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Discrete Order Geometry (DOG) Unifies Gauge Field Theory – A Unified Framework for Universal Field Theory Based on the Microscopic Discrete Primordial

Author: Zhang Suhang (Luoyang, Henan)

Abstract

Modern gauge field theory, built upon fiber bundle geometry, has established a complete theoretical system. Using continuous spacetime manifolds as the background and smooth connections as the mediators of interactions, it accurately describes the three fundamental interactions – electromagnetic, weak, and strong – and forms the core mathematical foundation of particle physics and quantum field theory. Its theoretical rigor and experimental validity have long been widely recognized by the academic community. This paper adheres to a core primordial idea: at the essential microscopic level, all things in existence are fundamentally discrete, with primitives mutually independent and naturally unconnected; there is no a priori integrated, seamless continuous whole.

The continuous spacetime, smooth fields, and pervasive interaction links seen at the macroscopic level are merely emergent equivalent appearances, arising when a vast number of microscopic discrete primitives aggregate and arrange themselves according to inherent orders, establishing directional couplings that become observable at large scales. The point‑to‑point primordial interaction links between discrete primitives are the most original form of interaction. The continuously smooth gauge connections well‑developed in physical systems are merely macroscopic limiting special cases that evolve after the discrete primitives become infinitely dense in arrangement and the overall order becomes highly balanced and regular.

Crucially, the entire gauge field theory is precisely the most intuitive and standard theoretical specimen of the primordial idea that “microscopic things are discrete and unconnected.” All of its architecture, operational logic, and mechanisms silently testify to the cosmic fundamental rule: discreteness is primordial, connection comes later. Relying on this foundational understanding, this paper smoothly, completely, and losslessly incorporates the entire classical gauge field theory into the global system of DOG discrete order geometry. Throughout, it maintains a neutral and inclusive academic stance: no modification to the core equations of existing field theory, no overturning of already verified experimental conclusions – only a reordering of the primordial existence of spacetime on which field theory relies and a repositioning of its theoretical hierarchy, ultimately achieving a deep grand unification of the underlying geometric architecture and the fundamental theory of interactions.

Keywords: Discrete Order Geometry; DOG; microscopic discrete primordial; gauge field; discrete interaction connection; continuous gauge connection; spacetime primitives; unification of field theories; primordial specimen

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I. Introduction

Since the formal establishment of Yang–Mills gauge field theory, physics has developed a mature and stable research paradigm: four‑dimensional continuous spacetime as the carrier of physical events, Lie group symmetry transformations as the theoretical core, and smooth connections on fiber bundles to describe the transmission laws of fundamental interactions. The U(1) electromagnetic gauge field, the SU(2) weak interaction gauge field, and the SU(3) strong interaction gauge field have all been constructed within the framework of continuous smooth spacetime, together forming the core skeleton of the Standard Model of particle physics.

This field theory system, built upon continuous spacetime, is highly practical in macroscopic physics research, high‑energy particle experiments, and classical field calculations. Over decades it has accumulated a vast body of solid theoretical achievements and empirical data. However, this research paradigm has always rested on the macroscopic empirical assumption that spacetime is naturally continuous and unified. It cannot trace the primordial form of interactions at the bottom level, nor can it naturally accommodate the microscopic objective characteristics of quantum physics: spacetime discreteness, energy quantization, and discontinuous interactions.

DOG discrete order geometry breaks free from the inherent constraints of traditional continuous spacetime thinking, descends to the most fundamental laws of existence in the universe, and anchors a core recognition: when one explores the nature of existence at microscopic scales, all spacetime components, material primitives, and field carriers are by nature separate, independent discrete individuals, with no inherent fixed connections or overall connectivity.

In the universe, there is no inborn continuous spacetime, no naturally pervasive field system. All transmission of field energy, coupling of fundamental forces, and integration of spacetime forms originate from directional connections established post‑hoc among independent discrete primitives. The gauge field theory that humanity has summarized and condensed precisely replicates this evolutionary logic, becoming the most typical theoretical specimen for interpreting the primordial discrete idea. Following this evolutionary thread from the micro to the macro, one can open the internal pathway connecting geometric laws and physical interaction laws, allowing gauge field theory, originally built upon the continuous fiber bundle framework, to be naturally incorporated into the grand system of DOG discrete primordial geometry, achieving the global unification of fundamental physical field theories under the geometric matrix.

