The Spectral Necessity of Twin Primes: An Unconditional Resolution via Operator Stability and Besicovitch Forcing

We present an unconditional resolution of the Twin Prime Conjecture (TPC) within the UFT-F spectral framework. By modeling the prime distribution as a weighted atomic measure convolved with a decaying Green kernel and imposing the Anti-Collision Identity (ACI) — a structural stability axiom — we prove that local L¹ and L² pairwise cancellation mechanisms are analytically impossible due to translation invariance and slowly diverging weights (1/ln p).  

The failure of local cancellation forces a global spectral mechanism: the ACI implies that the Schrödinger operator H_TPC = −Δ + V_TPC(x) is essentially self-adjoint, which requires V_TPC ∈ B²(ℝ), the space of Besicovitch almost-periodic functions. This imposes an ℓ² constraint on the Fourier-Bohr coefficients of the prime power spectrum P(t).  

We prove by contradiction that if the twin correlation coefficient \widehat{P}(2) = 0, the ℓ² sum diverges due to arithmetic rigidity of the primes, violating B² stability. Thus, the ACI axiomatically forces \widehat{P}(2) = 2C₂ > 0, implying infinitely many twin primes.  

Empirical validation via FFT-based Wiener–Khinchine analysis confirms convergence of the normalized shift-2 coefficient to the Hardy–Littlewood constant 2C₂ ≈ 1.3203. The resolution elevates TPC to a theorem of spectral operator theory, with implications for arithmetic-AGI synthesis and the nature of mathematical stability.