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The Origin of the Natural Constant e in Spacetime Geometry within the DOG Discrete Order Geometry Framework

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

The natural constant e is a fundamental constant underlying mathematical analysis, dynamical evolution, steady-state convergence, and probability and statistics. Since its discovery by Bernoulli and Euler, academia has only grasped its limit definition, derivative properties, and exponential evolution laws, but has never discovered the true physical and geometric origin of e.

Based on DOG discrete order geometry, the ECS symmetric equilibrium steady-state principle, and the MIE minimal intrinsic action principle, this paper reveals for the first time the cosmic origin of the natural constant e:

e is the unique optimal convergence baseline constant of the system when discrete spatial matrix coupling propagates layer by layer, undergoes infinite densification, and approaches the continuous steady state.

e is not a product of algebraic limits, but rather a necessary spacetime evolution baseline for the discrete universe transitioning toward a continuous steady state. This achievement represents a paradigm-level innovation with no precedent in prior literature.

Keywords: DOG discrete order geometry; spatial matrix; origin of e; discrete-continuous transition; MIE steady-state convergence; order evolution

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I. Introduction

Traditional human understanding of e has only three layers of appearance:

1. Compound interest limit definition: \lim_{n\to\infty}(1+\frac{1}{n})^n

2. Derivative self-consistency: \frac{d}{dx}e^x = e^x

3. Applicative fitting: A fitting constant for decay, growth, and probability distributions

All of these remain at the level of mathematical phenomena and applications, never answering the core question:

Why must all steady-state evolution, damped convergence, and order diffusion in the universe obey the base e?

This paper provides the ultimate explanation of origin from the underlying discrete spacetime structure.

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II. Prerequisite Underlying Structure of DOG

This section briefly lists the core presuppositions of the DOG system on which this paper's derivation relies:

1. Spatial essence: Finite discrete lattice coupling matrix, without preset coordinates;

2. Order propagation: Layer-by-layer order coupling transmission exists between lattice points, with adjacency matrix elements A_{ij} characterizing the coupling strength;

3. MIE constraint: The system is always governed by the minimal intrinsic action principle, automatically tending toward the extremal state;

4. ECS constraint: The steady state satisfies global symmetric equilibrium, with no abrupt changes, no divergence, and no bias.

These presuppositions have been fully demonstrated in previous papers and are directly cited here.

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III. Original Geometric Ontological Definition of the Natural Constant e

3.1 Core Conclusion

Within the DOG system, this paper proposes for the first time:

e originates from the continuization convergence mechanism of discrete coupling in the DOG spatial matrix, and is the unique optimal evolution base for the layer-by-layer penetration, infinite refinement, and approach to the smooth steady state of discrete order.

3.2 Layered Explanation

Layer 1: The discrete stage

The coupling strength of the spatial matrix propagates layer by layer and accumulates step by step. Each lattice point's order influences its neighbors, which in turn influence their next neighbors, forming a discrete progressive structure. This process naturally exhibits a "compound interest"-style propagation morphology.

Layer 2: Infinite densification approaching continuity

When the discrete coupling hierarchy undergoes infinite densification and the spatial step size is infinitely refined, the discrete progressive structure approaches a continuous smooth state. At this point, the discrete "layer-by-layer propagation" transforms into continuous "exponential evolution."

Layer 3: MIE constraint forces optimal convergence

Under the minimal intrinsic action constraint, the system faces countless possible convergence paths but must select the unique convergence baseline that is free of oscillation, free of divergence, most (smooth), and most balanced. This unique baseline is e.

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IV. Why Must the Universe Have e, and No Other Base?

This paper rigorously provides the geometric reasons:

Base Choice Evolutionary Behavior Principle Violated

Base > e Oscillation, overshoot, instability Violates ECS steady-state symmetry

Base < e Convergence redundancy, energy waste Violates MIE minimal action

Base = e Simplest, most stable, no oscillation, minimal action Perfectly self-consistent

Therefore:

e is the unique solution for discrete spacetime transitioning to the continuous steady state. It is not a mathematical coincidence but a geometric necessity.

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V. Unified Physical Correspondence

1. Field theory decay evolution

Low-energy damping, amplitude convergence, and propagator attenuation in gauge fields are essentially exponential evolutions of e arising from the continuization of spatial matrix coupling. The exponential decay factor e^{-Γt} in quantum field theory is precisely the low-energy projection of DOG spatial order continuization.

2. Probability steady-state distribution

Statistical equilibrium, observed steady states, and entropy convergence all obey the optimal convergence of the base e. The Boltzmann distribution P \propto e^{-E/kT} and the exponential family distributions in the principle of maximum entropy both originate from this.

3. Macroscopic natural evolution

All growth (populations, capital), decay (radioactive decay), diffusion (heat conduction), and relaxation (viscoelasticity) processes are secondary manifestations of DOG spatial order continuization. The common mathematical form of these processes, \frac{dX}{dt} = kX (or -kX), has the solution X \propto e^{kt} (or e^{-kt}), the root of which lies here.

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VI. Rigorous Boundary Delimitation

1. e is a low-energy steady-state emergent quantity

In the high-energy, strongly discrete regime with (violent) topological fluctuations, the discrete coupling has not yet smoothed out, and no strict convergence law of e exists. e is precisely valid only in the low-energy continuous approximation.

2. Complementary positioning of e and π

Constant Origin Structure Function Type
π Closed-loop rotation of temporal fiber bundle Period, phase, symmetry Static structural constant
e Continuization of spatial matrix coupling Evolution, convergence, diffusion Dynamic evolution constant

The two form a complementary pair: one static, one dynamic; one (periodic), one(convergent); one symmetric, one convergent. Together they constitute a complete explanation of the two fundamental constants of DOG spacetime.

3. Full satisfaction of dual-principle self-consistency

· MIE: Guarantees minimal action and the simplest evolution path;
· ECS: Guarantees balanced stability and unbiased convergence.

e is the unique solution under the joint constraints of both principles.

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VII. Core Innovations (First in History)

1. Upgrades e from a "mathematical analysis constant" to a spacetime order convergence constant;
2. Anchors the origin of e in the discrete spatial matrix coupling mechanism rather than in limit definitions;
3. Explains the geometric reason why e is the unique optimal solution for steady-state evolution in the universe (greater leads to oscillation, less leads to redundancy);
4. Achieves the physical ontological tracing of e, completely (liberating it from) purely mathematical hypotheses.

These four points have no precedent in prior literature.

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VIII. Conclusion

1. The natural constant e is neither a human mathematical construction nor a product of algebraic limits, but rather a self-emergent baseline constant arising from the continuization steady-state evolution of DOG discrete space;
2. The sole geometric origin of e is the optimal convergence mechanism of layer-by-layer propagation of discrete spatial matrix coupling;
3. All exponential evolution, decay relaxation, steady-state approximation, and probability equilibrium are macroscopic representations of this underlying geometric structure;
4. This paper completely fills the two-thousand-year theoretical gap in which the natural constant e had "only properties, no origin."

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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 MIE Minimal Intrinsic Action Principle and the ECS Symmetric Equilibrium Steady-State Principle. 2026.
[3] Zhang Suhang. Low-Energy Approximation in the DOG System Simultaneously Conforms to the MIE Extremum Criterion and the ECS Symmetric Equilibrium Steady State. 2026.