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The Origin of π in Spacetime Geometry within the DOG Discrete Order Geometry Framework

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

Since ancient times, humanity has understood the circumference ratio π only through mathematical representations such as plane geometry, limit approximations, periodic functions, and complex analysis, always treating it as a purely mathematical ratio constant, never finding its physical spacetime origin.

Based on DOG discrete order geometry, the ECS symmetric equilibrium principle, and the MIE minimal intrinsic action principle, this paper proves for the first time that π is not a man-made mathematical constant, but rather a closed-loop symmetric order constant that spontaneously emerges in the low-energy steady state of discrete spacetime.

The sole ontological origin of π is the global angular uniform partitioning symmetry of the local closed-loop rotation of the temporal fiber bundle.

This interpretation has no precedent in prior literature and represents a paradigm-level溯源 (tracing-to-origin) innovation in fundamental mathematics and physics.

Keywords: DOG discrete order geometry; temporal fiber bundle; origin of π; closed-loop symmetry; ECS symmetric equilibrium; low-energy steady state

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I. Introduction

Traditional mathematics and physics have confined their understanding of π to the phenomenal level:

1. Geometric level: The ratio of circumference to diameter of a circle;

2. Analytical level: Trigonometric function periods, infinite series, integral results;

3. Physical level: Quantum phase factors, gauge field rotations, periodic oscillation systems.

All prior research suffers from a fatal flaw: they know only "how to use" π, but not "why π exists at the fundamental level of the universe."

The core questions that have never been answered:

· Why must the universe's rotations, phases, and periodic structures give birth to this fixed constant π?

· Is π a "factory setting" of the universe, or the emergent result of some deeper order?

This paper, through the underlying structure of DOG spacetime, directly provides the ultimate geometric origin.

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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 ontology: Discrete lattice order coupling matrix, without preset coordinates;

2. Temporal ontology: Each spatial lattice point independently possesses an attached temporal fiber bundle;

3. Fiber dynamics: The fiber undergoes unitary periodic oscillation with discrete time steps; the oscillation rate is defined as the eigenfrequency ν of that point;

4. ECS principle: Global Coupled Symmetric Equilibrium system — the steady-state system tends toward structural symmetry, coupled equilibrium, and situational stability;

5. MIE principle: Minimal Intrinsic Action principle — the system automatically converges to the state of minimal intrinsic action.

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

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III. Geometric Ontological Definition of π (Original to This Paper)

3.1 Core Conclusion

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

π is the unique baseline constant that emerges from the global uniform symmetric partitioning of the closed-loop rotation of temporal fibers in the low-energy steady state.

π is not a mathematical invention, not a spatial geometric ratio, but rather the self-expression of spacetime order in the steady state.

3.2 High-Energy Discrete State: No π

In the high-energy, strong-fluctuation, topologically unsmoothed regime:

· Temporal fiber oscillation consists of discrete stepwise jumps;

· Rotation angles are non-uniform, with no perfect closed loop;

· Phase evolution exhibits aperiodic or quasiperiodic characteristics;

· At this level, the precise standard constant π does not exist.

This insight explains that π is not an absolute a priori constant, but rather a steady-state emergent quantity. At the Planck scale or in the high-energy limit, π may be imprecise or even undefined.

3.3 Low-Energy Steady State: ECS Symmetric Equilibrium Drives the Emergence of π

In the low-energy approximation, the ECS symmetric equilibrium principle dominates:

· Discrete lattice fluctuations subside;

· Temporal fiber oscillations become highly synchronized, uniform, and closed-loop;

· Rotational trajectories satisfy isotropic symmetry;

· The phase evolution over one full cycle achieves global uniform angular partitioning.

This geometric constraint of "closed-loop uniform symmetric partitioning" uniquely locks in the constant π.

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IV. Self-Consistency of π with Phase Factors and Gauge Fields

4.1 External Implantation in Traditional Physics

In standard quantum mechanics and field theory, the time evolution operator is written as:

```

e^{-i2πν}

```

Traditional treatments directly implant π as a known constant, constituting an external mathematical assembly, never explaining where π comes from.

4.2 The True Mechanism Revealed by DOG

The mechanism revealed in this paper is:

1. The π in the phase factor is not artificially inserted;

2. It arises because the temporal fiber naturally possesses the ECS closed-loop symmetric structure;

3. The steady-state symmetry requires the existence of a uniform closed-loop baseline constant, and that constant is π.

Therefore:

Quantum phases, gauge field rotations, and periodic field evolutions are all secondary consequences of the spacetime geometric origin of π, not the source of π.

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V. Rigorous Boundary Delimitation

This section clarifies the scope of applicability and boundary conditions of π:

1. π does not originate from spatial circular geometry (the locus of previous errors)

The macroscopic circle is merely a投影 (projection) representation of spacetime closed-loop symmetry onto the spatial dimension. The true origin of π is the local closed-loop symmetric topology of the temporal fiber bundle, not the Euclidean geometric definition of π.

2. π is a low-energy steady-state product, not a high-energy absolute constant

Energy Regime Existence State of π Reason

High-energy discrete state No precise π Topology not smooth, asymmetric, no perfect closed loop

Low-energy steady state π precisely emerges ECS symmetric equilibrium requires closed-loop uniform partitioning

3. π fully satisfies the dual constraints of ECS + MIE

· ECS: Guarantees rotational symmetry, uniformity, and absence of bias;

· MIE: Guarantees the system converges to the simplest steady-state structure, with π as the unique baseline constant of that steady state.

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VI. Innovations of This Paper (First in History)

1. Upgrades π from a "mathematical ratio constant" to a spacetime symmetric order constant;
2. Demonstrates that π originates from temporal fiber closed-loop topology rather than spatial geometry;
3. Distinguishes the hierarchical boundary of no π at high energies, emergence of π at low energies;
4. Incorporates π into a unified physical spacetime architecture (DOG + ECS + MIE), achieving a grounding of its physical origin.

These four points have no precedent in prior literature.

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VII. Conclusion

1. π is neither a human mathematical invention nor an a priori cosmic constant, but rather a self-emergent symmetric baseline quantity of DOG discrete spacetime in its low-energy steady state;
2. The sole origin of π is the global uniform partitioning symmetry of the closed-loop rotation of the temporal fiber bundle;
3. All circular geometry, periodic functions, quantum phases, and gauge field rotations are secondary representations of this spacetime geometric origin;
4. This theory concludes the two-thousand-year theoretical gap in which π had "only numerical properties, no cosmic 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. Native Derivation of Yang-Mills Equations within the DOG Framework. 2026.
[3] Zhang Suhang. The ECS Symmetric Equilibrium Stability Principle and the MIE Minimal Intrinsic Action Principle. 2026.

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