Digit Gaps in Prime Series Products: A Number-Theoretic Analysis of Mod-6 Prime Classes

All primes greater than 3 belong to exactly one of two residue classes modulo6: the 5-series (p ≡ 5 mod 6) or the 7-series (p ≡ 1 mod 6). We prove thatcumulative products of primes from each series exhibit specific digit gaps —scales at which no product exists. For the 5-series, no product has exactly 60or 61 decimal digits. For the 7-series, no product has exactly 62 or 63 digits.We further prove that any integer filling these gaps must contain 3 as a factor.These results follow from elementary modular arithmetic and directcomputation of prime products. To our knowledge, these specific digit gaps have not previously been documented.