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 computational anchor values at the relevant n (Tables 1–2 digit counts, verified to 200-digit precision via mpmath) with monotonicity of the digit-length sequence (Lemma 5). Lemmas 1–4, 6, 7 are elementary algebraic statements about residue classes and coprimality, requiring no high-precision computation. The theorems are exact computational verifications rather than first-principles structural derivations (§4 status note). What is shown (OBSERVED). Under the §4.0 divisibility convention (mod-3 inclusion via mod-6 completion; mod-15 escalation at coincident clusters via joint-absence; one among alternatives, with the falsifier specified in §6.3), Propositions 1–2 verify by exhaustive enumeration that the minimum positive integer k satisfying the convention is 3 at boundary 1 (for both series), 3 at boundary 2 for the 5-series, and 15 at boundary 2 for the 7-series. The transition from boundary 1 to boundary 2 — staggered → coincident gap structure with the 3 → 15 factor escalation at the 7-series — reflects digit-arithmetic at coincident gaps rather than newly-activating modular constraints (§6.1, §6.2). Scope. This paper treats the digit-arithmetic content at the first cluster only. Empirical generalisation across the cluster sequence D ∈ [90, 17600] — including a 23-cluster comparative tabulation, a six-instance bounded merged regime sequence, and the empirical execution of the §6.3 F5 alternative-convention probe — is in companion paper [6]. Large-scale digit-gap statistics across four orders of magnitude are in [7]. The χ₅ tight-oscillation regime in 5-series primes, co-located at the same arithmetic scale as the first cluster but consistent with chance under random placement (p = 0.42; not a structural connection), is in [8]. A physics interpretation is in [9].