Odd-Frobenius decrease in leaf counts

Prove that the number of leaf numerical semigroups with Frobenius number F is greater than the number with Frobenius number F+1 whenever F is odd and F≥5.

Background

The computed fixed-Frobenius data display a proposed alternating behavior in the number of leaf semigroups. The conjecture specifically concerns transitions from odd Frobenius numbers to the following even values.

References

$lns_{F}> lns_{F+1} \text{, for } F \text{ odd, and } F\geq 5.$

— Internal numerical semigroups  (2608.19984 - Casas et al., 20 Aug 2026) in Section 5, Internal numerical semigroups with fixed Frobenius number