AbstractWe present a component-wise descent scheme for the Collatz map based onthe odd accelerated map and an anti-9 lift that sends every odd non-multiple of 3to an odd multiple of 3 in the same connected component of the Collatz graph.The core step is a well-founded descent on the odd parameter u associated withodd multiples 3u: for every odd u > 1, we construct a strictly smaller odd integeru∗ < usuch that 3u and 3u∗ belong to the same connected component. The descentis analytic in the regimee(u) := v2(9u +1) ≥ 7,and is discharged in the complementary regime e(u) ≤ 6 by a finite family of dyadicleaf certificates. The finite regime is organized through the 6-bit hierarchyMt =9·214+6t,whose role is to preserve the relevant 2-adic residue control under refinement. Thecertificate tables at scale M = 9 · 220 are checked by a deterministic verifier usingexact integer arithmetic. A new uniformity lemma (Lemma 8.1) proves analyticallythat the descent condition w(yk) < u holds for every element of the leaf, not merelyfor the tested representative; this closes the gap between finite verification anduniform validity on the whole class. Combined with strong induction, this yieldsconvergence of every positive integer to the cycle (1,2,4)
Julian REDERO (Fri,) studied this question.