Toward a Synchronization Obstruction Theory for the Collatz Problem: Safe Corridors, 2-adic Hearts, Integer Liftability, and a Finite Arithmetic Witness

We develop a structural obstruction framework for the Collatz conjecturecentered on the notion of integer liftability. The paper introduces the integer-liftability ratio ρ_L = log₂(|s_L|+1)/A_L,where s_L is the centered representative of the Heart residue b_L mod 2^{A_L}.This ratio measures the gap between 2-adic symbolic compatibility and actualpositive-integer realizability. Principal contributions: 1. Integer-Liftability Necessity Lemma (Theorem): any integer-liftable infinite safe corridor must satisfy ρ_L → 0. 2. Numerical evidence (Observation): over 3,000 high-tension corridors, ρ_L → 1 with corr(A_L, log|s_L|) = 0.9991 and P(ρ ≥ 0.95) = 0.977. 3. Finite Arithmetic Witness (Exact Theorem): the survivor set S_k of odd residue classes (mod 2^14) that can sustain the Beatty corridor satisfies S_9 = ∅. No integer can follow the Beatty word for 9+ steps. Proof is by complete exhaustive computation with no floating-point arithmetic. 4. Zero-Tail Exclusion Problem (Open): ruling out infinite corridors with ρ_L → 0 is identified as the precise remaining barrier. This paper forms a companion to:Moon, K. (2026). Collatz Normal Form: Time as Degree-of-Freedom Elimination and the Trace-Compressed Engine. Zenodo. https://doi.org/10.5281/zenodo.18233316 The two papers describe the same obstruction from multiplicative (cocycle equation2^K = 3^E · C) and additive (2-adic residue congruence) perspectives respectively. This paper does not claim a proof of the Collatz conjecture.It develops a rigorous obstruction program with a precisely identified open barrier.