Twin-Prime Starts, Additive Block Coverage, and a Finite Automaton in Goldbach Space

We present a finite-range experimental note on an additive structure induced by twin- prime starts in Goldbach space. Let a twin-prime start be an integer a such that both a and a + 2 are prime, and define the start-sum set S = {a + c : a, c are twin-prime starts}. From each such sum we associate the three-term block {s, s + 2, s + 4}, and we study the even integers covered by the union of all such blocks. The note has two main goals. First, we document that twin-prime starts behave as a nontrivial additive memory inside the tested Goldbach range. Second, we show that the resulting presence pattern compresses naturally into a finite automaton with a dominant stable phase, a small boundary layer, and a localized defect mechanism. In computations up to 106, all non-total states are confined to a finite low region, and beyond 4208 every tested even integer lies in the total state. The main empirical defect law is exact in the tested range. The uncovered even integers consist of exactly 33 values grouped into 11 consecutive triples, and these triples coincide exactly with the interiors of the 11 consecutive gaps of length 12 in the set S. Equivalently, in the verified range, a consecutive gap (s, s + 12) in S occurs if and only if it produces the uncovered triple {s + 6, s + 8, s + 10}. This note is purely experimental. It does not claim a proof of Goldbach’s conjecture, nor of the infinitude of twin primes, nor of any asymptotic theorem. Its purpose is only to isolate and document, in finite verified range, an exact empirical relation between twin-prime-start sums, additive block coverage, and the finite automaton that emerges from their presence pattern.