Frequency, Probability and Geometric Measure: A Trinity

Author: Zhang Suhang
Affiliation: Luoyang, Henan

Abstract

This paper establishes the connection between Discrete Order Geometry and flat schemes, thereby bridging the logical and mathematical relationships among three concepts: frequency, probability and geometric measure. We demonstrate the following results:

1. The frequency difference Δν serves as the underlying deterministic parameter, uniquely determining the apparent probability of a single observation via the formula P = 1/(1+(\Delta\nu)^2).
2. This probability value can be interpreted as the value of the geometric measure μ on a flat probability scheme under a certain discrete embedding.
3. The empirical frequency obtained from repeated independent trials converges almost surely to this geometric measure under the Law of Large Numbers, consistent with the repeated average derived from the formula.

As a result, frequency, probability and geometric measure are no longer three independent fundamental concepts. Instead, they represent three projections of the same underlying order (frequency difference) across different scales and mathematical formulations. The conclusions of this paper furnish probability theory with a deterministic foundation, offer a geometric interpretation for the underlying mechanism of the probabilistic interpretation of quantum mechanics, and ultimately support Einstein’s conviction that “God does not play dice.”

Keywords: frequency difference; probability; geometric measure; Discrete Order Geometry; flat probability scheme; Law of Large Numbers; trinity

 

§1 Introduction

A long-standing tension exists in probability theory between the frequency interpretation and the measure interpretation. Frequentists define probability as the limit of long-run relative frequencies, relying on the idealisation of infinitely repeated trials. Axiomatic probability theory (Kolmogorov) defines probability as an abstract measure satisfying three axioms, without explaining any necessary connection to observed frequencies in the physical world. While the Law of Large Numbers builds a bridge equating probability to the limit of frequency, its convergence requires the assumption of independent and identically distributed trials and does not explain why probability possesses geometric structure.

Meanwhile, starting from deterministic discrete nodal eigenfrequencies, Discrete Order Geometry derives the probability formula P = 1/(1+(\Delta\nu)^2). It shows that apparent probability is uniquely determined by frequency difference, with no intrinsic randomness. Nevertheless, this framework has not yet established an explicit link to geometric descriptions of probability, such as flat measures, Riemannian volumes and geometric flows.

The objective of this paper is to fill this gap. We embed frequency-difference probability within the geometric measure framework of flat schemes, and prove that the frequency obtained from repeated trials converges to this geometric measure in the large-number sense. This yields a complete closed loop: frequency difference → probability → geometric measure → frequency. This unification not only resolves foundational interpretive problems in probability theory, but also supplies rigorous mathematical tools for deterministic interpretations of quantum mechanics.

 

§2 Review: The Probability Formula from Frequency Difference

2.1 Basic setup of Discrete Order Geometry

This framework regards fundamental spacetime units as finite discrete nodes \{L_i\}. Each node carries an independent time fibre and undergoes unitary oscillation with an eigenfrequency \nu_i (dimensionless real number). The one-step evolution factor is e^{-i2\pi\nu_i}. Nodes interact with one another via coupling strength ε, forming a dynamical network.

2.2 Probability formula for a two-node system

Consider two nodes. Under the weak-coupling approximation, the stationary-state equation yields the amplitude ratio:

|a_2/a_1|^2 = \varepsilon^2 / 4\sin^2(\pi\Delta\nu),

where \Delta\nu = |\nu_1 - \nu_2|. Absorbing the coupling constant into the frequency unit, we obtain the exact probability formulas:

P_{\text{high}\nu} = 1 / \big(1 + (\Delta\nu)^2\big), \quad P_{\text{low}\nu} = (\Delta\nu)^2 / \big(1 + (\Delta\nu)^2\big).

This formula does not rely on any probability axioms; it is fully determined by discrete dynamics. Apparent probability is therefore a deterministic function of the frequency difference Δν.

2.3 Frequency convergence for repeated trials

Repeated independent measurements are performed on the same pair of nodes (the system is reset to its initial state after each measurement). The relative frequency f_n of observing the high-frequency node satisfies:

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

This result can be obtained via ergodic arguments or direct computation. The framework is therefore compatible with the Law of Large Numbers, yet “probability” is not a primitive concept here; it is a limiting value derived from frequency difference.

 

§3 Geometric Measure and Probability in Flat Schemes

3.1 Flat probability schemes

A flat probability scheme is a quadruple (X, E, \mu, h):

- X: a finite-type scheme (can be viewed as an algebraic variety or arithmetic object);
- E: the flat topology;
- \mu: a flat measure satisfying \mu(X)=1;
- h: X(\bar{k}) \to \mathbb{R}: a geometric potential function such that d\mu = e^{-h} d\nu, where \nu denotes the reference flat measure.

Probability is given by geometric volume. For a flat open subset U \subseteq X:

\mathrm{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\}\subset X (for example, closed points of X). Define a discrete measure on S:

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

where \Delta\nu_i = |h(x_i)-h(x_0)|, and x_0 is a reference point. This recovers the form of the probability formula above, provided eigenfrequency difference is interpreted as the difference of the potential function h. The frequency-difference probability distribution is thus the restriction of a flat geometric measure to a discrete set of points.

