Derivation of the Existence of an Upper Limit to Sequence Identity Required to Maintain Replication Function

In this study, we derive the critical sequence identity Q required to maintain replicationfunction in a minimal model of early life. Unlike classical quasispecies models that assumea static fitness landscape, our model evaluates fitness based on relative similarity to theimmediate parent (a moving reference frame). Using the Poisson approximation to the binomial mutation count, we obtain the exactcondition for the critical threshold Q as follows: Q = maxq ∈ {0, 1/n, 2/n,. . . , 1 | Γ (1 + n − ⌊nq⌋, np) /Γ (1 + n − ⌊nq⌋) ≧ 1/2} where Γ (a, x) is the upper incomplete gamma function and ⌊x⌋denotes the floor func-tion. This result implies that the required sequence identity per generation must be at mostthe limit set by Q. In the regime where the expected number of mutations np is sufficientlylarge, this analytical threshold simplifies to Q ⋍ 1 - p. Consequently, the fact that there isan upper limit (Q) to the identity required to maintain replication suggests that lineagesmust tolerate ongoing divergence to persist.