Structural Decomposition of Collatz Dynamics: Valuation Collapse, Residue Expansion, and Suppression of Expansion Chains

This work proposes a structural decomposition of the Collatz dynamics into three interacting components: valuation collapse, residue-based expansion, and suppression of sustained expansion chains. We reformulate the Collatz problem as a global gain–loss inequality problem. The analysis proceeds in three stages: 1. A strict valuation law describing collapse dynamics in terms of v2 (3n+1). 2. A residue-based branching structure governing expansion behavior, including a level-shifting mechanism for valuation a=2 transitions. 3. An empirical study of expansion chain suppression, showing that admissible residue classes supporting long expansion streaks form a rapidly thinning filtration under dyadic refinement. Extensive computational experiments up to modulus 2¹2 indicate that expansion streaks of length ≥5 are not supported by any residue class in the tested range. These results suggest that sustained expansion is suppressed not by monotone collapse intensity, but by progressive loss of admissible re-entry states in the residue space. This work is intended as a structural contribution and does not claim a complete proof of the Collatz conjecture. Version 2. 0: - Added valuation stability lemma (fully proven) - Introduced macro-block drift framework- Established finite expansion budget (empirical + structural bound) - Reformulated Collatz as gain–loss inequality problem