The Replication–Deletion Primorial Sieve: A Generative Stage-Based Sieve (No Prelisted Inputs) and an Unconditional Twin-Prime Theorem

We introduce a primorial stage-lift sieve designed to track twin-admissible residue classes and to propagate them through successive primorial moduli. Unlike classical sieves that begin from a prelisted interval of integers (or a precomputed list of primes), our construction generates the integers under consideration stage-by-stage by replication from the previous stage, and the next sieving prime is extracted intrinsically from the replicated set. At stage i the state is a set of residue classes modulo Fi for which both a and a+2 are coprime to Fi. The update from Fi to Fi+1 = Fi·p(i+1) is fully explicit: each class lifts to p(i+1) fibers and exactly two fibers are deleted, corresponding to divisibility of n or n+2 by the new prime p(i+1). We give numerical data for the initial stages and define a cumulative prediction function for the number of twin primes up to Fi+1 derived from the stage sizes. Analytically, we develop exact Fourier identities for the lift–deletion transition, introduce a Selberg-weighted discrepancy energy Wi(D) at the natural cutoff D = floor(sqrt(Fi+1)), and prove a one-step normalized energy recursion. Combining these bounds with a Selberg lower bound at the cutoff yields a uniform positive proportion of survivors for all sufficiently large stages, and therefore establishes the infinitude of twin primes. Along the way, the paper serves as a self-contained introduction to the stage-lift sieve dynamics and their structural properties.