Frequency Difference Determines Probability: Geometric Proof of God Does Not Play Dice from the Perspective of Discrete Order Geometry

Author: Zhang Suhang

Address: Luoyang, Henan, China

Abstract

The intrinsic randomness of probability in quantum mechanics lies at the core of the long-standing debate between Niels Bohr and Albert Einstein. Based on the Discrete Order Geometry (DOG) framework, this paper derives the observational probability formula P = 1/(1+(\Delta\nu)^2) strictly from the eigenfrequencies of discrete nodes, where \Delta\nu denotes the frequency difference between two nodes. The formula demonstrates that probability is not an outcome of God playing dice, but a deterministic projection stemming from discrete frequency discrepancy. Einstein’s conviction that God does not play dice gains rigorous geometric validation, while Bohr’s principle of complementarity and the concept of intrinsic randomness are reduced to ambiguous appearances under continuous approximation. This paper further illustrates quantum measurement collapse and quantum entanglement nonlocality can be uniformly interpreted via order defect degree and topological phase transition within DOG theory. Quantum mechanics is demoted to a low-energy effective theory, bringing about a paradigm-shifting upheaval in fundamental physics.

Keywords: frequency difference; origin of probability; Discrete Order Geometry; God does not play dice; foundations of quantum mechanics

1. Introduction

In 1926, Max Born proposed the probabilistic interpretation of wave function, embedding randomness into the core of quantum mechanics. Albert Einstein opposed this notion throughout his life, putting forward the famous statement: God does not play dice. Niels Bohr responded with the principle of complementarity, arguing that probability is fundamental and irreducible to deeper physical laws. For nearly a century, this profound controversy has remained unresolved on fundamental physical grounds.

Constructed by the author, the Discrete Order Geometry (DOG) framework reconstructs physical space-time with finite discrete nodes and intrinsic eigenfrequencies. Without introducing any axiom of probability, the fundamental observational probability formula is naturally deduced. It explicitly proves that probability value is exclusively governed by frequency difference \Delta\nu, eliminating the existence of intrinsic randomness. This work validates Einstein’s physical intuition and reveals the inherent limitations of Bohr’s fundamental randomness hypothesis. This article elaborates the scientific implications of the probability formula and discusses its revolutionary impacts on the foundations of quantum mechanics.

2. Origin of Probability under DOG Framework

2.1 Discrete Nodes and Eigenfrequencies

DOG reconstructs space-time into a set of finite discrete nodes \{\mathcal{L}_i\}. Each node is attached with an independent time fiber, oscillating unitarily at its inherent dimensionless eigenfrequency \nu_i. The single-step evolutionary form satisfies e^{-i2\pi\nu_i}. No innate connection exists between discrete nodes, and interactive coupling strength \varepsilon is determined by geometric order distribution.

2.2 Derivation of Probability Formula

For a two-node physical system under weak coupling approximation, the discrete evolutionary equations are written as:

\begin{aligned}

\psi_1(n+1) &= e^{-i2\pi\nu_1}\psi_1(n) + \varepsilon \psi_2(n), \\

\psi_2(n+1) &= e^{-i2\pi\nu_2}\psi_2(n) + \varepsilon \psi_1(n).

\end{aligned}

Solving for stationary states yields the amplitude ratio:

\left|\frac{a_2}{a_1}\right|^2 = \frac{\varepsilon^2}{4\sin^2(\pi\Delta\nu)} \approx \frac{(\varepsilon')^2}{(\Delta\nu)^2}

where \varepsilon' = \varepsilon/(2\pi). Absorbing coupling constant into frequency unit normalization, the definitive probability formulas are obtained:

\boxed{P_{\text{high}\nu} = \frac{1}{1+(\Delta\nu)^2}},\qquad

P_{\text{low}\nu} = \frac{(\Delta\nu)^2}{1+(\Delta\nu)^2}

Consequently, probability is no longer an elementary random property, but a deterministic function determined solely by frequency difference \Delta\nu.

