Rooted Interlocks for Paradoxical Collatz First Excursions

Let T(n)=cases(3n+1)/2,&n odd,\\ n/2,&n even,cases be the shortcut Collatz map. Rozier and Terracol call a finite segment paradoxical when its multiplicative coefficient is less than one although its endpoint is not below its start. We study a more rigid object: a paradoxical first excursion that is attached to a rooted Mersenne-type gate through a capped inverse tree, a suffix-debt condition, a finite $3$-adic label, and a dyadic threshold. We first prove that the translated map $F(n+1)=T(n)+1$ identifies right-cusp first excursions exactly with paradoxical first excursions. We then exhibit an explicit infinite family of locally debt-compatible phase hits, and in fact infinitely many inverse depths supporting such families. Thus neither the inverse branch nor suffix debt nor dyadic phase geometry alone can exclude an attachment; the terminal first-excursion word is essential. For the basic rooted branch ᵏ we derive a mixed system consisting of one congruence modulo 3ᵏ, one congruence modulo 2L, and an exact terminal deficit identity 2L3ᵏ(H-y)=GLt-UL. For every fixed inverse depth, terminal word, and threshold exponent, the admissible first-passage interval has length strictly smaller than the mixed CRT period 2L3ᵏ. Hence there is at most one arithmetic candidate. An independent exact audit of the 593 paradoxical sequences classified by Rozier and Terracol reduces them to 80 right-cusp first excursions, then to 13 rooted anchors and 18 gate instances; all 72 debt-compatible node-label pairs miss the terminal threshold. Combining this audit with their exclusion theorem, we obtain the conditional-on-their-computational-theorem consequence that any canonical rooted attachment must start above 2.8×10¹⁹+1 in shifted coordinates and have terminal length at least 301,994. No universal nonexistence theorem, no exclusion of all Collatz counterexamples, and no proof of the Collatz conjecture is claimed.