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 distribution ≈ 68:26:6; the cumulative synchronisation budget is 9.97% (z = −0.27 vs 10.58 ± 2.25% null); the staggered-to-coincident transition itself is generic. A prime-specific bounded regime (load-bearing result): within a 50k–300k digit-scale window, the spacing of synchronisation clusters follows a power law with exceptional fit quality — at every individual regime scale (50k, 100k, 200k, 300k), the real-prime R² exceeds the maximum across 100 PNT-density null trials, giving zero null exceedances per scale and therefore zero joint exceedances across the four scales (R² = 0.984, 0.986, 0.979, 0.979 versus null maxima of 0.923, 0.861, 0.856, 0.886). The structure is specifically a property of the mod-6 partition: a shuffle-test control that preserves prime-prime correlations but reassigns primes to random pseudo-classes (100 trials) does not reproduce it — the shuffle-R² distribution sits on the density-matched null at every regime scale (z_BC ranges −0.04 to +0.27, |z_BC| ≤ 0.27 everywhere), while real-mod-6 R² remains separated from the shuffle distribution (z_AB ranges +2.16 to +2.22 across 50k, 100k, 200k, 300k). The exponent β ≈ 0.637 is itself null-indistinguishable in any single base (z = +0.53 at 200k base-10). Across five positional encodings (base-2, 5, 6, 7, 10), β stays within 0.020 of itself and within ±0.02 of ln(2)/ln(3) = 0.6309; the two bases coprime to 6 (5 and 7) land within 0.003 and 0.0002 of base-10 respectively, eliminating any factor-of-2 dependence. Zero of 100 null trials reach the real R² in any base. These five encodings are correlated reparametrisations of the same prime sequence rather than statistically independent samples; the consistency under reparametrisation is a robustness statement. The regime ends in a clean phase boundary at 300k–500k digits, observed in both real-mod-6 and shuffle populations. A second suggestive phenomenon (open): above the 300k–500k phase boundary, real-prime cluster sequences are more scattered than null (R²_real = 0.196 vs R²_null_mean = 0.570 at 1M digits, z = −1.89; weaker at 2M and 3.25M, with |z| < 2). Primes are cleaner than null inside the bounded regime, more scattered than null above it. The maximum |z| does not clear the paper's |z| ≥ 2 bar; we label this suggestive rather than prime-specific. The mechanism is open. Observational, folklore, and closed-structurally results. The §6.2 cluster-count disparity at 200k (z = −1.37) is reported as marginal-to-null under honest two-tailed framing (parametric p ≈ 0.17). The §5.3 inter-cluster alternation observation (z = 2.22 to 7.28 across thresholds, small counts) is reported observationally, not as a pre-registered confirmatory test. The mod-6 class-membership exclusion principle (§3) generalises §3's mod-3 result and yields as one corollary the structural impossibility of factor 105 as a coincident-gap escalation at any scale (closing a v2.6 open question). The residue N mod 3 of any RSA modulus N = pq classifies whether its prime factors belong to the same or different mod-6 classes — elementary folklore formalised here within the digit-gap framework, consistent with the independently-posted observation of Gocgen [6]. Python verification scripts in the companion repository confirm all results; the principal script completes in approximately 32 seconds using only the standard library.