Title: The Geometric Positioning of the Probability Formula P = 1/(1+(\Delta\nu)^2) in DOG Spacetime

Author: Zhang Suhang

Address: Luoyang, Henan

---

Abstract

Within the framework of Discrete Order Geometry (DOG), the observed probability formula P = 1/(1+(\Delta\nu)^2) is not a mere algebraic construct but has a clear geometric ontological origin. This paper explicitly demonstrates that the constant term “1” in the formula originates from the order coupling strength between nodes encoded in the spatial matrix, while the frequency-difference term (\Delta\nu)^2 arises from the difference in fiber oscillation frequencies of the temporal fiber bundle. Probability, as an observational outcome, is the statistical projection resulting from the combined action of the spatial matrix and the temporal fiber bundle. This interpretation elevates the probability formula from a formal expression to a direct corollary of the geometric structure of DOG spacetime.

---

I. Problem Statement

In the paper “Frequency as the Origin of Probability,” starting from DOG discrete dynamics, we derived the observed probability formula for a two-node system:

P = \frac{1}{1+(\Delta\nu)^2}
\]

where \Delta\nu = |\nu_1 - \nu_2| is the dimensionless intrinsic frequency difference, and the constant “1” in the denominator results from absorbing the coupling constant into the unit of frequency. Readers may ask: what geometric realities in DOG spacetime correspond to the two parts of this formula — the “1” and the (\Delta\nu)^2 ? This paper provides a focused answer to this question.

---

II. The Spatial Matrix Determines the “1”

According to the DOG view of spacetime (Zhang Suhang, Spatial Matrix and Temporal Fiber Bundle), space is not a continuous manifold but is constituted by a finite set of discrete lattice points and their order coupling matrix:

· Lattice point set \{\mathcal{L}_i\} : discrete, finite, without preset coordinates.
· Adjacency matrix A : A_{ij} represents the order coupling strength between lattice points i and j .

For a two-node system, the coupling matrix is:

A = \begin{pmatrix}
0 & \varepsilon \\
\varepsilon & 0
\end{pmatrix}
\]

where \varepsilon is a small constant determined by geometric topology. In the probability formula P = 1/(1+(\Delta\nu)^2) , the “1” in the denominator is precisely the manifestation of this coupling strength \varepsilon after being absorbed into the frequency unit.

Geometric meaning: Without spatial coupling ( \varepsilon = 0 ), there is no probability transfer, and the system is completely deterministic. The magnitude of the coupling strength sets the reference scale in the probability formula.

---

III. The Temporal Fiber Bundle Determines (\Delta\nu)^2

In the same spacetime view, time is defined as the unitary oscillation of fibers:

· Each lattice point \mathcal{L}_i is attached with a fiber space \mathcal{F}_i .
· The fiber state undergoes unitary rotation with discrete step n , termed “oscillation.”
· The oscillation rate is defined as the eigenfrequency \nu_i of that node, with the single-step evolution operator given by e^{-i2\pi\nu_i} .

When two nodes have different oscillation frequencies, their difference \Delta\nu = |\nu_1 - \nu_2| directly enters the probability formula, appearing squared in the denominator.

Geometric meaning: The larger the frequency difference, the more asynchronous the oscillation rhythms of the two fibers, and the smaller the probability of amplitude transfer from the high-frequency node to the low-frequency node (i.e., the smaller the probability of observing the high-frequency node).

---

IV. Unified Geometric Positioning of the Probability Formula

In summary, the probability formula

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

can be rewritten as:

P_{\text{high}\,\nu} = \frac{\varepsilon'^2}{\varepsilon'^2 + (\Delta\nu)^2}, \quad \varepsilon' = \frac{\varepsilon}{2\pi}
\]

· Numerator \varepsilon'^2 : coupling strength from the spatial matrix (squared).
· First term in denominator \varepsilon'^2 : the same coupling strength, serving as the normalization reference.
· Second term in denominator (\Delta\nu)^2 : from the squared frequency difference of the temporal fiber bundle.

Probability is thus the outcome of competition between “spatial coupling strength” and “temporal frequency difference”:

· Strong spatial coupling (large \varepsilon' ) and small temporal frequency difference (small \Delta\nu ) → probability approaches 1.
· Weak spatial coupling (small \varepsilon' ) and large temporal frequency difference (large \Delta\nu ) → probability approaches 0.

When \varepsilon' is chosen as the frequency unit, \varepsilon' = 1 , the formula simplifies to P = 1/(1+(\Delta\nu)^2) .

---

V. Conclusion

This paper explicitly answers the question: “What does this small formula correspond to in DOG spacetime?”

Term in Formula Geometric Origin in DOG
Constant “1” (i.e., \varepsilon'^2 ) Order coupling strength of the spatial matrix
(\Delta\nu)^2 Oscillation frequency difference of the temporal fiber bundle
Fractional structure \frac{1}{1+(\Delta\nu)^2} Observational projection from competition between spatial coupling and temporal rhythm difference

Therefore, the probability formula is not an extraneous mathematical assumption but a natural output of the geometric structure of DOG spacetime. Readers may regard this paper as a companion geometric commentary to “Frequency as the Origin of Probability.”

---

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. A New View of Spacetime in Discrete Order Geometry (DOG): Spatial Matrix and Temporal Fiber Bundle. 2026.