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Heisenberg Uncertainty Relation Derived as a Special Limiting Case of a Single Elementary Phase Cycle (v.02).

Hennadii Kovalov

Published Jun 12, 2026• 0 citations
Publisher

Abstract

The proposed geometric model is not a return to 19th-century classical mechanics, but rather builds on the latest trends in Planck-scale deterministic cellular automata ('t Hooft, 2021), the theory of internal quantum clocks (Donadi, 2023), and the nonlinear quantum dynamics of asymmetric rotors (Micheli and Stone, 2022). The model directly resolves the topological crisis of describing light nuclei as saturated manifolds, documented in the current literature (Manton, 2022; Lau, 2020). We derive a direct, non-linear relation between the internal phase time per radian , the macroscopically observable coordinate time , and the relative energy fluctuation . Through rigorous differentiation and integration of these phase relations, we demonstrate that for topologically synchronized systems, the product of energy and observable time constitutes a strict dynamic process invariant: . Moving beyond phenomenological interpretations, we deduce the classical Heisenberg uncertainty relation without invoking independent quantum postulates, showing it to be a special boundary limit governing a single elementary phase cycle () under critical fluctuation. Furthermore, we prove that Louis de Broglie’s matter wave equation () is not an independent axiom but emerges naturally as the spatial displacement executed by a localized particle during a single complete internal chronal rotation. Finally, we bridge the gap between microscopic quantum reversibility and the macroscopic thermodynamic arrow of time by analyzing the dynamics of synchronization breakdown and phase slippage () using the latest advancements in deterministic quantum mechanics.

DOI
10.5281/zenodo.20662343

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We derive a direct, non-linear relation between the internal phase time per radian , the macroscopically observable coordinate time , and the relative energy fluctuation . Through rigorous differentiation and integration of these phase relations, we demonstrate that for topologically synchronized systems, the product of energy and observable time constitutes a strict dynamic process invariant: . Moving beyond phenomenological interpretations, we deduce the classical Heisenberg uncertainty relation without invoking independent quantum postulates, showing it to be a special boundary limit governing a single elementary phase cycle () under critical fluctuation. Furthermore, we prove that Louis de Broglie’s matter wave equation () is not an independent axiom but emerges naturally as the spatial displacement executed by a localized particle during a single complete internal chronal rotation. To establish the physical carrier of this wave, we introduce the Rotor-Photon () as a unified electromagnetic-chronal field operator. Finally, we bridge the gap between microscopic quantum reversibility and the macroscopic thermodynamic arrow of time by analyzing the dynamics of synchronization breakdown and phase slippage ().
We derive a direct, non-linear relation between the internal phase time per radian , the macroscopically observable coordinate time , and the relative energy fluctuation . Through rigorous differentiation and integration of these phase relations, we demonstrate that for topologically synchronized systems, the product of energy and observable time constitutes a strict dynamic process invariant: . Moving beyond phenomenological interpretations, we deduce the classical Heisenberg uncertainty relation without invoking independent quantum postulates, showing it to be a special boundary limit governing a single elementary phase cycle () under critical fluctuation. Furthermore, we prove that Louis de Broglie’s matter wave equation () is not an independent axiom but emerges naturally as the spatial displacement executed by a localized particle during a single complete internal chronal rotation. To establish the physical carrier of this wave, we introduce the Rotor-Photon () as a unified electromagnetic-chronal field operator. Finally, we bridge the gap between microscopic quantum reversibility and the macroscopic thermodynamic arrow of time by analyzing the dynamics of synchronization breakdown and phase slippage ().
We derive a direct, non-linear relation between the internal phase time per radian , the macroscopically observable coordinate time , and the relative energy fluctuation . Through rigorous differentiation and integration of these phase relations, we demonstrate that for topologically synchronized systems, the product of energy and observable time constitutes a strict dynamic process invariant: . Moving beyond phenomenological interpretations, we deduce the classical Heisenberg uncertainty relation without invoking independent quantum postulates, showing it to be a special boundary limit governing a single elementary phase cycle () under critical fluctuation. Furthermore, we prove that Louis de Broglie’s matter wave equation () is not an independent axiom but emerges naturally as the spatial displacement executed by a localized particle during a single complete internal chronal rotation. To establish the physical carrier of this wave, we introduce the Rotor-Photon () as a unified electromagnetic-chronal field operator. Finally, we bridge the gap between microscopic quantum reversibility and the macroscopic thermodynamic arrow of time by analyzing the dynamics of synchronization breakdown and phase slippage ().
We derive a direct, non-linear relation between the internal phase time per radian , the macroscopically observable coordinate time , and the relative energy fluctuation . Through rigorous differentiation and integration of these phase relations, we demonstrate that for topologically synchronized systems, the product of energy and observable time constitutes a strict dynamic process invariant: . Moving beyond phenomenological interpretations, we deduce the classical Heisenberg uncertainty relation without invoking independent quantum postulates, showing it to be a special boundary limit governing a single elementary phase cycle () under critical fluctuation. Finally, we bridge the gap between microscopic quantum reversibility and the macroscopic thermodynamic arrow of time by analyzing the dynamics of synchronization breakdown and phase slippage ().
We derive a direct, non-linear relation between the internal phase time per radian , the macroscopically observable coordinate time , and the relative energy fluctuation . Through rigorous differentiation and integration of these phase relations, we demonstrate that for topologically synchronized systems, the product of energy and observable time constitutes a strict dynamic process invariant: . Moving beyond phenomenological interpretations, we deduce the classical Heisenberg uncertainty relation without invoking independent quantum postulates, showing it to be a special boundary limit governing a single elementary phase cycle () under critical fluctuation. Finally, we bridge the gap between microscopic quantum reversibility and the macroscopic thermodynamic arrow of time by analyzing the dynamics of synchronization breakdown and phase slippage ().
We derive a direct relation between the internal phase time , the observable coordinate time , and the relative energy change . Through a rigorous integration of differential relations, we demonstrate that for synchronized systems, the product of energy and time constitutes a process invariant . From this dynamical principle, and without invoking independent postulates, the Heisenberg uncertainty relation is derived as a special limiting case of a single elementary cycle. Furthermore, we analyze the dynamics of synchronization breakdown , which serves as a bridge linking quantum phase evolution to the thermodynamic arrow of time in the macroscopic world.