Formal Verification of Sieve Sequence Stages and Their Transitions

This article defines a sieve sequence stage as the increasing sequence of integers accepted by a finite prefix of prime filters. The accepted pattern is periodic modulo the product of those filters, so one finite list of positive gaps reconstructs the infinite stage. It formally verifies strict increase, completeness, block-period shift, gap-cycle reconstruction, exact transition counts, and the local copy-or-merge rule for next gaps. The result has explicit boundaries: next-head primality is conditional on a square bound supplied mathematically by Bertrand's postulate, and direct construction of the next cycle remains a separate composition problem. The article establishes verified finite-stage sieve specification and transition semantics, not a new prime-sieving algorithm or persistence theorem for a particular short-window gap.