Historical Reconvergence in a Parallel Rewriting System

Abstract — We study the propagation of endpoint multiplicities in a minimal parallel rewriting system with rules A → AB or BA and B → A or B. A history is a complete sequence of local rewrite choices, while an endpoint is the word obtained after a fixed number of parallel steps. Distinct histories may therefore reconverge to the same endpoint. We quantify this reconvergence through the endpoint collision count c_t = ∑_W n_t(W)², where n_t(W) is the number of histories ending at W. Our first result expresses the next-step collision count as a quadratic form in an exact common-successor kernel K_1(U,V) = |S(U) ∩ S(V)|. For this rewrite system, we derive an explicit formula: if U and V have common successor length L, then K_1(U,V) = 2^L · 2^(-|E(U) ∪ E(V)|), where E(U) records the starting positions of the two-letter outputs forced by A-symbols in U; otherwise the overlap is zero. We then express the kernel as a positive subset-feature hierarchy expansion and obtain an exact sum-of-squares representation of the collision count, together with a non-negative remainder for every finite-order truncation. Independently, we prove an exponentially decreasing upper bound on the probability that two independently sampled histories have the same endpoint: q_t ≤ 2^(-⌈(t+1)/2⌉) for t ≥ 1. We also show that the full microscopic common-successor kernel has infinite rank. These results are exact for the stated rewriting system. They do not establish finite-dimensional closure, asymptotic independence, a physical interpretation of the hidden historical sector, or an information-first ontology.