Deduction of the Fine-Structure Constant and Quantum–Atomic–Relativistic Unification through the Constant A= 0.86

Abstract

Modern physics faces an unresolved structural tension between quantum randomness and deterministic laws. In this work, we introduce the Principle of Equilibrium (PE) as a fundamental law establishing that probability is not merely a statistical tool, but a generative magnitude that regulates the stability of physical systems. By normalizing the system to unity, the PE reveals a fundamental identity between energy and structural probability, where space and time act as iterative variables of interaction. The PE postulates that no probabilistic event can remain indefinitely unrealized or occur without limit, imposing bounded occurrence as a universal physical constraint. From this principle, universal stability thresholds emerge, defined by γ ≈ 0.5772 and the saturation limit (1-1/e). From first principles, the PE allows the non-empirical deduction of fundamental constants. The fine-structure constant (α) necessarily emerges from the intersection of the probabilistic limit γ and the atomic structural limit (Z = 118), while the strong coupling constant (α_s) arises from the nuclear saturation condition (n = 8). A Structural Consistency Criterion (SCC) is introduced, demonstrating that gauge unification is not a geometric coincidence but a necessary consequence of probabilistic stability, uniquely selecting the structure of the MSSM. The PE redefines quantum superposition as probabilistic symmetry and explains state reduction as an endogenous phenomenon where matter and information emerge as stable equilibrium states. Finally, a deep connection with general relativity is established, where γ appears as a geometric limit in the photon sphere of a black hole. The PE thus constitutes a unifying framework in which fundamental constants and physical laws emerge as necessary consequences of a universal principle of probabilistic equilibrium. Wave function collapse occurs naturally upon reaching an equilibrium threshold, without the need for observer intervention, providing an objective and physically grounded interpretation of the phenomenon.

Keywords: Probability Equilibrium (PE); bounded randomness; Euler–Mascheroni convergence; probability field