Digit Gaps at the First Cluster: Structural Analysis of the Mod-6 Prime Series Products

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). The cumulative products Π₅(n) = ∏ p₅,ᵢ and Π₇(n) = ∏ p₇,ᵢ exhibit digit gaps — decimal scales at which no product exists. The present paper analyses the first cluster in the cluster-and-gap structure of these digit-jumps, treating its small-n features structurally. What is true (DERIVED). Theorems 1–3, supported by Lemmas 1–7: the 5-series skips digits 60–61, the 7-series skips digits 62–63, both series skip digits 80–81, and the modular constraints on potential lifting factors are determined by the active-constraint sets of each series. The theorems combine exact…