Dickson's conjecture versus superadditivity of prime-constellation counts: explicit certificates, and an unconditional refutation of a recent compatibility claim

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.