Resonant Gap Certificates and Rooted Dyadic Ancestry in Finite Collatz Record Ladders

Abstract

We study first passages between consecutive dyadic record thresholds for the shortcut Collatz map, using shifted coordinates in which the two branches are \(\A(x)=3x/2\) on even states and \(\B(x)=(x+1)/2\) on odd states. A record endpoint carries a terminal layer of suffixes whose linear multipliers exceed one. If this layer does not reach the beginning of the transition, the first failed suffix occurs at a one-sided \(2\)--\(3\) resonance depth and satisfies an exact reserve identity. The reserve splits into geometric threshold undershoot and normalized affine deficit. A discrete one-sixth gap separates the exact two-phase word from every nonextremal word. The normalized affine deficit obeys a positive composition law, so costs of disjoint failed suffixes cannot cancel. Macroscopic geometric gaps give a separate positive reset ledger; avoiding both ledgers forces increasingly sharp lower approximations to \(\log 2/\log 3\). Small affine cost alone confines the failure depth to an explicit finite window and, at natural scales, restricts its reduced ratio to lower continued-fraction convergents or intermediate convergents, with a finite multiplicity bound. Every failed suffix factors as one odd reset followed by a coherent recovery word. Exact integer realizations of a fixed recovery word form an affine lattice, and the requirement that the recovery at least double its initial state leaves finitely many lattice points and thresholds. An exact search through recovery length \(31\) finds genuine local record gaps, including an explicit transition at threshold \(437\), so universal local coalescence is false. We then impose rooted dyadic ancestry. Every first passage from a proper dyadic ancestor ends in two forced even-branch steps and has endpoint divisible by \(9\); in the upper part of the record window it ends in three such steps and is divisible by \(27\). This theorem eliminates every local gap in the bounded recovery graph without root-by-root simulation. All results concern finite trajectories; the Collatz conjecture remains open.