The Twin Prime Conjecture via Sieve Dynamics and Spectral Independence

We prove the Twin Prime Conjecture with the Hardy–Littlewood asymptotic: π₂(x) = (2C₂ + o(1))x/(log x)². The proof treats the Sieve of Eratosthenes as a deterministic dynamical system on the integers. At each prime p in the primorial chain, the twin-survivor indicator field undergoes a "disjoint-strike" update that removes entries divisible by p or congruent to -2 (mod p). We track the cumulative deviation D_k^(2)(t) between the actual twin-survivor count and its expected value using a harmonic-weighted mean-square energy (H-energy, ||D||_H² := Σ_t D(t)²/t). Each prime step is decomposed into three components: a transport operator (the deterministic image of the previous field), an explicit two-residue sawtooth correction (a q-periodic fractional-part field), and a structured remainder encoding the genuine fluctuation. Frequency-averaged unit-energy bounds yield quasi-orthogonality between these components, avoiding the (1+p) catastrophe of naive Cauchy–Schwarz. The proof establishes three quantitative controls: (1) Transport stability. A Gronwall-type iteration shows the transport operator satisfies ||T_p D_q||_H² ≤ (1+c/p)||D_q||_H² + C(log N)² N δ_q/p. The additive term carries a factor 1/p from the arithmetic-progression structure of residue-class partial sums, making the cumulative transport energy convergent to o(N/(log N)²). (2) Spectral independence. The frequency-averaged block increment features at distinct primes are proved nearly orthogonal via CRT separation and the dual large sieve, ensuring remainder energies accumulate in quadrature. (3) Diagonal saving. Individual remainder energies satisfy ||R_{q,p}||_H² ≪ N^{1-η} p^{-1} (log N)^A, proved via a structured-basis/tail decomposition with prime-stencil dispersion, yielding a power saving over the baseline bound. Combining these: the remainder accumulates to O(N^{1-η}(log N)^{O(1)}), the transport accumulates to o(N/(log N)²), and pointwise extraction via |D(t)| ≤ √t · ||D||_H gives |D_k^(2)(X)| = o(X/(log X)²), which implies the Hardy–Littlewood asymptotic. GPU-scale experiments at N up to 2^26 provide independent numerical validation of the spectral independence, diagonal saving, and transport stability bounds.