Prefix Cylinder Laws for the Generalized Collatz Family 3n + c: Algebraic Structure Preservation and Computational Verification

We generalize the Prefix Cylinder Law and exact negative-binomial distribution established for the Collatz map (c = 1) to the family of generalized maps 3n + c for c ∈ {1, 5, 7, 11}. The central algebraic result (Theorem 6) proves by induction that for each admissible c, any finite valuation sequence corresponds to a unique odd residue class modulo 2S_r+1, characterized by the formula a_k(c) ≡ 3⁻¹(2^k - c) (mod 2ᵏ⁺¹). Corollary 9 establishes that the exact negative-binomial distribution of the prefix sum S_N is preserved identically for all admissible c. Computational validation covers more than 2,000,000 orbits across four values of c, achieving 100% congruence accuracy in all cases. A matrix of 48 experiments (M ∈ {64, 128, 256}, N ∈ {20, 40, 60, 80}, c ∈ {1, 5, 7, 11}) confirms that valuation frequency errors remain below 0.04 and KS statistics below 0.10 in every configuration. The key insight is that the algebraic prefix structure of the Collatz map is not special to c = 1: it is a generic property of the entire 3n + c family for any odd admissible c. This separates the algebraic difficulty of the Collatz problem from its combinatorial difficulty, and identifies the extension from the prefix regime to the full orbit as the genuine open frontier, independent of c.