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

Abstract. We present a component-wise descent scheme for the Collatzmap based on the odd accelerated map and an “anti-9” lift that sendsany odd non-multiple of 3 to an odd multiple of 3 in the same connectedcomponent of the Collatz graph. The core step is a well-founded descenton the odd parameter u associated to odd multiples 3u: for every oddu > 1 we produce a strictly smaller u∗ < u such that 3u and 3u∗belong to the same component. The descent is analytic when e(u) :=v2(9u +1) ≥ 7 and is certified by a finite table of dyadic leaf certificateswhen e(u) ≤ 6. The remaining regime e(u) ≤ 6 is discharged by afinite leaf-wise verification. The certificate tables covering all dyadicleaves modulo M = 9 · 220 with e(c) ≤ 6 have been constructed andverified by a deterministic verifier; all four reference logs end with RESULT:OK