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r-Adic Stabilization of Hyperoperations: Affine Laws of Hyperoperation, Phase Freezing, and Transcendence of the Attractor

Year: 2026
Type: preprint
Abstract: This paper develops a unified theory of r-adic stabilization across the Knuth-Grzegorczyk hyperoperation hierarchy, focusing on the convergence of terminal digits in integer power towers and higher hyperoperations. For ordinary right tetration, we establish exact p-adic valuation laws for consecutive increments using Lifting-the-Exponent techniques. This yields a complete trichotomy for the congruence position, determines explicit finite transient thresholds, and characterizes the periodic oscillations of congruence speed in non-squarefree radices, fully resolving the case of perfect power bases. For higher hyperoperations of level 5 and above, we prove that the modular phase of the tower height freezes unconditionally. Consequently, the congruence position becomes strictly affine with respect to an intrinsic discrete time, and the normalized difference derivative stabilizes to an exact universal rational constant in any number base. We apply this framework to obtain exact, closed-form expressions for the stable digits of the Ackermann-Péter sequence and Graham's number in general radices. For left-associative tetration, we prove that the congruence position remains stationary. For general mixed-chirality signatures, we classify asymptotic behaviors through a finite five-element transition monoid and prove the existence of an explicit 2-adic contracting attractor. Finally, we establish that the tetration attractor in the p-adic integers is transcendental over the rationals, except for an explicit family of rational fixed points of the form a = v^v. The paper is accompanied by a self-contained Python script implementing the deterministic evaluation algorithm without materializing large power towers.
DOI: 10.5281/zenodo.24135758
Language: English
Open Access status: green
OpenAlex ID: W7225785551
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