Exact Rooted Phase Closure and Mersenne Cusp Peeling in Dyadic Collatz First Passages

We study finite first passages for the shifted shortcut Collatz map F(z)=cases3z/2,&z even,\\(z+1)/2,&z odd,cases from an odd rooted seed τ3ˢ to a dyadic threshold τ2ᵛ with the same odd phase τ. The terminal segment begins at the last pre-endpoint multiplier record and ends in a multiplier-record failure. For a fixed prefix word, terminal word, root label, and threshold exponent, we prove an exact phase-closure theorem: at most one odd phase can realize the prescribed word and terminal crossing, and the candidate is given by an explicit divisibility equation and a quotient--remainder window. We then show that first-passage minimality and canonical last-record status factor into two independent finite tests: an affine barrier envelope and a left--right multiplier cusp. Every noninitial odd-seed cusp has the form pʳ w with r≥2 and w beginning in . Peeling the block ʳ yields a finite suffix-debt language, the exact gate 2ʳ⁺¹t-1\,ʳ\,3ʳt, automatic removal of the gate from the first-passage barrier, and one explicit residue class for the peeled predecessor and rooted phase. Together these results give a complete exact decision procedure for every fixed word datum. They do not classify unbounded word lengths, exclude all rooted certificates, exclude repeated record failures, or prove the Collatz conjecture.