The Bernoulli Barrier: How Parallel Fixation Violates the Law of Large Numbers

Parallel fixation—the simultaneous selection of multiple beneficial mutations—is frequently invoked to explain how evolutionary change can occur faster than sequential fixation would permit. We identify a fundamental statistical constraint that has been overlooked in this argument: the Law of Large Numbers. As the number of simultaneously segregating beneficial loci increases, the variance in individual fitness decreases relative to the mean, asymptotically approaching zero. Selection requires variance to operate; parallel fixation destroys the variance it needs. We demonstrate that for 157,000 loci to fix in parallel (the number required to account for human-chimpanzee divergence within available generations), the fitness differential between the "best" and "worst" genotypes in a population of 10,000 is only 14.7×—while the required differential is 1,570×, a shortfall exceeding 100-fold. The probability of finding a single individual carrying all beneficial alleles is 10⁻⁴⁷'²⁶², requiring 10⁴⁷'¹⁸² universe-volumes of human beings to yield one expected "optimal" genotype. We further show that heterogeneous allele frequency distributions do not relax this constraint, as the "active zone" of intermediate-frequency alleles imposes a hard limit on pipeline capacity of approximately 230 simultaneous sweeps. Genetic drift independently eliminates 98% of beneficial mutations before they reach this zone, compounding the throughput limitation. The parallel fixation mechanism is self-defeating: it invokes a process that eliminates the conditions necessary for its own operation.