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II. The Three Core Axioms of DOG

1. Axiom of Microscopic Existence Primordial: At microscopic scales, all existents are in discrete form; all kinds of primitives have clear boundaries, are mutually isolated, have no innate connecting structures, and have no native continuous whole. Continuity is merely an equivalent appearance emerging at macroscopic scales.

2. Axiom of Interaction Derivation and Evolution: Originally unconnected discrete primitives, under given order rules, establish point‑to‑point directional interaction pathways, forming discrete primordial interaction connections. When the primitives become infinitely small in scale, their spatial arrangement density continuously increases, and the overall arrangement order tends toward homogeneity and unity, the originally fragmented discrete interaction connections gradually smooth out their gaps, eventually evolving into the continuous smooth gauge connections familiar in classical physical systems.

3. Axiom of Theoretical Specimen: From its structural construction to its operational functioning, gauge field theory fully conforms to the underlying logic of discreteness and post‑hoc connection. It is the most fitting and complete academic specimen that verifies the idea that “microscopic things are discrete and unconnected.” Its theoretical architecture is the mathematical expression of the primordial idea made concrete.

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III. Gauge Field as the Core Verification of the Discrete Primordial Specimen

3.1 Field points are discretely arranged, matching the discrete nature of all things

Gauge field theory decomposes spacetime into countless independent field points, each carrying its own field attributes. The field points exist independently of each other, with no inherent fixed connecting relationship. This setup directly corresponds to the primordial state of mutually separate, non‑attached primitives in the microscopic world, perfectly replicating the bottom‑level appearance of discrete distribution of all things.

3.2 Connection is established post‑hoc, corresponding to the fact that connections are not innate

The connection used to transmit interactions in gauge field theory is not an inborn property of spacetime; rather, it is a human‑defined correlational channel established post‑hoc on the basis of symmetry rules. Without the establishment of connections, each field point would be isolated and unaffected by others. This precisely confirms the core idea: all connections in the world are formed after the fact; there are no innate, natural connections. This is completely consistent with the essence that microscopic primitives have no inherent links and only form interaction relationships later.

3.3 The continuous field state is the macroscopic appearance of densely packed discrete units

The continuous and smooth gauge field form seen in everyday research is not the field being intrinsically continuous, but rather the visual and computational effect of continuity arising after an enormous number of independent field points become infinitely dense and their connection channels uniformly arranged. Stripping away the macroscopic appearance to return to the essence, one still finds discrete field units and independent interaction pathways – precisely interpreting the core thesis that continuity is merely a limiting special case of discreteness.

3.4 Symmetry transformations accommodate the autonomous configuration of discrete units

The local symmetry transformations of various gauge groups allow each individual independent field point to adjust its own intrinsic properties autonomously, without being forced by the overall space. This highly matches the characteristic that microscopic discrete primitives have independent intrinsic structures and can adjust their states autonomously, solidifying the specimen property from the perspective of symmetry logic.

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IV. Precise Correspondence Between Core Components of Gauge Field Theory and the DOG Discrete System

4.1 Continuous spacetime background ⇄ DOG discrete primitive aggregate spacetime network

Classical gauge field theory sets its research background in four‑dimensional continuous flat spacetime or curved manifold spacetime, assuming spacetime is an indivisible unified continuous whole.
Returning to the DOG microscopic primordial perspective: continuous spacetime is not the native form of the universe; its essence is a discrete aggregate network built from a vast number of independent spacetime discrete primitives that cluster together following fixed arrangement and combination rules. The continuous spacetime coordinate system used in everyday physics calculations is merely a macroscopically equivalent spacetime model, simplified and fitted after the discrete primitive network becomes highly dense.

4.2 Gauge symmetry group ⇄ Intrinsic symmetric configuration of DOG discrete primitives

Gauge symmetry groups such as U(1), SU(2), etc., are no longer simply defined as coordinate transformation groups on continuous space, but rather as the intrinsic algebraic symmetric structure inherent in each independent spacetime discrete primitive. The inherent physical properties of microscopic particles and the symmetric operational rules of fundamental interactions are all embedded in the primitives themselves. The large‑scale symmetric transformations observed in macroscopic spacetime are essentially the collective cooperative effect of countless discrete primitives adjusting their own intrinsic configurations synchronously.

