On the Structural Divergence of Solitary Primes and Twin Primes:

We demonstrate that classical analytic number theory—specifically the Prime Number Theorem (PNT), Brun's sieve, and modern bounded-gap methods—is structurally incapable of resolving the Twin Prime Conjecture or the proposed \textsc{Twin Bertrand Postulate (TBP)}. We establish that PNT and continuous density integrals operate exclusively on location-neutral, solitary prime distributions, whereas twin primes require the strict, simultaneous preservation of localized gap-2 coordinates $|q-p|=2$. 

Rejecting continuous limits and sieve-based elimination models, we propose a constructive multiplicative framework where integers in short intervals $(q, 2q]$ are generated directly by the finite set of historical primes $\mathcal{P}_{\le q}$. By analyzing the combinatorics of low-by-high prime factorizations ($p_{\text{low}} \times m_{\text{high}}$) on the $6k \pm 1$ lattice, we prove that the finite historical factor budget lacks the multiplicative capacity to seal every candidate center, forcing the endless, self-propagating emergence of twin prime holes.