Genus monotonicity for numerical semigroups

Let ng denote the number of numerical semigroups of genus g. We prove that ng+1 ≥ ng for every g ≥ 0, settling the weak monotonicity form of the Bras-Amorós conjecture. The computer-assisted proof constructs an injection outside a specified family of leaves and bounds this exceptional family by a collection of unused targets. Generating-function estimates establish the comparison for g ≥ 837; exact transfer computations bound relaxed counts of the exceptional sources for 1 ≤ g ≤ 836. The analytic estimates distinguish genuine semigroups from a larger two-pivot relaxation, whose coefficients can be computed by transfer along paths and cycles.