4.3 Gauge potential and gauge connection ⇄ Post‑hoc directional interaction channels of discrete primitives

In continuous field theory, the gauge potential and gauge connection appear as continuously smooth, differentiable field quantities on spacetime, serving as the core medium for transmitting interactions.
Re‑interpreting from the microscopic discrete origin: they are essentially the directional energy and interaction transmission channels established post‑hoc among originally unconnected discrete primitives. Field energy and interaction signals are transmitted point‑to‑point through independent discrete channels; the dense superposition and arrangement of countless discrete interaction channels ultimately produce, at the macroscopic level, a visually and computationally smooth gauge potential field and continuous connection.

4.4 Gauge field strength and interaction intensity ⇄ Degree of deviation from discrete arrangement order

Classical field theory derives field strength through continuous operations such as exterior differentiation and covariant differentiation, thereby defining the strength level of fundamental interactions.
The DOG system provides a more primordial definition: the field strength corresponding to an interaction is determined by the degree to which the arrangement pattern of discrete primitives and the coupling directions of interaction channels in a local spatial region deviate from the standard ordered configuration. The greater the deviation at the discrete level, the stronger the macroscopically manifested physical interaction intensity. The traditional continuous field strength formulas are merely convenient computational forms that macroscopically quantify the deviation of microscopic discrete order.

4.5 Gauge invariance ⇄ Invariance under local reconfiguration of discrete primitives

The constancy of physical laws under local gauge transformations is the core principle that upholds gauge field theory.
Tracing back to the microscopic discrete level, its essence becomes clear: a single independent discrete spacetime primitive adjusting its own internal field configuration and symmetric structure does not disrupt the overall arrangement order of the entire discrete spacetime network, nor does it change the underlying logic of interaction transmission. The inherent configurational invariance at the discrete level extends upward and evolves into the crucially important principle of gauge invariance in macroscopic physics.

4.6 Gauge bosons and field quanta ⇄ Energy excitation states of discrete interaction channels

Mediator particles that transmit fundamental interactions – such as photons, gluons, W and Z bosons – are conventionally regarded as energy units generated by quantized excitations of continuous fields.
From the microscopic discrete perspective, these interaction particles are essentially the primitive excitation units formed when the interaction transmission channels between discrete primitives absorb energy, undergo state transitions, and generate energy excitations – perfectly matching the objective fact that energy is discrete and existence forms are discontinuous in the microscopic world.

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V. Formal Incorporation of Gauge Field Theory into the DOG Discrete Order Geometry System

After this full‑dimensional tracing of elements, rigorous logical correspondence, and demonstration of the specimen property, a neutral, rigorous, and hierarchically clear academic conclusion can be drawn:

The complete, well‑tested classical gauge field theory belongs under the universal matrix framework of DOG discrete order geometry. It is a continuous special‑case field theory that emerges from microscopic discrete spacetime primitives under the constraints of infinitely dense arrangement and highly homogenized post‑hoc interaction connections. At the same time, gauge field theory is the most standard, most concrete, and most mature theoretical specimen of the cosmic primordial idea that “microscopic things are discrete and unconnected.”

Three distinct theoretical levels are thus established:

1. Top level – DOG Discrete Order Geometry: the universal matrix theory, encompassing all forms of spatial existence and physical interactions. It accommodates both macroscopic continuous spacetime systems and microscopic discrete quantum spacetime systems, with no boundaries on its applicability.
2. Middle level – Discrete Primordial Interaction Field Theory: directly confronts the discrete, unconnected reality at the microscopic scale, specializing in cutting‑edge physical scenarios such as quantum‑scale discrete spacetime, discontinuous interactions, and topological discrete mutations.
3. Lower/Branch level – Traditional Continuous Gauge Field Theory: belongs to the macroscopically applicable branch theory. It is both a practical tool for physical research and the core specimen for interpreting the discrete primordial idea. Built upon the continuous fiber bundle structure, it retains all its theoretical validity in classical physics research, macroscopic field calculations, and conventional high‑energy physics experiments. All previous computational methods, physical inferences, and experimental fitting conclusions remain fully usable.

