Critical Survivor Capacity and Integer Realizability in Accelerated Collatz Dynamics
Year: 2026
Type: preprint
Abstract: A persistent obstruction in the Collatz problem is the gap between finite-depth compatibility and realization by one fixed positive integer. For nested nonempty positive-integer source fibres, this preprint proves an exact realization criterion: a common positive integer exists if and only if the least source remains bounded, equivalently eventually stabilizes. It also proves an abstract critical-survivor capacity theorem. If a survivor family has entropy rate H, arithmetic scale rate Xi/2, independent information saving alpha, and state-window exponent mu, then linear survivor demand is impossible whenever mu * 2(H - alpha log 2) / Xi < 1. Under one explicitly stated dense-periodic calibration, this yields the sector-specific threshold alpha_* = 0.493074483908… bits per coarse step. Finally, an exact cyclic-rotation transport identity shows that same-cycle target equations cannot be repeatedly counted as independent entropy reductions once the realized segment data are fixed. These results isolate structural and quantitative obstructions; they do not prove the Collatz conjecture or exclude all positive nontrivial cycles.
Author: Moon, Kyung-Up
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212190374
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