Maximum-Entropy Spacings and Hierarchical Dilution of Prime Constellations

Abstract

We report a high-precision phenomenological study of the spacing statistics of prime constellations (twin primes, prime triplets, quadruplets, and higher admissible tuples) up to 1.1×108 , with a replication window at 1010. Five structures emerge. (i) A provable residue class locking lemma: conditioned on a twin-prime centre c, the probability that c+6g is again a twin centre factorises over small primes into an explicit three-case product formula, accurate to 1% pointwise at small lags and to the sampling noise floor beyond. (ii) Combined with an empirically validated hazard structure (memory corrections ≤ 4%), the product formula reconstructs the full gap distribution of primes and of twin centres end-to-end with total variation 0.025 and 0.014 respectively—two orders of magnitude better than the geometric baseline. (iii) Constellation spacings are characterised by a constrained maximum-entropy law: fixing the mean and roughly K∗ ≈ 3.2 λ¯(x) short-range constraints determines the entire distribution, including the tail; the healing length K∗ grows proportionally to the mean spacing. (iv) Higher constellations nest into lower ones, and the number of parent-skeleton steps between successive higher-order constellations is geometric to within an entropy deficit of 0.03%–0.27%: a hierarchical geometric dilution law, whose dilution ratios are closed-form ratios of Hardy–Littlewood singular series times ln x, matching measurements to 0.2% after logarithmic-integral correction. (v) The rigidity of the point process decays with constellation order, from hyperuniform (primes) through near-Poisson (twin centres) to Poisson (triplets), as quantified by number variance and the structure factor; the rational Bragg peaks of the prime structure factor obey the closed-form height S(2π/q) = N ρ/(q − 1)2 . Several natural alternatives (renewal structure, percolation criticality, a gel-point transition, constant ecological-style dilution) are tested and falsified. We make no progress on the twin-prime conjecture itself: all distributional laws are stated and verified as phenomenology resting on the numerically supported Hardy–Littlewood main terms.