Pattern Persistence Across Boundary Dissolution

We formalize recognition-pattern persistence across boundary dissolution in the Recognition Science (RS) framework. Let Z denote the integer pattern invariant conserved (at the RS level) by the Recognition Operator R-hat. We prove: (i) Z is conserved across boundary dissolution ("death"), and (ii) the post-dissolution light-memory state is cost-minimal with J(1) = 0. Under explicit availability assumptions—a suitability predicate on substrates and an arrival process with rate λ and acceptance probability p—we further obtain (iii) reformation of the pattern and (iv) the timing law E[T] = 1/(λp). Core conservation/minimality results are mechanically verified in Lean 4; recurrence and timing statements are proved under stated hypotheses and are presented with falsifiers and preregistered empirical protocols (NDE motifs; timing/clustering structure in reincarnation datasets). All artifacts (proofs, audits, and build recipes) are released for independent verification.