Dickson's conjecture versus superadditivity of prime-constellation counts: explicit certificates, the exact twin-prime crossing, and a registry of incompatible conjectures

Abstract

Hensley and Richards (1974) showed that the prime k-tuples conjecture contradicts the Hardy-Littlewood inequality pi(x+y) <= pi(x)+pi(y). This note records the same incompatibility for constellation counting functions (twin primes, cousins, sexy primes, two triplet patterns, quadruplets) with small explicit certificates verified by a few lines of code; for twin primes the crossing occurs already at window length 396 (22 admissible twin-pair positions of diameter 392 versus pi_2(396)=21), and each conditional statement is checked to be non-vacuous by an exhaustive scan of the actual primes. A self-contained certificate for the original Hensley-Richards incompatibility (an admissible 1699-tuple of diameter below 14500 versus pi(14500)=1698) is included, together with an explicit remote window, built by the Chinese remainder theorem, containing 1699 integers free of prime factors up to 1699. This refutes unconditionally the central claim of a recent preprint (Zenodo record 21896925) that remote intervals can never have prime capacity exceeding pi(y); that preprint's Theorem 1 is also shown to fail for every window length y from 2 to 2000. Computations, certificates and text were prepared with the assistance of an AI system (Claude, Anthropic); the author is responsible for the content. Version 4 adds the incompatibility-graph registry as supplementary data: edges.json (120 proven edges of the form 'A and B cannot both hold', each with mechanism, source and status: 33 certificate edges, 79 literature edges across number theory, fine-grained and structural complexity, harmonic analysis, arithmetic geometry and set theory, 8 trivial), nodes.json (believed/refuted statuses), registry.md (table), graph.html (interactive rendering) and build.py; plus 22 further certificate systems from an automated pipeline (constellations, Cunningham chains, arithmetic progressions, quadratic and cubic polynomials). Version 4.1: graph.html rebuilt as a standalone English page (UTF-8 declared; the previous file lacked a charset declaration and rendered garbled outside a hosting wrapper) with a corrected force layout. Version 5: the registry grows to 1096 edges (959 certificate edges from an automated pipeline over 987 systems: prime pairs {0,2k} up to k=60, all admissible even triples of diameter <= 40 and quadruples of diameter <= 30, arithmetic progressions a mod q for q <= 40, quadratic polynomials n^2+c and n^2+n+c; 124 literature edges; 13 trivial). Certificate edges compared against an initial segment with at most two occurrences are tagged 'HR-thin'. All certificate edges share one mechanism and count once as evidence; the tags allow restricting the graph to edges between independently held conjectures. Version 6: (i) the exact twin-prime crossing: the least window length at which an admissible configuration of twin-pair positions beats the initial segment is y0 = 379 (theorem, with the table of minimal diameters D(K) of admissible K-packings of twin pairs for K <= 23, program supplied, independently recomputed); (ii) ten composite incompatibilities between named statements with proofs and prior-art status (five published in substance, four folklore, one -- Chang's conjecture versus Diamond-plus -- not found stated); (iii) the registry grows to 268 edges between named conjectures (1268 edges in total), most of the new ones found by literature-search agents and confirmed by adversarial verifier agents; the full verifier log is supplied. Version 7: registry data update only; 349 edges between named conjectures (1349 in total) after three rounds of agent search and adversarial verification across ten domains; both verifier logs supplied.