A Component-wise Descent Framework for Collatz-type Maps Ta,b via Anti-a2 Lifts and Dyadic Certificates

We present a component-wise descent framework for the family of odd affine Collatz-typemapsTa,b(n) = n/2if n ≡ 0 (mod 2),an+b if n≡1 (mod 2),under the general admissibility assumptionsa ≥3,a odd,b odd,The paper is deliberately written at two levels.gcd(a,b) = 1.First, it develops a general framework valid for all admissible pairs (a,b): componentinvariance of terminal behavior, anti-a2 reduction to source vertices au, affine formulas foraccelerated odd iterates, and a dyadic uniformity theorem propagating descent from theminimal representative of a leaf to the whole leaf.Second, it isolates a more restrictive but cleaner regime in which stronger global conclusions may be drawn, notably when a = p is an odd prime and the anti-p2 lift is universal.The resulting framework is hybrid and conditional: once a complete verified certificatefamily and a finite kernel are available, one obtains a well-founded component-wise descent.This yields eventual periodicity on the covered set, with no uniqueness of the terminal cycleassumed or claimed. In particular, the conclusion is compatible with the presence of severaldistinct terminal cycles, as can occur for maps such as 5x + 1