A Modular Residue Framework for Orbitwise 2-Adic Valuations in Collatz Sequences: Exact Prefix Laws, a Density-One Bound, and Large-Scale Validation

We study the distribution of 2-adic valuations Kⱼ = v₂ (3·Tₒddʲ (n) +1) along deterministic Collatz orbits, combining exact algebraic results with large-scale computational validation. Rigorous results: For each k ≥ 1, the event Kⱼ = k is equivalent to a unique congruence xⱼ ≡ aₖ (mod 2^ (k+1) ), where aₖ ≡ 3^ (-1) (2ᵏ - 1) (mod 2^ (k+1) ). An exact prefix cylinder law is proved by induction: any finite valuation sequence (k₀,. . . , kₑ-₁) corresponds to a unique odd residue class modulo 2^ (Sᵣ+1). The exact negative-binomial distribution of the prefix sum SN follows. A density-one prefix discrepancy theorem is established via Hoeffding and Chernoff bounds: for any η 0, the fraction of odd n < 2M violating |pₖ - 2^ (-k) | ≤ C·log (N) /√N in the first ⌊ηM⌋ steps tends to zero as M → ∞. Open problem: Extending from the prefix regime N ≤ η·log₂ (n) to the full orbit remains open. Computational validation: A matrix of 12 experiments (M ∈ 64, 128, 256, N ∈ 20, 40, 60, 80, 10, 000 orbits each) shows valuation frequency errors below 0. 002 and autocorrelations below 0. 009 at all lags. Exhaustive verification over 4, 999, 944 orbits (n ≤ 10⁷) finds no violation of the bound with empirical P95 constant C₉5 = 0. 300.