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

This work presents a termination approach for 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) = \bigl(t_{\min}(n), n\bigr),M(n)=(tmin(n),n),

and demonstrating its strict descent under iteration, the paper explores termination for all positive odd integers.

An attempted proof based on this data-dependent block length and lexicographic potential is presented. However, critical lemmas remain unproven by experts, leaving the argument incomplete. This paper is offered as a contribution to ongoing discourse rather than a conclusive resolution. Verification and validation by the mathematical community are essential before the approach can be accepted.