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The Dual Constant Dynamic-Static Unification Principle in the DOG System

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

Within the complete framework of DOG discrete order geometry, the universe's two most fundamental mathematical constants, π and e, are no longer isolated mathematical tools but rather endogenous order representations corresponding to two underlying cosmic laws.

Through systematic integration, this paper establishes a rigorous dual relationship:

· π corresponds to the ECS symmetric equilibrium steady-state principle, representing the static structural order of the universe;

· e corresponds to the MIE minimal intrinsic action principle, representing the dynamic evolutionary order of the universe.

One static, one dynamic; one form, one tendency; one structure, one evolution. The two constants and two laws mutually align and achieve complete self-consistency, jointly constituting the ultimate foundational cornerstone of all physical laws, field structures, statistical probabilities, and steady-state evolutions in DOG spacetime.

Keywords: DOG discrete order geometry; ECS symmetric equilibrium; MIE minimal intrinsic action; π; e; dynamic-static duality; origin of order

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I. Introduction

In traditional mathematical and physical systems, π and e have long existed in a state of "having properties but no origin, no归属 (systematic affiliation), and no unified logic." Academia has only been able to grasp their limit definitions, series expansions, functional applications, and numerical laws, but has never been able to answer two fundamental questions:

1. Why must the universe necessarily possess these two fixed baseline constants?

2. Why must all natural motion, symmetric structures, and evolutionary convergence obey the mathematical rules of these two constants?

Previous series of DOG specialized papers have respectively accomplished:

· Construction of DOG spacetime structure

· Unification of the four forces

· Native derivation of Yang-Mills equations

· Geometric positioning of probability formulas

· Tracing the origin of π

· Tracing the origin of e

This paper does not重复 (repeat) those derivations, but rather performs top-level convergence of the system, establishing the dynamic-static general outline of the entire theory, and completing the ultimate unified closed loop of the underlying logic of mathematics and physics.

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II. Static System: π — ECS Symmetric Equilibrium Stability

The core meaning of ECS is: the system spontaneously tends toward a symmetric, balanced, regular, unbiased, and stable structure. It is the underlying constraint law of the universe's structural form.

The entire geometric origin of π fully conforms to the static order of ECS:

1. π originates from the closed-loop uniform symmetric partitioning of the temporal fiber bundle;

2. Closed-loop, periodicity, phase, and rotational isotropy are geometric expressions of极致 (extreme) ECS symmetry;

3. The regular structure of steady-state spacetime, the phase基底 (foundation) of gauge fields, and the symmetric configuration of spacetime are all scaled by π.

Therefore, a unique correspondence can be established:

π is the mathematical incarnation of the ECS symmetric steady-state structure.

It is not responsible for motion, evolution, or energy convergence. It is only responsible for shaping spacetime structure into a symmetric, balanced, and stable standard form.

It is a static universal constant that "定形、定序、定结构" (determines form, determines order, determines structure).

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III. Dynamic System: e — MIE Minimal Intrinsic Low-Energy Convergence

The core meaning of MIE is: system evolution always tends toward minimal intrinsic action, highest efficiency, smoothest return, and simplest path of motion. It is the underlying driving law of the universe's dynamical evolution.

The entire evolutionary origin of e fully conforms to the dynamic order of MIE:

1. e originates from the optimal convergence process of continuization of discrete coupling in the spatial matrix;

2. All decay, diffusion, relaxation, steady-state回归 (return), probability equilibrium, and low-energy field evolution are natural consequences of the base e;

3. Among all evolutionary bases, only e can achieve oscillation-free, redundancy-free, minimal-action, and highest-efficiency system stabilization.

Therefore, a unique correspondence can be established:

e is the mathematical incarnation of the MIE low-energy optimal evolution.

It is not responsible for structural symmetry, spatial configuration, or static equilibrium. It is only responsible for allowing all systems to naturally return to the low-energy steady state and complete the simplest evolution.

It is a dynamic universal constant that "顺势、归稳、趋极小" (follows tendency, returns to stability, approaches the minimum).

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IV. Complete Dynamic-Static Dual Architecture of the DOG Universe

Through the systematic integration of this paper, the underlying order of DOG forms a complete dichotomy, perfect symmetry, and gap-free closed loop:

Static Dimension (Structural Layer)

· Law: ECS Symmetric Equilibrium Stability

· Constant: π

· Function: Constructs spacetime symmetric structure, orders gauge phase, establishes steady-state geometric form

Dynamic Dimension (Evolutionary Layer)

· Law: MIE Minimal Intrinsic Action

· Constant: e

· Function: Drives system optimal evolution, realizes low-energy convergence, maintains natural relaxation equilibrium

Core Dual Conclusion

ECS–π governs the static, determining the form of the universe;

MIE–e governs the dynamic, enacting the tendency of the universe.

All physical phenomena in the universe arise from the coupled superposition of:

Static symmetric structure + Dynamic optimal evolution.

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V. Unified Confirmation in Field Theory and Statistical Systems

1. Gauge field structure

Phase rotation relies on π (ECS symmetry); field quantity evolution relies on e (MIE convergence). The complete form of the Yang-Mills gauge field is precisely the combined expression of the two dual constants.

2. Probabilistic observation system

Spatial coupling steady state is guaranteed by symmetric structure; frequency difference evolution is constrained by convergence rules. The probabilistic steady-state distribution simultaneously carries the structural order of π and the evolutionary order of e.

3. Macroscopic natural laws

All periodic structures归于 (return to) π; all growth and decay归于 (return to) e. All laws of the natural world fall within the DOG dynamic-static dual framework.

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VI. Summary of Innovations

1. For the first time in history, upgrades π and e from purely mathematical constants to dual representations of the universe's underlying order;

2. For the first time in history, accomplishes a strict binding of the ECS–π static system and the MIE–e dynamic system;

3. Completely打通 (opens up) the global unification of geometric structure, physical laws, and fundamental mathematical constants;
4. Ends the long-standing academic gap in which π and e had "properties but no origin or systematic affiliation."

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VII. Conclusion

Within the unified framework of DOG discrete order geometry:

· π is the structural constant of the universe's symmetric steady state, corresponding to the ECS law;
· e is the convergence constant of the universe's low-energy evolution, corresponding to the MIE law.

One static and one dynamic構築 (construct) the foundation of spacetime; one form and one tendency govern all laws of the universe.

At this point, the origin of mathematical fundamentals, the foundation of physical laws, and the underlying structure of spacetime have achieved global self-consistency and a complete closed loop.

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References

[1] Zhang Suhang. A New View of Spacetime in Discrete Order Geometry (DOG): Spatial Matrix and Temporal Fiber Bundle. 2026.
[2] Zhang Suhang. The Origin of π in Spacetime Geometry within the DOG Discrete Order Geometry Framework. 2026.
[3] Zhang Suhang. The Origin of e in Spacetime Geometry within the DOG Discrete Order Geometry Framework. 2026.
[4] Zhang Suhang. The ECS Symmetric Equilibrium Stability Principle and the MIE Minimal Intrinsic Action Principle. 2026.

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