A computer-assisted well-founded descent for Collatz components under dyadic leaf certificates in the 6-bit hierarchy Mt = 9 · 214+6t

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)