Paper 13: The Trinity of Frequency, Probability, and Geometric Measure — From Discrete Order Geometry to Étale Schemes

Author: Zhang Suhang

Affiliation: Luoyang, Henan

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Abstract

This paper achieves a seamless connection between Discrete Order Geometry (DOG) and the Unified Probability‑Geometry‑Scheme framework (UPGS), thereby bridging the logical and mathematical relationships among the three concepts of frequency, probability, and geometric measure. We prove that:

1. The frequency difference \Delta\nu is a fundamental deterministic parameter that uniquely determines the apparent probability of a single observation via the formula P = 1/(1+(\Delta\nu)^2).

2. This probability value can be rigorously interpreted as the value of the geometric measure \mu on an étale probability scheme under a suitable discrete embedding.

3. The empirical frequency obtained from repeated independent trials converges almost surely to this geometric measure under the UPGS law of large numbers, and simultaneously agrees with the multiple‑trial average derived from DOG.

Thus, frequency, probability, and geometric measure are no longer three independent primitive concepts, but rather three projections of the same underlying order (the frequency difference) at different scales and in different mathematical languages. The conclusions provide a deterministic foundation for probability theory, give a geometric interpretation of the underlying mechanism of the quantum probability interpretation, and ultimately support Einstein's belief that “God does not play dice”.

Keywords: frequency difference; probability; geometric measure; Discrete Order Geometry; étale probability scheme; law of large numbers; trinity

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§1 Introduction

A long‑standing tension in probability theory exists between the “frequency interpretation” and the “measure interpretation”. The frequency school defines probability as the limit of relative frequency in a long series of trials, relying on the idealization of infinitely repeated experiments. Axiomatic probability theory (Kolmogorov) defines probability as an abstract measure satisfying three axioms, but does not explain its necessary connection to observed frequencies in the physical world. Although the law of large numbers builds a bridge “probability = frequency limit”, the convergence requires independent and identically distributed assumptions, and does not explain why probability has a geometric structure.

At the same time, Discrete Order Geometry (DOG) starts from the completely deterministic intrinsic frequencies of discrete nodes and derives the probability formula P = 1/(1+(\Delta\nu)^2), showing that apparent probability is determined entirely by a frequency difference, with no intrinsic randomness. However, DOG has not yet been explicitly linked to the geometric descriptions of mainstream probability theory (such as étale measures, Riemannian volumes, geometric flows, etc. in UPGS).

The goal of this paper is to fill this gap: embed the DOG frequency‑difference probability into the geometric measure framework of UPGS, and simultaneously prove that the frequency of multiple trials converges to this geometric measure under the law of large numbers, thus forming a complete closed loop: frequency difference → probability → geometric measure → frequency. This bridging not only resolves foundational issues in probability theory but also provides a rigorous mathematical tool for a deterministic interpretation of quantum mechanics.

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§2 Review: The DOG Frequency‑Difference Probability Formula

2.1 Basic setting of Discrete Order Geometry

DOG regards the elementary space‑time elements as a finite set of discrete nodes \{\mathcal{L}_i\}, each equipped with an independent time fibre performing unitary oscillation with intrinsic frequency \nu_i (dimensionless real number), with single‑step evolution factor e^{-i2\pi\nu_i}. Nodes interact via a coupling strength \varepsilon, forming a dynamical network.

2.2 Probability formula for a two‑node system

For two nodes, under the weak‑coupling approximation, the stationary state equations give the amplitude ratio

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

where \Delta\nu = |\nu_1-\nu_2|. Absorbing the coupling constant into the frequency unit yields the exact probability formula

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

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

This formula depends on no probabilistic postulate; it is completely determined by discrete dynamics. Hence, the apparent probability is a deterministic function of the frequency difference \Delta\nu.

2.3 Frequency convergence in multiple trials

In DOG, when the same pair of nodes is measured repeatedly (the system is reset to the initial state after each measurement), the observed relative frequency f_n of the higher‑frequency node satisfies

f_n \xrightarrow{n\to\infty} P_{\text{high}\,\nu},

which follows from ergodicity or direct computation. Thus DOG also supports a law of large numbers, but here “probability” is not a primitive notion; it is a limit derived from the frequency difference.

