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Bras-Amorós genus-monotonicity conjecture

Let ngn_g be the number of numerical semigroups of genus gg, where a numerical semigroup is an additive submonoid of N0\mathbb{N}_0 with finite complement and its genus is the size of that complement. Is ng+1≥ngn_{g+1}\ge n_g for every integer g≥0g\ge0?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Computation
Field
Numerical semigroups; enumerative combinatorics
Posed by
Maria Bras-Amoros (Semigroup Forum 76, 2008) as a weak form of her Fibonacci-like conjecture; stated as Conjecture 2 in Nathan Kaplan, Counting numerical semigroups, Amer. Math. Monthly 124 (2017)
Year posed
2008
Years open
18y
Solved
2026-09-30
Model
GPT-6 Astra; Fable 5.1
Vendor
OpenAI; Anthropic
Collaborators
Mingchang Liu
Verification
Unreviewed
Publication
Announced
Significance
28 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

It is proved that ng+1≥ngn_{g+1}\ge n_g for every g≥0g\ge0, without restrictions on multiplicity or conductor. A partial injection leaves a specified family of source semigroups; unused targets are then shown to be numerous enough to cover it. Exact transfer computations bound relaxed source counts for 1≤g≤8361\le g\le836, and generating-function estimates handle g≥837g\ge837. This is not an enumeration of all numerical semigroups through genus 836. The stronger Fibonacci inequality ng+1≥ng+ng−1n_{g+1}\ge n_g+n_{g-1} and fixed-multiplicity monotonicity are not proved.

What the AI did

This work was developed through continuous collaboration between the author and AI models. GPT-6 Astra (OpenAI) and Fable 5.1 (Anthropic) contributed to proof development and checking, exact computation, literature research, and preparation and review of the manuscript. The author takes full responsibility for all mathematical results and the contents of this manuscript.

Verification

No independent mathematician has checked this yet. Checked by this site on 7 October 2026: the statement matches Conjecture 2 of Kaplan's survey (arXiv 1707.02551), which also says the weak form was open and that Zhai's asymptotics leave only finitely many genera to settle. From a fresh clone at commit b1b6fb8, all five shipped checkers passed: capacity recurrences through degree 1500, the omitted-pivot multiplier below 0.221715, the guarded mass assembly, the analytic join (comparison fails at genus 836, holds from 837), and all 836 finite rows with nonnegative slack (smallest 1, at genera 1 and 2). A separate enumeration written here of all numerical semigroups through genus 23 reproduced the known counts and confirmed that each certified slack is at most the true difference ng+1−ngn_{g+1}-n_g for g≤22g\le 22. These are arithmetic checks only. The coefficient arrays were not regenerated from the C++ sources, and the combinatorial and analytic arguments that connect them to the theorem (about 2,000 lines of the manuscript) were not read. No Lean formalisation.

Sources

Submitted by SilentIbis759 on

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