Quotient--Remainder Phase Locking for Terminal Paradoxical Certificates in Rooted Dyadic Collatz Ladders

For the shifted shortcut Collatz map, we study multiplier-record failures inside first passages from an odd root seed to a dyadic threshold with the same odd phase. A last-record reduction produces a canonical terminal suffix whose multiplicative coefficient is strictly below one at every nonempty prefix, although its endpoint exceeds its start. We prove that each fixed suffix lies on an exact affine lattice ray and that a suffix crossing $H=\tau2^v$ must have length $L>v$. Writing $q=L-v$, we then prove an exact quotient--remainder locking theorem: its occurrence index on the affine ray is $\floor{\tau/2^q}$, while the remainder $\tau\bmod 2^q$ lies in an explicit word-dependent interval. This yields a phase-dependent coefficient bound and, in every nonprimitive quotient channel, forces the maximal number of expanding letters compatible with contraction. Finally, we transfer the published Rozier--Terracol census through the shifted conjugacy and verify all canonical rooted embeddings in the certified range; none occur. The result is finite in its computational part and does not exclude certificates outside that range, exclude repeated failures, or prove the Collatz conjecture.