Digit Gaps and Synchronisation Clusters in Mod-6 Prime Products: Analysis to 3.25 Million Digits with Null Model Verification

All primes greater than 3 belong to exactly one of two residue classes modulo 6: the 5-series (p ≡ 5 mod 6) or the 7-series (p ≡ 1 mod 6). We study the cumulative products of primes from each series across four orders of magnitude in digit-length, from 10³ to 10⁶ digits. These products exhibit digit gaps — decimal scales at which no product exists — whose joint behaviour transitions from staggered to coincident as scale increases. We establish results in four categories, each tested against a 100-trial null model of pseudo-primes with prime-number-theorem density. Universal properties (generic to logarithmic sequences): the coincident-gap spacing lies in {1, 2, 3} for 99.9998% of 2.35 million spacings, with…