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Discrete Order Geometry (DOG): A New View of Spacetime — Space as Matrix and Time as Fiber Bundle

Author: Zhang Suhang (Luoyang, Independent Researcher)

Abstract: Discrete Order Geometry (DOG) fundamentally reconstructs the picture of physical spacetime. This paper explicitly articulates two core propositions of spacetime under the DOG framework: space is a matrix (the order-coupling matrix among discrete lattice points directly encodes spatial structure); time is the canonical evolution of a fiber bundle (the internal swinging of fibers defines the passage of time). Continuous geometries (Euclidean, Riemannian) are interpreted as emergent approximations of discrete order at macroscopic scales. This view of spacetime provides a background‑free, discrete geometric foundation for quantum mechanics, gauge field theory, and quantum gravity.

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I. Introduction

Traditional geometry relies on continuous backgrounds: the flat space of Euclidean geometry, the curved manifolds of Riemannian geometry. Although these presuppositions are extremely successful at macroscopic scales, they encounter difficulties at the Planck scale, short‑range weak interactions, quantum gravity, and other frontiers. Discrete Order Geometry (DOG) proposes that the ontology of physical spacetime is a finite, countable set of discrete ordered lattice points, an adjacency matrix, and fiber bundle evolution — not a continuous manifold.

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II. Space as Matrix

DOG abandons the definition of space as "point set + distance" and replaces it with:

· Lattice point set \{\mathcal{L}_i\}: finite, countable, with no preset coordinates.

· Adjacency matrix A: A_{ij} represents the order‑coupling strength between lattice points i and j. Non‑zero entries define "spatial connections".

· Evolution matrix M: \boldsymbol{\Psi}_{k+1} = M \boldsymbol{\Psi}_k, incorporating both spatial coupling and internal dynamics.

All geometric properties of space (adjacency relations, distance, topology, curvature) are uniquely determined by the algebraic structure of the matrix (zero‑entry pattern, eigenvalues, rank). Space is not a "background" for the matrix; space is the matrix itself. This is the fundamental watershed between DOG and all continuous geometries.

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III. Time as the Swinging of a Fiber Bundle

In traditional DOG, time appears only as a discrete step index k. Here we further geometrize time:

· Attach a fiber space \mathcal{F}_i (e.g., spinors, gauge group representations, chirality markers) to each lattice point \mathcal{L}_i.

· The complete state of the system is a section \boldsymbol{\Psi}_k, taking an element in the fiber at each lattice point.

· The evolution matrix M in the equation \boldsymbol{\Psi}_{k+1} = M \boldsymbol{\Psi}_k acts simultaneously on lattice indices (spatial coupling) and fiber indices (internal transformations).

The essence of time: the unitary rotation of fiber states with each step — we call this "swinging". The difference in fiber state between adjacent steps, \Delta \boldsymbol{\Psi}, defines an intrinsic "rate of time flow". In the continuous limit, the frequency of this swinging maps onto the ordinary time parameter.

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IV. Continuity as a Special Case of Discreteness

Euclidean geometry, Riemannian geometry, and smooth manifolds are all emergent approximations of DOG discrete spacetime under the following conditions:

· The lattice spacing \Delta x is much smaller than the observational resolution (\Delta x \ll \lambda_{\text{obs}}).

· The swinging frequency of fibers is much higher than the observational time scale (\Delta t \ll T_{\text{obs}}).

· The lattice arrangement is nearly uniform, and the fiber swinging is nearly uniform in rate.

Under these conditions, the discrete lattice averages into a continuous background, and the accumulated phase of the fiber swinging becomes a continuous time coordinate. Therefore, continuous geometry is not a first principle, but a special case of DOG under macroscopic approximation.

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V. Unified Picture and Significance

Traditional View DOG New View

Space = point set + distance Space = discrete lattice + adjacency matrix

Time = external parameter Time = unitary swinging of fibers

Continuity is default Continuity is a special case of discreteness

Background spacetime independent of matter Space matrix + fiber bundle together form a spacetime‑matter unity

This view of spacetime naturally incorporates quantum mechanics (spinor fibers), gauge theory (group representation fibers), and weak interactions (chirality fibers), and naturally leads to discrete spacetime effects (e.g., Lorentz invariance violation, energy spectrum staircases). Computation is not the focus of this paper; the focus is that DOG provides a background‑free, self‑consistent, and computable underlying model of spacetime geometry.

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VI. Conclusion

DOG reduces space to a matrix, time to fiber swinging, and continuity to a limiting special case of discreteness. This is a picture of physical spacetime that is radically different from mainstream geometry, yet internally consistent and more fundamental. Readers need not accept it immediately, but it is worth serious consideration: perhaps spacetime is not a membrane, but a discrete order built from matrices and swinging.

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References (omitted)

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