Exact Rooted Phase Closure and Mersenne Cusp Peeling in Dyadic Collatz First Passages
Year: 2026
Type: preprint
Abstract: We study finite first passages for the shifted shortcut Collatz map $F(z)=\begin{cases}3z/2,&z\text{ even},\\(z+1)/2,&z\text{ odd},\end{cases}$ from an odd rooted seed $\tau3^s$ to a dyadic threshold $\tau2^v$ with the same odd phase $\tau$. 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 $p\B\A^r\mid w$ with $r\ge2$ and $w$ beginning in $\B$. Peeling the block $\B\A^r$ yields a finite suffix-debt language, the exact gate $2^{r+1}t-1\xrightarrow{\,\B\A^r\,}3^rt,$ 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.
Authors: Ueoka, Yoshiki, Nagi, Akari, Sui
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7172296992
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