Source, scope, and significance assessment

Revision 1.2 assessment, September 16, 2026

Theorem 6.1 establishes a growing-ordinarization limit on the scale r^3/g under r^2/g -> 0. It uses necessary multiplicity-chain conditions to bound every valid tuple outside the bulk and a hypergeometric collision argument inside the bulk. This goes beyond a pointwise fixed-r expansion and supplies an explicit transition law. The theorem and its proof, source comparison, and remaining limitations are reviewed in uniform-review.md.

The result still does not settle either all-genus counting conjecture. It does not treat r proportional to g, does not give a relative exponential estimate at infinite r^3/g, and does not control errors at the scale needed to prove consecutive differences positive. The full publicly well-known conjecture remains unresolved, so the thread goal remains active. The publication is an internally audited, AI-assisted preprint revision and carries no independent correctness or exhaustive novelty certification.

The following assessments are retained as records of earlier revisions.

Revision 1.1 assessment, September 16, 2026

The revised manuscript answers its own previously stated question about the constant beta_r. Theorem 5.1 gives a closed formula for every positive r. Its proof classifies all contributions from a single additive equality, shows that remaining configurations have at least two independent equalities, and evaluates the resulting lattice volumes. It also determines the growth beta_r/A_r ~ -r^3/2. This last identity concerns coefficients and is not a uniform genus estimate.

The credited r=1 and r=2 formulas remain prior results; agreement with them is a check, not a novelty claim. The new contribution is the arbitrary-r evaluation. The September 16 search rechecked the primary Cyrusian–Kaplan article and queries combining ordinarization, second coefficient, quasipolynomial, and 2026. No matching general closed formula was located in the inspected primary material. The earlier r=3 lead remains an access/priority limitation; this is not an exhaustive novelty certificate.

The full Bras-Amorós Fibonacci inequality, total monotonicity at every genus, and refined all-genus monotonicity remain unproved here. This extension does not meet the thread's stopping condition of solving a publicly well-known, significant conjecture. It is an internally audited, AI-assisted preprint revision, not an independently refereed result. See the revision proof review.

Original-version assessment (historical)

Date: September 9, 2026. This assessment was performed by the originating AI agent, not an independent expert.

Explicit problem and primary evidence

Sogol Cyrusian and Nathan Kaplan, Ordinarization numbers of numerical semigroups, Communications in Algebra (2026), 1–24, DOI, arXiv v2. The publisher records online publication on February 11, 2026. The arXiv version header is dated March 10, 2026; its HTML document date differs. The version identifier and retrieved bytes, rather than an inferred latest-edit date, identify the audited source.

The audit read the introduction, the fixed-ordinarization section including the proof of Theorem 3.4, the small-case formulas, and Remark 3.7. The latter explicitly asks whether the fixed-r quasipolynomial holds at all positive genera. The source's Conjecture 1.5 states the stronger all-genus monotonicity conjecture. Its r=1 and r=2 results precede this manuscript. The relevant source PDF pages 8–9 were also visually inspected.

Beck and Robins, Computing the Continuous Discretely, second edition, publisher, supply standard rational Ehrhart theory. Theorem 3.23 was checked in the author-hosted draft. No third-party source PDF is redistributed here.

Proposed advance

The manuscript answers Remark 3.7 for arbitrary fixed positive r by keeping strict inequalities homogeneous and including every mixed-sum closure constraint. It then determines the universal leading coefficient and the parity oscillation of the next coefficient. These yield eventual strict monotonicity for each fixed r, a partial result toward Conjecture 1.5.

Cyrusian and Kaplan already use rational polyhedra and, for r=2, a bulk with exceptional additive equalities. Those ideas are credited antecedents. The proposed additional argument works for arbitrary r and observes that the relevant exceptional hyperplanes pass through the origin, forcing their top-dimensional correction coefficient to be constant. This controls consecutive differences; an unstructured error estimate would not suffice.

Priority search and limitation

Queries on September 9, 2026 included ordinarization with quasipolynomiality, all positive genus, eventual monotonicity, strict monotonicity, and leading coefficient. No general-r result matching the manuscript was located. Absence from these searches is not a proof of priority.

A ResearchGate citing-paper listing attributes Numerical Semigroups of Ordinarization Number Three (July 2026) to Akul Kumar. Its abstract reports an r=3 solution and the leading coefficient 7/7680. The full text, stable bibliographic record, and publication status could not be retrieved; link attempts failed. This is a discovery lead, not an audited mathematical dependency. No novelty claim is made for the r=3 case alone. The listing was encountered on the ResearchGate page for Beck–Robins.

The related primary preprint Braun–O'Neill–Park, Eventually nondecreasing quasi-polynomials, arXiv:2607.19207v1, was screened; it gives general criteria for eventual monotonicity. Its retrieved full HTML contains no ordinarization occurrence. The present proof explicitly computes the leading consecutive differences and does not depend on that paper.

Significance and scope

The strongest reason for a distinct paper is the combination of an explicit published question with an all-r partial result on semigroup enumeration monotonicity. It provides a reusable separation between a full-dimensional counting region and exceptional additive loci. Potential further work includes full periods, the constant part of the next coefficient, effective thresholds, and all-genus injections.

This remains a focused contribution, not a resolution of total semigroup-count monotonicity or the full fixed-ordinarization conjecture. No citation-count forecast, independent correctness certification, journal acceptance, or confirmed historical priority is asserted. One consolidated manuscript is counted; its individual theorems are not separate papers.