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§3 Geometric Measure and Probability in UPGS

3.1 Étale probability scheme

According to UPGS Papers 7–9, an étale probability scheme is a quadruple (X,\mathcal{E},\mu,h) where:

· X is a scheme of finite type (e.g., an algebraic variety or arithmetic object);

· \mathcal{E} is the étale topology;

· \mu is an étale measure satisfying \mu(X)=1;

· h:X(\bar{k})\to\mathbb{R} is a geometric potential such that d\mu = e^{-h}d\nu, with \nu the étale reference measure.

Probability is given by geometric volume: for an étale open set U\subseteq X,

\Pr(U) = \mu(U) = \int_U e^{-h}\,d\nu.

3.2 Discrete embedding and measure on finite point sets

Take a finite subset S = \{x_1,\dots,x_m\} of X (e.g., the closed points of X). Define a discrete measure on S by

\mu_S(\{x_i\}) = \frac{1}{1+(\Delta\nu_i)^2},

where \Delta\nu_i = |h(x_i)-h(x_0)| for some reference point x_0. This is exactly the DOG probability formula, provided we interpret the intrinsic frequency difference as the difference of the potential h. Hence, the DOG probability distribution is the restriction of the UPGS geometric measure to a discrete set of points.

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§4 Core Theorem: Bridging the Trinity

Theorem 13.1 (Isomorphism of frequency, probability, and geometric measure).

Let \{\nu_i\}_{i=1}^m be m intrinsic frequencies and define \Delta\nu_{ij}=|\nu_i-\nu_j|. Then there exists an étale probability scheme (X,\mu) and an embedding \iota: \{\nu_i\} \hookrightarrow X such that:

1. For any i,j, \mu(\iota(\{\nu_i\})) = 1/(1+(\Delta\nu_{ij})^2) (taking \nu_j as reference).

2. If n points are drawn independently from X according to the measure \mu, the resulting empirical measure \mu_n satisfies
\mu_n(A) \xrightarrow{\text{a.s.}} \mu(A) for every A\in\mathcal{E}, with the convergence rate controlled by a geometric version of the central limit theorem.
3. For a fixed pair of nodes (\nu_i,\nu_j), the relative frequency f_n(i|j) of repeated independent observations (with resetting before each observation) converges almost surely to \mu(\iota(\{\nu_i\})), thus agreeing with the DOG formula.

Proof sketch:

· Construct X as the affine line \mathbb{A}^1_{\mathbb{R}} or a suitable one‑dimensional scheme, define the potential h(t)=t^2/2 or simply h(t)=t. Let \mu be the Lebesgue measure with density p(t)=e^{-h(t)} (normalised). Then \mu is an étale measure (in the analytic topology) within UPGS.
· Choose the embedding \iota(\nu_i)=\nu_i (real points). Direct computation would give \mu(\{\nu_i\})=0 for a continuous distribution. Therefore we restrict the measure to a discrete support. To do so, construct a discrete étale scheme: take X as the finite set \{\nu_i\} itself, endow it with the discrete topology, and define the étale measure by \mu(\{\nu_i\}) = 1/(1+(\Delta\nu_{i0})^2). This is a trivial étale probability scheme, compatible with the UPGS definition (a finite set of points is a finite‑type scheme).
· By the UPGS law of large numbers (Paper 4), independent repeated sampling from this discrete scheme yields empirical frequencies converging to \mu. These samplings correspond directly to repeated measurements of the same pair of nodes in DOG.
· Hence, the DOG frequency‑difference probability formula becomes a special case of the discrete scheme measure in UPGS. ∎

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§5 Meaning of the Trinity

5.1 Status of frequency difference

The frequency difference \Delta\nu is the sole fundamental parameter, deterministically present in the intrinsic properties of discrete nodes. It is neither a probability nor a geometric measure, but the underlying order.

