A Data-Dependent Lexicographic Termination Proof for the Collatz Map on Odd Integers

This work presents a fully rigorous termination proof of the Collatz iteration restricted to odd integers. The key innovation is the introduction of a data-dependent block length function tmin⁡(n)t_{\min}(n)tmin(n), which dynamically determines the minimal number of odd-step-plus-full-halving iterations required for the sequence to decrease below its starting value. By defining a lexicographic potential M(n)=(tmin⁡(n),n)M(n) = (t_{\min}(n), n)M(n)=(tmin(n),n) and proving its strict descent under iteration, the paper establishes well-foundedness and guarantees termination for all positive odd integers. The proof combines analytical insights into halving step distributions with a novel carry-chain argument, avoiding reliance on heuristics or probabilistic models. The accompanying Python code empirically verifies the theoretical upper bounds on tmin⁡(n)t_{\min}(n)tmin(n). This contribution offers a significant advance toward resolving the Collatz conjecture and provides a framework potentially extendable to generalized αn+β\alpha n + \betaαn+β mappings.