Resonant Terminal Layers and Critical Interfaces in Finite Collatz Bridges

Abstract

We study finite orbit segments of the shortcut Collatz map in shifted coordinates. A terminal word is called multiplier-supercritical when its linear multiplier exceeds one. For a bridge ending at height $k$ with overshoot $U$, we prove an exact depth-by-depth criterion for terminal supercriticality. The criterion is governed by the resonance $2^\ell\approx 3^{\floor{(\log 2/\log 3)\ell}}$, yields a terminal depth that diverges whenever $k\to\infty$ and $U/k\to0$, and is attained by explicit finite orbit segments at infinitely many first-failure depths. At fixed letter counts, the classical extremal affine word is the two-phase block $\B^r\A^h$. Starting from this known extremum, we express the deficit from the extremal affine constant as a positive weighted inversion sum. Every nonextremal word has relative deficit strictly greater than $1/6$; hence a near-threshold terminal suffix is forced to equal $\B^r\A^h$ exactly. We then show that finite $2$-adic and $3$-adic congruence requirements never obstruct embedding a prescribed finite prefix, although bounded displacement selects at most one point in each resulting displacement lattice. Compressing the prefix into two affine coordinates gives an exact balance identity and a dichotomy: either the complete power ratio approaches neutrality, or the prefix contains a low odd state. In the latter deep-reset regime, the return to endpoint scale contains an unbounded ladder of record endpoints, each carrying a multiplier-supercritical terminal layer, immediately before a one-letter supercritical/subcritical interface and the exact two-phase block. All statements concern finite bridges; the Collatz conjecture remains open.