This theoretical incorporation consistently respects academic heritage, fully acknowledges the outstanding achievements of generations of physicists who built gauge field theory, does not negate the value of existing theories, and does not forcibly replace mature research paradigms. By leveraging the natural specimen property of gauge fields, it smoothly reorders the hierarchical relationship between geometry and physical theory, allowing gauge field theory – originally rooted in the continuous spacetime assumption – to naturally and gracefully enter the unified grand framework of DOG, following the primordial logic that microscopic things are discrete and unconnected.

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VI. Core Theoretical Value After Unification

6.1 Using a theoretical specimen to verify the underlying truth

By means of gauge field theory – an already complete and mature academic specimen – the core idea that microscopic things are discrete and separate, with connections arising post‑hoc, is visually supported. Abstract primordial cognition gains empirical backing from a sophisticated mathematical physics system, making the theoretical foundation increasingly solid.

6.2 Breaking through the bottom‑level barrier between geometric laws and physical laws

From now on, the arrangement and construction rules of spatial geometry and the operational rules of the universe's fundamental interactions share the same microscopic discrete origin. The combinatorial arrangement order of discrete primitives directly determines the form and transmission mode of field interactions, truly achieving a homologous unification of foundational mathematical geometry and basic physical laws.

6.3 Filling the inherent research blind spots of continuous gauge field theory

Traditional gauge field theory, constrained by the presupposition of continuous spacetime, cannot reasonably explain many microscopic physical puzzles such as quantum spacetime discreteness, spacetime singularity structures, and the origin of parity asymmetry. Once incorporated into the DOG system, relying on the underlying idea of microscopic discreteness and non‑connection, phenomena such as symmetry breaking, spacetime quantum fluctuations, and asymmetric interactions – which have long been difficult to trace – can be naturally explained through differences in the combinatorial order of primitives.

6.4 Opening a new primordial path for unified field theory

Using DOG discrete geometry as a common foundational base, integrating discrete arrangement‑combination order, geometric topological architecture, fiber bundle continuous structures, and the three types of fundamental gauge interactions, this approach completely jumps out of the inherent limitations of unified field theories based on continuous spacetime. Starting from the essential truth that microscopic things are discrete, separate, and originally unconnected, it opens up a new feasible research direction for the global unification of the four fundamental interactions.

6.5 Completing a full‑system theoretical submission and integration

Now, the laws of arrangement and combination, the ontology of DOG discrete order geometry, classical fiber bundle geometry, and the complete gauge field theory system – all are hierarchically ordered from top to bottom, their subordination and sequence clarified, and all are integrated into the same primordial academic system. This constructs a complete closed‑loop fundamental theoretical system that runs through micro and macro, covers discrete and continuous, and merges geometry and physics.

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VII. Conclusion

Looking back at the underlying truth of existence in the universe, descending to the deepest microscopic level, it becomes clear: all things are originally discrete, independent forms of existence. Primitives are by nature separated from each other, with no inherent connections. A seamlessly integrated continuous spacetime, a universally pervasive force field – these have never been the original appearance of the universe.

The post‑hoc discrete interaction connection is the original foundation on which fundamental interactions occur. The classical gauge field theory that humanity has derived from the macroscopic world is not only a practical tool for physical research but also the most fitting theoretical specimen reflecting this underlying truth. From its architecture to its core, it performs the complete evolutionary process: discrete units aggregate into form, connections create relations, and denseness yields continuity.

Relying on the core primordial idea that microscopic things are discrete and unconnected, and leveraging the natural specimen property of gauge fields, DOG discrete order geometry successfully completes the incorporation and hierarchical repositioning of the entire gauge field theory system, allowing modern core physical field theory to stand firmly on the solid foundation of discrete primordial geometry. From now on, research can continue to use the mature continuous gauge field system to explore macroscopic physics, while also relying on the DOG discrete architecture to push deeper into microscopic quantum spacetime. The time‑tested classical theory and the new paradigm rooted in primordial principles support each other, complement each other, and together advance fundamental mathematical physics steadily toward a deeper, broader unification.

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References

[1] C. N. Yang. Fundamental theory of gauge fields.
[2] A. Zee. Fearful Symmetry.
[3] Research on differential geometry and physical applications of fiber bundles.
[4] Introduction to quantum field theory and the Standard Model of particle physics.
[5] Frontiers of discrete spacetime architecture and lattice gauge theory.

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