A DYADIC, COMPONENT-WISE WELL-FOUNDED DESCENT FOR THE 5x+1 MAP (ANALOGY WITH 3x+1)

Abstract. We study the Collatz-type map T5 defined by T5(n) = n/2 for even n andT5(n) = 5n+1 for odd n. We develop a dyadic leaf-certificate descent framework in analogywith the 3x + 1 case: odd acceleration, the 25u + 1 parametrization on sources 5u, 2-adiccylinders attached to valuation words, a 220-block refinement hierarchy linked to ord25(2) = 20,and an anti-25 lifting loop. We give a fully explicit forward/backward proof that every oddcomponent intersects an odd source 5u. We also prove an automatic descent in the large regimee(u) = v2(25u +1) ≥ 21. In the small regime e(u) ≤ 20, we formalize a terminating, leafwiseverifiable certificate framework. Under this hypothesis we obtain a well-founded descent onsources and a conditional multi-attractor convergence conclusion (classification into a finite setof terminal cycles)