Collatz Conjecture — Reduction to 0.415 Criterion and Coupling Inequalities

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<0.415P1−P3<0.415; (2) identifying and critiquing three common logical traps; (3) establishing a precise classification of local dynamics (three engine states, the consecutive S1S1 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 can be verified line by line. It does not complete a proof, but precisely marks the coordinates that future attacks must target, and records a falsified path.