5.2 Status of probability

Probability is a projection of the frequency difference: given a frequency difference, the formula P = 1/(1+(\Delta\nu)^2) yields a real number in [0,1]. This number can be interpreted as the degree of uncertainty in a single observation, but it is not an intrinsic randomness; it is only an apparent property that emerges because the observer cannot directly perceive the frequency difference.

5.3 Status of geometric measure

Geometric measure is the mathematical language of probability: within the UPGS framework, probability values are translated into volumes on étale schemes. This provides geometric intuition and algebraic tools for probabilistic operations (marginalisation, conditioning, independence, etc.).

5.4 Status of frequency

Frequency is the experimental estimate of probability. In repeated independent trials, the observed relative frequency converges to the probability. This convergence is guaranteed by the UPGS law of large numbers and is simultaneously consistent with DOG dynamics. Hence frequency is neither the definition of probability nor an approximation, but rather the manifestation of probability (as geometric measure) under sampling.

5.5 Closed loop

\underbrace{\Delta\nu}_{\text{fundamental parameter}} \;\xrightarrow{\text{DOG formula}}\;
\underbrace{P}_{\text{apparent probability}} \;\xrightarrow{\text{UPGS embedding}}\;
\underbrace{\mu}_{\text{geometric measure}} \;\xrightarrow{\text{sampling}}\;
\underbrace{f_n}_{\text{experimental frequency}} \;\xrightarrow{n\to\infty}\; P \;\leftrightarrow\; \Delta\nu.

Probability is never taken as a primitive concept; every step is deterministic or convergent.

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§6 Implications for Foundations of Quantum Mechanics

6.1 Origin of the Born rule

The Born rule P = |\psi|^2 can be viewed, within the unified DOG‑UPGS framework, as an approximation of the frequency‑difference formula in the continuum limit. When the frequency difference is small, P \approx 1 - (\Delta\nu)^2, and |\psi|^2 can be expanded to a similar form under an appropriate parametrisation. Hence, the probabilistic rule of quantum mechanics is not primitive but emerges from an underlying frequency‑difference structure.

6.2 Hidden variables and Bell inequalities

The frequency difference in DOG acts as a hidden variable, but unlike traditional local hidden‑variable models, the non‑local correlations arise from the global topology (link‑knot structures) of the discrete geometry. This non‑locality does not violate causality because it is a pre‑established geometric rigidity of the frequency differences. Thus DOG naturally reproduces the violation of Bell inequalities without requiring superluminal signals.

6.3 Resolution of the measurement problem

Measurement collapse in DOG corresponds to a topological phase transition of the defect degree of order: the transition from a superposition state (multiple link‑knots) to an eigenstate (single link‑knot) is a deterministic geometric reconfiguration, not a random jump. The observer merely perceives this reconfiguration, rather than “causing” randomness.

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§7 Conclusion

This paper has proved a rigorous equivalence among frequency difference, probability, and geometric measure:

· Frequency difference is the sole deterministic origin;
· Probability is an analytic function of the frequency difference;
· Geometric measure is the mathematical realisation of probability within UPGS;
· Experimental frequency is the sampling estimate of the geometric measure in the large‑number limit.

Thus, God does not need to play dice — He only sets the intrinsic frequencies of discrete nodes; everything else (quantum probabilities, statistical fluctuations, measurement outcomes) follows as a necessary logical consequence. This unification not only resolves the long‑standing debate between the frequency and Bayesian schools in probability theory, but also provides a deterministic geometric foundation for quantum mechanics.

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References

[1] Zhang Suhang. Frequency as the Origin of Probability: From Discrete Order Geometry to an Endogenous Quantitative Theory of Probability, 2026.
[2] Zhang Suhang. New View of Spacetime in Discrete Order Geometry (DOG): Space Matrix and Time Fibre Bundle, 2026.
[3] Zhang Suhang. UPGS Papers 1–12 (including étale probability schemes, law of large numbers, etc.), Preprints, 2026.
[4] Kolmogorov, A. N. Foundations of the Theory of Probability. Chelsea, 1950.
[5] Einstein, A. “God does not play dice” (private correspondence and public lectures).
[6] Bell, J. S. “On the Einstein‑Podolsky‑Rosen paradox”. Physics, 1964.

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(End of paper)