Asymptotic orders for invariants of Erdős-Rényi numerical semigroups

For $p\in(0,1)$, let ${\mathcal S}(p)$ denote the Erdős-Rényi numerical semigroup obtained by independently selecting each positive integer as a generator with probability $p$. Write $\F({\mathcal S}(p))$ for its Frobenius number, $\e({\mathcal S}(p))$ for its embedding dimension, and ${\mathrm E}$ for expectation. The asymptotic-order conjecture. As $p\to0$, the expected Frobenius number and expected embedding dimension satisfy

$${\mathrm E}[\F({\mathcal S}(p))]\asymp\frac{1}{p}\log\left(\frac{1}{p}\right), \qquad {\mathrm E}[\e({\mathcal S}(p))]\asymp\log\left(\frac{1}{p}\right).$$

The paper's bounds establish these quantities only up to a polylogarithmic factor, while experiments motivate the sharper orders conjectured here. The proposed asymptotics remain open in the source.