Resonant Terminal Layers and Critical Interfaces in Finite Collatz Bridges

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^≈ 3^(log 2/log 3), yields a terminal depth that diverges whenever k→∞ and U/k→0, 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 ʳʰ. 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 ʳʰ 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.