From the 0.415 Criterion to Coupling Inequalities — A Binary Exploration of the Collatz Conjecture

This paper is not a proof of the Collatz conjecture. It is an exploratory note that performs a precise problem reduction of the conjecture. Its core contributions are: (1) strictly equating the convergence of the conjecture to a precise numerical inequality P1 - P3 < 0.415; (2) identifying and critiquing three common logical traps; (3) establishing a precise classification of local dynamics (three engine states, the consecutive S1 lockdown theorem); (4) honestly marking the boundaries that current mathematical tools cannot cross; (5) honestly documenting an optimistic conjecture that was falsified by the LTE lemma, and distilling deeper insights from it. All derivations in this paper can be verified line by line by any trained mathematical reader. It does not complete a proof of the conjecture, but it precisely marks the coordinates that future attacks must target, and honestly records a falsified path.