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 K_j = v_2(3·T_odd^j(n)+1) along deterministic Collatz orbits, combining exact algebraic results with large-scale computational validation. Rigorous results: For each k ≥ 1, the event {K_j = k} is equivalent to a unique congruence x_j ≡ a_k (mod 2^(k+1)), where a_k ≡ 3^(-1)(2^k - 1) (mod 2^(k+1)). An exact prefix cylinder law is proved by induction: any finite valuation sequence (k_0,...,kᵣ₋₁) corresponds to a unique odd residue class modulo 2^(S_r+1). The exact negative-binomial distribution of the prefix sum S_N follows. A density-one prefix discrepancy theorem is established via Hoeffding and Chernoff bounds: for any η < 1/2 and C > 0, the fraction of odd n < 2^M violating |p_k - 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 ≤ 10^7) finds no violation of the bound with empirical P95 constant C_95 = 0.300.