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

A source-compatible finite-certificate framework is developed for the odd affine Collatztype mapsTa,b(n) = n/2,n ≡0 (mod 2),an+b, n≡1 (mod 2),under the admissibility assumptionsa ≥3,a odd,b >0,b odd,gcd(a,b) = 1.The constants of the 3x + 1 proof are replaced by their natural affine analogues:9 becomes a2, the anti-9 lift becomes an anti-a2 lift, the exponent 6 becomes q = orda2(2),the contraction (u −7)/64 becomes (u−da,b)/2q with da,b = b(2q −1)/a^2, and the primitivecoefficient 6561 = 3^8 becomes a^8.The result is conditional in the mathematically necessary sense. For a general pair (a,b),the anti-a2 lift need not cover every odd residue class, and the finite frontier counts neednot equal the special 3x + 1 counts. The theorem is therefore stated on the covered set:the integers whose odd part is either already a source au or is anti-a2 liftable to such asource. Under a complete finite source-compatible certificate and a finite periodic kernel,every covered orbit is ultimately periodic. Moreover, all terminal cycles on the covered set arealready among the cycles reached by the finitely many kernel sources. Upward reconstructiondescribes basins of attraction and does not create new cycles