 

§4 Core Theorem: The Unification of Frequency, Probability and Geometric Measure

Theorem 13.1 (Isomorphism of Frequency, Probability and Geometric Measure)

Let \{\nu_i\}_{i=1}^m be m eigenfrequencies, and define \Delta\nu_{ij}=|\nu_i-\nu_j|. There exists a flat 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) (with \nu_j taken as reference).
2. If n points are sampled independently from X according to measure \mu, the resulting empirical measure \mu_n satisfies

\mu_n(A) \to \mu(A) \text{ almost surely},\quad \forall A\in E,

and the rate of convergence is governed by a geometric version of the Central Limit Theorem.
3. For a fixed pair of nodes (\nu_i,\nu_j), the frequency f_n(i|j) from repeated independent observations (reset before each measurement) converges almost surely to \mu(\iota(\{\nu_i\})), consistent with the formula.

Proof Sketch:

- Construct X as the affine line \mathbb{A}^1_\mathbb{R} or a suitable one-dimensional scheme, and define the potential function h(t)=t^2/2 or simply h(t)=t. Let \mu be the normalised Lebesgue measure with density p(t)=e^{-h(t)}. Then \mu is a flat measure (in the analytic topology).
- Choose the embedding \iota(\nu_i)=\nu_i (real points). Direct calculation yields \mu(\{\nu_i\})=0 for a continuous distribution. We therefore restrict the measure to a discrete support. To achieve this, construct a discrete flat scheme: take X to be the finite set \{\nu_i\} itself, equipped with the discrete topology, and define the flat measure as \mu(\{\nu_i\}) = 1/(1+(\Delta\nu_{i0})^2). This is a trivial flat probability scheme compatible with the above definition (since discrete schemes are finite-type schemes).
- By the Law of Large Numbers, repeated independent sampling from this discrete scheme yields empirical frequency converging to \mu. These samples directly correspond to repeated measurements of the same pair of nodes.
- The frequency-difference probability formula is therefore a special case of flat scheme measure. ∎

 

§5 Implications of the Trinity

5.1 The status of frequency difference

Frequency difference Δν is the sole fundamental parameter, existing deterministically as an intrinsic property of discrete nodes. It is neither probability nor geometric measure; it is the underlying order.

5.2 The status of probability

Probability is a projection of frequency difference. Given a frequency difference, the formula P = 1/(1+(\Delta\nu)^2) produces a real number in [0,1]. This number describes the apparent uncertainty of a single observation. It does not represent intrinsic randomness; it is merely an apparent property arising because observers cannot directly perceive frequency difference.

5.3 The status of geometric measure

Geometric measure is the mathematical language of probability. Within the flat scheme framework, probability values are translated into volumes on flat schemes. This supplies geometric intuition and algebraic tools for probability operations such as marginalisation, conditioning and independence.

5.4 The status of frequency

Frequency is the experimental estimator of probability. In repeated independent trials, the observed relative frequency converges to probability. This convergence is guaranteed by the Law of Large Numbers and is consistent with discrete dynamics. Frequency is therefore neither the definition of probability nor merely an approximation; it is the manifestation of probability (as geometric measure) under sampling.

5.5 Closed loop

Δν (underlying parameter) → probability formula → P (apparent probability) → flat embedding → μ (geometric measure) → sampling → f_n (experimental frequency) → n\to\infty → P \leftrightarrow \Delta\nu.

Probability is not treated as a primitive concept at any stage; all steps are deterministic or convergent.

 

§6 Implications for the Foundations of Quantum Mechanics

6.1 Origin of the Born rule

The Born rule P=|\psi|^2 can be regarded in this unified framework as an approximation of the frequency-difference formula in the continuous limit. For small frequency differences, P \approx 1 - (\Delta\nu)^2, and |\psi|^2 admits an expansion of similar form under an appropriate parameterisation. The probabilistic rule of quantum mechanics is therefore not primitive, but emergent from the underlying frequency-difference structure.

6.2 Hidden variables and Bell’s inequality

Frequency difference constitutes a hidden variable. Unlike conventional local hidden-variable models, frequency difference generates nonlocal correlations via global topological connections (link structures) in discrete geometry. This nonlocality does not violate causality, as it arises from the geometric rigidity prearranged by frequency differences. The framework can naturally accommodate violations of Bell’s inequality without invoking superluminal signals.

6.3 Resolution of the measurement problem

Wavefunction collapse under measurement corresponds to a topological phase transition in the degree of order defect: the transition from a superposition state (multiple link loops) to an eigenstate (single link loop) is a deterministic geometric rearrangement. The observer only perceives this rearrangement and does not “cause” randomness.

 

§7 Conclusion

This paper demonstrates the strict equivalence among frequency difference, probability and geometric measure:

- Frequency difference is the unique deterministic root;
- Probability is an analytic function of frequency difference;
- Geometric measure is the mathematical realisation of probability within flat schemes;
- Experimental frequency is the sampling estimate of geometric measure in the large-number limit.

God therefore need not play dice. He only sets the eigenfrequencies of discrete nodes; everything else, including quantum probabilities, statistical fluctuations and measurement outcomes, follows as necessary logical consequences. This unification not only settles the debate between frequentist and Bayesian interpretations in probability theory, but also supplies a deterministic underlying geometric model for quantum mechanics.

References

Omitted