What sets the critical genome length for sympatric speciation? A closed form and asymptotic theory

In the Derrida--Higgs model of sympatric speciation, a sexually reproducing population with finite binary genomes can mate only when the genetic overlap between two individuals exceeds a threshold $q_{\min}$. Depending on the genome length $L$, the population either remains genetically connected or fragments into reproductively isolated species. A central problem is therefore to predict the critical genome length $L_c$ at which fragmentation begins. At finite $L$, fluctuations broaden the overlap distribution and allow genetically distant regions of the population to remain connected. A previously proposed transient variance criterion captures this effect, but its evaluation requires numerical iteration of coupled moment equations. Here we first correct the unrestricted moment equations by removing a previously implicit assumption and then derive an explicit closed form expression for $L_c$. The resulting formula shows that the critical genome length is determined, at the time the mean overlap reaches $q_{\min}$, by the competition between deterministic separation from the unrestricted equilibrium and the transient genealogical variance of the overlap distribution. This expression permits a systematic asymptotic analysis. When the deterministic contribution dominates, $L_c$ becomes essentially independent of the population size $M$ and scales as $μ^{-2}$. When the transient genealogical variance dominates, $L_c$ grows as $M^{3/2}$ or as $\sqrt{M}/μ$, depending on how $M$ and the mutation rate $μ$ jointly vary. Simulations support all predicted behaviors. Our results identify transient genealogical variance as the principal mechanism linking finite genome fluctuations to the onset of reproductive fragmentation and provide a practical analytical prediction for the critical genome length.