Internal–Leaf Inequalities and a Positive Limiting Leaf Density for Numerical Semigroups with Frobenius Number 2m+3
Year: 2026
Type: preprint
Abstract: We study numerical semigroups of multiplicity m >= 6 and Frobenius number F=2m+3, using leaves and internal nodes of the full numerical-semigroup tree. In this restricted infinite family, we establish I_m >= 3 L_m for every m >= 6 and I_m >= 5 L_m for every m >= 12. The former coefficient is attained at m=6; 12 is the minimal threshold for validity for all subsequent m, although the second inequality also holds at m=10. The leaf proportion tends to a strictly positive limit c with an explicit exponential error bound and certified enclosure 0.1314883 < c < 0.1446435. The error bound is conservative for small and moderate m. This is a fixed-family strengthening of Conjecture 41 of Casas, Madrid and Rosales, Internal numerical semigroups (2026), and does not solve the unrestricted conjecture. The total N_m=7*2^(m-4) is already known and is not claimed novel. Full-tree leaves differ from the leaves of specially pruned leaf-discriminating enumeration trees. Independent AI mathematical audit: PASS within its author-reported scope. Windows PowerShell 5.1 reproduction: Python and C++17 PASS, based on the author-supplied audit record, which reports matching the original transcript SHA256. The private transcript was not re-inspected in publication preparation and is excluded from all public files. Its metadata-only SHA256 is b3d94afab5686bedeb23b1f961b5b69830faa008284127ff776e2677d0b49fee. During publication preparation, the supplied Python, C++17 and exact rational interval programs were rerun on Linux and passed. REFERENCES PARTIAL: bibliographic metadata and selected original-source passages were checked; exhaustive theorem-level overlap checking is incomplete. NOVELTY INCONCLUSIVE. Not human peer-reviewed. All rights reserved, including the supplied verification sources. Author: Ryutaro Yonezu — Independent Researcher. Version: v1.0. Date: 2026-11-05. Publication type: Preprint. GitHub repository: https://github.com/yonezaemon1-hub/no56-numerical-semigroup-internal-leaf-density GitHub Release v1.0: https://github.com/yonezaemon1-hub/no56-numerical-semigroup-internal-leaf-density/releases/tag/v1.0
Author: Yonezu, Ryutaro
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225309349