A Phase Boundary Theorem for Forced-Prefix Feasibility in Collatz Lift Geometry

This paper establishes a sharp phase boundary for forced-prefix feasibility in the accelerated Collatz lift geometry under the rounded-critical regime

a=round(p∗m),p∗=1log⁡23.a = \mathrm{round}(p^* m), \quad p^* = \frac{1}{\log_2 3}.a=round(p∗m),p∗=log231.

We prove that the first infeasible prefix depth r∗r^*r∗ satisfies

r∗m→2(1−p∗)as m→∞,\frac{r^*}{m} \to 2(1 - p^*) \quad \text{as } m \to \infty,mr∗→2(1−p∗)as m→∞,

with fluctuations of order O(m−1/2)O(m^{-1/2})O(m−1/2).

The proof is entirely combinatorial and relies on:

• A triangular 2-adic prefix automorphism

• An odd-unit residue permutation induced by the OptA lift congruence

• Exact binomial prefix counting under the mathematical lift measure

• A saddle-point analysis of linear barrier crossing

No stochastic independence assumptions are required at any stage.

The limiting constant

2(1−p∗)=2−2log⁡23≈0.738140492857…2(1 - p^*) = 2 - \frac{2}{\log_2 3} \approx 0.738140492857\ldots2(1−p∗)=2−log232≈0.738140492857…

arises deterministically from feasibility geometry and entropy-neutral prefix density under the lift congruence.

This result isolates the large-scale geometry governing forced-prefix survival depth and provides a structural constant for the lift process in the rounded-critical regime.