Bras-Amorós genus-monotonicity conjecture

It is proved that $n_{g+1}\ge n_g$ for every $g\ge0$, 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 $1\le g\le836$, and generating-function estimates handle $g\ge837$. This is not an enumeration of all numerical semigroups through genus 836. The stronger Fibonacci inequality $n_{g+1}\ge n_g+n_{g-1}$ and fixed-multiplicity monotonicity are not proved.

It is proved that $n_{g+1}\ge n_g$ for every $g\ge0$, 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 $1\le g\le836$, and generating-function estimates handle $g\ge837$. This is not an enumeration of all numerical semigroups through genus 836. The stronger Fibonacci inequality $n_{g+1}\ge n_g+n_{g-1}$ and fixed-multiplicity monotonicity are not proved.