Tail-Index Renormalization after Posts in Classical Sequential Growth

Classical sequential-growth models of causal sets transform their projective coupling sequences after a post by a known binomial map. This preprint proves a uniform adjacent-ratio theorem for sequences whose ratios are eventually positive, eventually nonincreasing, and regularly varying with index −α < 0. The effective power index is −α/(α+1) throughout every subscale precursor window. When the precursor size is proportional to the balance scale, the normalized response instead approaches the unique positive solution of x(c+x)^α = 1. Constant-ratio, slowly varying, finite-support, and sparse boundaries delimit the result. The proof uses regular variation and geometric localization of positive triangular sums. The theorem is conditional on a supplied algebraic coupling tail and does not select a CSG law, establish post occurrence or recurrence, recover a continuum, or predict a physical parameter.