Twin Primes as a Phase II Problem: Why Variance Is Necessary but Not Sufficient

We argue that the Twin Prime Conjecture is a Phase II problem: it requires structural control beyond RH-level variance rigidity. Variance constraints stabilize global prime statistics and exclude large-scale spectral defects, but they do not generate the bilinear, local correlations needed to enforce persistent prime pairing at gap 2. This note clarifies the necessity–insufficiency split: variance is a necessary background regularity condition, yet twin primes demand additional coupling principles (sieve-level or correlation-level mechanisms). The result is scope-defining rather than proof-oriented: it identifies the precise structural step from linear variance control to constructive pairing.