Empirical and Analytical Observations on Twin Prime Distribution via Quadratic Interval Partitioning

We study the distribution of twin prime centers — integers n such that n−1 and n+1 are both prime — within the quadratic intervals Im = m², (m+1)² for m ≥ 3. Every such center satisfies 6 | n, and among multiples of 6 in Im (which can only end in 0, 2, 4, 6, or 8), those with last digit 4 or 6 are excluded by an elementary divisibility argument, leaving exactly three of the five possible residue classes and (2m+1)/10 + O(1) candidates per interval. We prove an elementary non-repetition theorem (Theorem 4.1) for minimal obstructing primes — the smallest prime factor of 6k±1 when composite — showing that no prime p ≥ 5 can obstruct two consecutive candidate positions. This result establishes a necessary structural condition for the independence of obstructing primes and, together with empirical near-zero autocorrelation in the obstructor sequence (lag-1 = −0.040), motivates Conjecture 4.3 on asymptotic independence after renormalization. We contrast this partition with Gallagher’s classical intervals of length ~log n: while the Poisson parameter for Gallagher intervals tends to zero, for quadratic intervals λ(m) → ∞, so P(no twin prime in Im) → 0. We formulate two conjectures whose joint proof would imply the twin prime conjecture as a corollary, and identify the precise logical gaps that remain open.