A Modular Sieve Balance Model and Computational Verification for the Twin Prime Conjecture

In this paper, we present a structural modular sieve framework in ℤ/30ℤ to establish the infinitude and distribution bounds of twin primes. By systematically eliminating residue classes divisible by 2, 3, and 5, twin primes greater than 5 are uniquely partitioned into three complementary pairs of arithmetic progressions of the form B = A + 2. Over any finite interval (√x, x], we deduce an exact algebraic balance identity linking the count of twin prime pairs PG(x) to the composite collision density within the multiples of the primary sequence, expressed as PG(x) = (kⱼₓ − k₀ₓ) · MA, where k₀ₓ denotes the critical survival threshold and kⱼₓ represents the compound sieve density. Because the shift parameter h = 2 admits no odd prime divisors, the compound density kⱼₓ is governed unconditionally by canonical Mertens sieve factors (p − 1)/p. We benchmark theoretical predictions against high-precision empirical counts executed via industrial programmable logic controller (PLC) automation up to x = 10⁹ and x = 2²⁸, matching empirical measurements within a relative agreement exceeding 98.7%. Finally, a second-order asymptotic Taylor expansion shows that the balance gap Δ(x) = kⱼₓ − k₀ₓ = (C + 14.0625)/ln²(x) > 0 remains strictly positive for all x ≥ 5, rigorously confirming the non-vanishing of twin primes in (√x, x] and recovering the classical Hardy-Littlewood asymptotic order of magnitude.