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

Abstract. Modern physics faces a fundamental dichotomy between quantum randomness and the determinism of its laws. This work postulates that chance is intrinsically bounded and that the stability of a system emerges from the entanglement of the cumulative probability functions of two complementary events. This principle forms the basis of the Law of Probability Equilibrium (LPE). The LPE reveals that energy accumulation, represented by photon frequency, maps onto the domain of probability measured in cycles, thereby establishing a formal equivalence between the energy of a photon (hν) and the probabilistic structure of light. The model postulates a duality between probability and information, where the equilibrium of a system also reflects the amount of information required to maintain its stability. The model introduces the unification constant (A ≈ 0.86), a value directly derived from the mathematical foundations of the theory and aligned with the relativistic velocity limit of the electron in Oganesson (Z = 118). It is demonstrated that the fine-structure constant (α) is the intersection between two universal limits: the probability accumulation limit (γ) and the material limit of the proton number (Z = 118). The manifestation of the LPE at the escape-probability limit of a black hole suggests a deep connection with General Relativity, where the limiting potential (γ) constrains geometry, and the constant A governs the informational stability of that boundary. Taken together, the LPE establishes a theoretical framework that unifies atomic, quantum, and relativistic physics.

Keywords: Probability Equilibrium Law (PEL); bounded randomness; Euler–Mascheroni convergence.