The most prominent count in a new mathematical preprint shows internal numerical semigroups outnumbering leaf semigroups at genus 26: 482,133 internal objects compared with 288,699 leaf objects, out of 770,832 in total. The figures come from a deterministic computational enumeration, not from observations of people, experiments or biological samples.
The result is part of a broader attempt to understand the tree structure of numerical semigroups. The paper separates objects by their position in that tree, develops procedures for counting them under fixed genus, Frobenius number or multiplicity, and compares internal and leaf counts under the same restrictions.
A map of the objects
In the paper, “internal” and “leaf” are not descriptions of the objects themselves but of where they sit in the tree. The full graph is described as a tree rooted at N. Its children are obtained by deleting a minimal generator x when x is greater than the semigroup’s Frobenius number.
That rule gives the enumeration a constructive route: starting from the root, the calculation follows permitted parent-child changes and records the resulting internal and leaf semigroups. The paper also proves that the internal family, denoted I, is a Frobenius variety, a structural result that supports treating the internal objects as a coherent family rather than as an unrelated list.
The genus count supplies the clearest test
The reported genus enumeration reaches g=26. At that point, the internal count is 482,133, the leaf count is 288,699 and the total count is 770,832. In this row, the internal group is therefore larger than the leaf group, although the calculation itself does not establish that the same ordering must hold at every possible genus.
The paper gives an exact counting identity alongside the table: the total number of numerical semigroups of genus k+1 equals the sum of μ over internal semigroups of genus k. Expressed this way, the next total is linked directly to the internal objects in the preceding genus, giving the tree-based enumeration a precise combinatorial relationship between successive layers.
The authors then propose a larger pattern in Conjecture 16. It states that internal counts should be at least the sum of the two preceding internal counts for g≥2, while leaf counts should satisfy the analogous lower bound for g≥6. The conjecture also proposes adjacent-sum ratios tending to 1 and successive-count ratios tending to φ, producing the Fibonacci-like and golden-ratio behavior highlighted by the paper.
Those growth statements are not established asymptotic results. They are conjectures inferred from finite computations, and the supplied analysis cautions that the reported tables cover finite ranges rather than all genera. Extending the calculation and proving or refuting the proposed limits remain open tasks.
The same ordering appears under other restrictions
A separate enumeration fixes the Frobenius number instead of the genus. At F=56, the paper reports 300,272,713 internal semigroups, 20,250,566 leaf semigroups and 320,523,279 semigroups in total. The internal count again exceeds the leaf count in the reported row.
The fixed-Frobenius section turns that observed ordering into further conjectures. Conjecture 25 is recorded as proposing that the internal count exceeds the leaf count for F≥1. The extraction also records a separate leaf-count comparison for odd F≥5, but warns that the notation of Conjecture 26 is ambiguous, so the exact printed inequality should be checked against the typeset source.
The fixed-multiplicity tables report computations through g=26 for m=5 and m=8. At g=26, the m=5 row contains 156 internal, 45 leaf and 201 total semigroups. The m=8 row contains 2,473 internal, 1,562 leaf and 4,035 total semigroups. Internal counts are higher than leaf counts in both reported strata.
Conjectures 32–34 propose that these fixed-multiplicity counts are nondecreasing across successive genera under stated conditions: total counts for m≥2 and g≥m−1, internal counts for m≥7, and leaf counts for m≥6. The tables provide computational evidence for the proposal, but the supplied analysis identifies these statements as conjectures rather than general proofs.
A more targeted tree algorithm
The paper also considers the more restrictive case in which both multiplicity and Frobenius number are fixed. There, the internal graph is described as a tree rooted at D(m,F). Its children are generated by adding special gaps x that meet the stated conditions: m<x<r(S), x is not F, and μ(S∪{x}) is not zero.
Within a restricted paired-invariant range, the paper reports a closed count: the extraction renders the number of internal semigroups as 2F − m − 1 and the number of leaf semigroups as zero. The supplied review notes that a possible superscript is not visibly preserved in this formula, so the exact notation requires verification against the original typeset version.
Conjecture 41 extends the internal-versus-leaf comparison to the paired setting. It proposes that the number of leaf semigroups is no greater than the number of internal semigroups whenever 2≤m≤F+1 and m does not divide F. Like the other broad comparisons, this is explicitly a conjecture, not a result proved for every allowable pair.
A finite map with a larger question behind it
Taken together, the calculations support the authors’ view that internal semigroups outnumber leaf semigroups in the examined fixed-invariant settings. They also suggest recurring growth patterns and identify cases where a compact formula may be available. But the computations do not prove that internal semigroups always outnumber leaves, nor do they establish the proposed Fibonacci-like behavior or limiting ratios beyond the ranges examined.
The main unresolved work is therefore mathematical rather than statistical: proving or disproving the genus, Frobenius-number, multiplicity and paired-invariant conjectures; extending the enumerations beyond the reported genus and Frobenius-number ranges; and checking whether the proposed asymptotic limits hold for the full sequences. The supplied analysis also flags notation and tabulation inconsistencies that should be resolved before relying on the formulas or inequalities.
For reproducibility, the paper states that its C++ implementation source code is available at a Google Drive URL. The supplied analysis notes that no independent computational validation or runtime analysis is reported, leaving reproduction of the finite tables as an important next check.
The document identifies itself as arXiv:2608.19984v1, dated 20 August 2026, with a version date of 19 August 2026. Its acknowledgment says the work was supported by a Departmental Collaboration Grant from Spain’s Ministry of Education, Vocational Training and Sports under the 2025/2026 call.
Paper data and sources
Original title: Internal numerical semigroups
Authors: Mario Casas, José A. Madrid, J. C. Rosales
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text