3. God Does Not Play Dice: Victory of Einstein

3.1 Deterministic Essence of Probability

Within DOG theory, identical frequency of two nodes \Delta\nu = 0 corresponds to absolute deterministic occupation with P_{\text{high}\nu}=1. When the frequency difference grows sufficiently large, the probability converges to zero. All probabilistic values are uniquely determined by fixed intrinsic parameters including eigenfrequency and coupling strength, with no indeterministic element involved. Physical results are predetermined once fundamental frequency parameters are settled, leaving no room for hypothetical dice-playing behavior of the universe.

3.2 Limitations of Bohr’s Complementarity Principle

Bohr maintained that probability constitutes the ultimate irreducible reality of nature. DOG framework proves probability is merely the projection effect of frequency deviation. Superposition state and probabilistic wave are invalid concepts at discrete fundamental scale. The principle of complementarity claiming particle property and wave property cannot be fully determined simultaneously is essentially an illusory phenomenon caused by missing fine frequency information in continuous approximation. Wave-particle duality vanishes at discrete node scale, leaving only intrinsic frequency and interactive coupling between physical nodes.

3.3 Conclusion of the Centennial Debate

The core divergence between Einstein and Bohr focuses on whether randomness is a fundamental attribute of the universe. DOG framework draws a definite conclusion: randomness is not elementary, while frequency difference serves as the fundamental physical quantity. The probability formula is derived purely from discrete dynamic equations without any probabilistic postulation. Einstein’s physical intuition stands valid, and Bohr’s fundamental randomness theory is proven to be a cognitive blind spot of conventional physical frameworks. The century-long academic dispute hereby reaches a definitive settlement.

4. Impacts on Traditional Quantum Mechanics

4.1 Geometric Interpretation of Quantum Collapse

Measurement process is defined as topological phase transition where order defect degree d declines to zero. Quantum superposition corresponds to multi-loop recursive evolutionary paths. Physical measurement compels the system to converge into a single stable path with zero defect degree. Wave function collapse is abandoned, replaced by inherent rearrangement of geometric topological order.

4.2 Topological Origin of Entanglement Nonlocality

Entangled quantum states correspond to independent components of topological links. Global topological invariants such as linking number govern entanglement entropy, which cannot be decomposed into local individual component quantities, naturally explaining the violation of Bell inequalities. Nonlocality does not imply superluminal information transmission, but reflects the rigid connectivity inherent to topological structure.

4.3 Quantum Mechanics Regarded as Effective Theory

Quantum mechanics emerges as macroscopic approximation of discrete fundamental geometry under continuous limit. Analogous to thermodynamics being statistical approximation of molecular kinematics, quantum mechanics is categorized as a low-energy effective theory derived from deeper discrete geometric laws. It will no longer be recognized as ultimate fundamental physical theory. The whole physics community faces an academic upheaval, which reclassifies the position of quantum mechanics in hierarchical physical systems rather than completely overturning existing quantum achievements.

5. Conclusions

1. Probability is uniquely determined by frequency difference following P = 1/(1+(\Delta\nu)^2), and intrinsic randomness does not exist in nature.
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2. Einstein’s viewpoint is verified: God does not play dice, and probability originates from projection of natural order discrepancy.
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3. Bohr’s complementarity principle requires re-evaluation; wave-particle duality is an approximate illusion under continuous space-time description.
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4. Quantum mechanics is demoted to effective theory, and its mysterious characteristics stem from incomplete cognition of discrete fundamental physical structure.

The rigorous mathematical derivation based on Discrete Order Geometry verifies that the universe operates in stable and inherent order rather than chaotic random fluctuation. The natural world follows definite physical laws with solid and consistent fundamental rules.

References

[1] Zhang S H. Frequency as Origin of Probability: From Discrete Order Geometry to Endogenous Quantitative Probability Theory, 2026.
[2] Zhang S H. New Space-time View in Discrete Order Geometry: Spatial Matrix and Temporal Fiber Bundle, 2026.
[3] Zhang S H. Low-energy Approximation Consistently Compliant with Core MIE Paradigm under DOG System, 2026.
[4] Einstein A. God Does Not Play Dice, Private Correspondence.
[5] Bohr N. Atomic Physics and Human Knowledge, The Commercial Press.

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