Fibonacci-type growth of internal and leaf semigroup counts

Establish the conjectured Fibonacci-type inequalities and asymptotic limits for the numbers of internal and leaf numerical semigroups by genus: prove $ins_g\geq ins_{g-1}+ins_{g-2}$ for g≥2, $lns_g\geq lns_{g-1}+lns_{g-2}$ for g≥6, and the four stated limits equal to 1, 1, φ, and φ, respectively.

Background

Computations suggest that the counts of internal and leaf semigroups exhibit behavior analogous to the known Fibonacci-like behavior of the total number of numerical semigroups. The conjecture combines lower bounds with asymptotic claims involving the golden ratio φ.