256fd3468860c0a6ad265b.

The selected TeX compiled successfully with the workspace Tectonic binary on 8 September 2026. Compilation verifies document construction, not mathematics. The parent task handles the Markdown conversion and final visual QA.

The supplementary fresh checker uses inclusion–exclusion modulo before exact division by , independently of the manuscript's generating-function proof. All 3,500 cases passed. Files: P06-independent-check.py, P06-independent-check-results.json. The inherited check_digits.py is retained as a separate earlier diagnostic; it is not substituted for the fresh audit.

Final artifact check

The root has now read the complete selected TeX, inspected every PDF page, and checked the matching complete Markdown conversion records. The source/PDF hashes and all resolved-reference, proof-body, and math-preservation checks are recorded in the collection manifest and per-paper reports. No internal mathematical finding remains open in this selected statement. Live GitHub visual rendering and human mathematical verification are separate from these local checks.