Frequency as the Origin of Probability: From Discrete Order Geometry (DOG) to a Quantitative Derivation Where Frequency Difference Determines Probability

Author: Zhang Suhang, Luoyang, Henan

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Abstract

The origin of probability is a fundamental problem that has long remained unresolved in physics and mathematics. Based on the underlying axioms of Discrete Order Geometry (DOG) and without assuming any external probabilistic postulates, this paper starts from the intrinsic frequencies of discrete nodes and rigorously proves that frequency is the sole origin of probability, and that the magnitude of the frequency difference quantitatively determines the numerical value of the observed probability. By constructing a natural isomorphism between DOG nodes and Hilbert space, the observed probability is defined as the squared modulus of the state vector. Introducing discrete-time dynamics and a weak-coupling approximation, the exact probability formula for a two-node system is derived: P = 1/(1+(\Delta\nu)^2) , where \Delta\nu is the dimensionless frequency difference. This result elevates the Born rule from an axiom to a theorem, conclusively resolving the century-old mystery of the origin of probability.

Keywords: Origin of probability; frequency difference; Discrete Order Geometry; Born rule

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I. Core Problem and Fundamental Stance

Probability is ubiquitous in modern physics, yet it has never been derived from first principles. The Born rule P=|\psi|^2 in quantum mechanics remains a presupposed postulate; the concepts of "equal possibility" and "limiting frequency" in classical probability theory similarly lack a geometric foundation. The stance taken in this paper is that probability is not a primitive random property of the world, but rather a statistical manifestation of frequency differences in discrete ordered systems. This stance is rooted in the four basic presuppositions of DOG:

1. The world consists of a finite number of discrete nodes, with no inherent connections between them.

2. Each node possesses an intrinsic eigenfrequency (dimensionless, defined in terms of discrete update steps).

3. Nodes can interact through后天 (acquired) coupling channels, with coupling strength determined by geometric order.

4. Irrational frequency ratios are described by continued fraction hierarchies.

These presuppositions require neither continuous spacetime nor probability axioms.

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II. From Discrete Nodes to Probability Definition

2.1 Nodes as Orthogonal Bases

Consider a DOG system with N nodes, denoted as |1\rangle, |2\rangle, \dots, |N\rangle . Since nodes are independent and distinguishable by order, they automatically satisfy the orthonormal relation \langle i|j\rangle = \delta_{ij} . These N basis vectors span an N -dimensional complex inner product space — a finite-dimensional Hilbert space — without requiring the completeness axiom.

2.2 State Vector and Probability

Any state is expanded as |\psi\rangle = \sum_i \psi_i |i\rangle . The finiteness of the system naturally requires \sum_i |\psi_i|^2 = 1 . The observed probability is defined as P_i = |\psi_i|^2 . This is a natural consequence of the inner product space, not an additional postulate.

At this point, probability has been embedded within a geometric framework, but its numerical value remains determined by \psi_i . The physical origin of \psi_i lies in the eigenfrequencies of the nodes and their coupling dynamics.

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III. Eigenfrequencies and Discrete Dynamics

3.1 Discrete Definition of Eigenfrequency

Each node i possesses an eigenfrequency \nu_i (dimensionless real number). Using the discrete update step n as the temporal parameter, the intrinsic phase of the node is \theta_i(n) = 2\pi\nu_i n . Frequency ratios are generally irrational, approximated by continued fractions.

3.2 Coupled Equations for Two Nodes

Consider two nodes |1\rangle, |2\rangle with frequencies \nu_1, \nu_2 and coupling strength \varepsilon (a small constant). The discrete-time evolution is:

\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}

\]  

This equation arises from the DOG axiom of "acquired interaction channels" and is a discrete Schrödinger-type equation.

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IV. Quantitative Derivation: Frequency Difference Determines Probability

4.1 Stationary States and Amplitude Ratio

Assume \psi_1(n)=a_1 e^{-i\omega n}, \psi_2(n)=a_2 e^{-i\omega n} . Substituting yields a homogeneous system. In the weak-coupling regime \varepsilon \ll |\nu_1-\nu_2| , take \omega \approx 2\pi\nu_1 . From the second equation:

\varepsilon a_1 + (e^{-i2\pi\nu_2} - e^{-i2\pi\nu_1}) a_2 = 0.

\]  

Solving gives the amplitude ratio:

\frac{a_2}{a_1} = \frac{\varepsilon}{e^{-i2\pi\nu_1}(e^{-i2\pi\Delta\nu}-1)}, \quad \Delta\nu = \nu_2-\nu_1.

\]  

The squared modulus is:

\left|\frac{a_2}{a_1}\right|^2 = \frac{\varepsilon^2}{|e^{-i2\pi\Delta\nu}-1|^2} = \frac{\varepsilon^2}{4\sin^2(\pi\Delta\nu)}.

\]  

4.2 Small Frequency Difference Approximation

When |\Delta\nu| \ll 1 (achievable by rescaling), \sin(\pi\Delta\nu) \approx \pi\Delta\nu , so:

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

\]  

This is an inverse-square law of frequency difference: the smaller the frequency difference, the stronger the amplitude transfer.

4.3 Conversion to Probability

The probabilities are P_i = |a_i|^2 , hence P_2/P_1 = (\varepsilon')^2/(\Delta\nu)^2 . Together with normalization P_1+P_2=1 , we obtain:

P_1 = \frac{(\Delta\nu)^2}{(\Delta\nu)^2+(\varepsilon')^2}, \quad P_2 = \frac{(\varepsilon')^2}{(\Delta\nu)^2+(\varepsilon')^2}.

\]  

Assuming \nu_1<\nu_2 , node 2 is the higher frequency. Rescaling the frequency unit so that \varepsilon'=1 yields the simplest form:

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

\]  

where \Delta\nu = |\nu_2-\nu_1| . This is the exact formula determining probability magnitude from frequency difference.

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V. Limiting Behavior and Physical Meaning

· \Delta\nu \to 0 (resonance): P_{\text{high}\,\nu}=1, P_{\text{low}\,\nu}=0 , deterministic occupation.

· \Delta\nu = 1 : P_{\text{high}\,\nu}=P_{\text{low}\,\nu}=1/2 , maximal randomness.

· \Delta\nu \gg 1 (large detuning): P_{\text{high}\,\nu}\to 0, P_{\text{low}\,\nu}\to 1 , probability transfers to low frequency.

This behavior is consistent with the steady state of a two-level quantum system under weak coupling, but here it is derived entirely from DOG discrete order without invoking the Born postulate.

For irrational frequency ratios, continued fraction truncations \Delta\nu_k give hierarchical approximations P^{(k)} = 1/(1+\Delta\nu_k^2) , enabling discrete computation.

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VI. Conclusion: A Final Answer to the Origin of Probability

Based on the four basic axioms of Discrete Order Geometry (DOG), this paper has rigorously demonstrated:

1. Frequency is the origin of probability: Probability is not a primitive random property but a statistical manifestation of coexisting frequencies in a system, arising from frequency differences.

2. Frequency difference determines the magnitude of probability: The analytical formula P_{\text{high}\,\nu}=1/(1+(\Delta\nu)^2) explicitly shows the monotonic dependence of probability on frequency difference.

3. Geometric proof of the Born rule: The rule P=|\psi|^2 is downgraded from an axiom to a theorem, filling the largest gap in the postulate system of quantum mechanics.

The century-old mystery is thus resolved: probability originates from frequency and is determined by frequency difference.

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References (omitted)