DOG-FCE and MOC-UCE: Two Equivalent Physical Representations of the Same Unified Field Essence

Author: Zhang Suhang, Luoyang (Independent Researcher)

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Abstract

Within the axiomatic DOG-MOC grand unified system, there exist two core dynamical equations: the DOG-FCE (Frequency Configuration Evolution) equation in the discrete matrix–frequency representation, and the MOC-UCE (Unified Curvature Evolution) equation in the continuous field–frequency representation. Each of these two equations can independently and completely unify the four fundamental interactions, differing only in mathematical representation and applicable scale. To avoid the historical dispute in early quantum mechanics over whether matrix mechanics or wave mechanics was “real,” this paper rigorously demonstrates that the two equations share the same physical origin, are mathematically equivalent, yield equivalent results, and have complementary representations. They are not two separate theories but two equivalent expressions of the same discrete-frequency spacetime ontology within the discrete iterative and continuous analytical mathematical frameworks. This paper clearly defines their respective domains of applicability, limiting-case correspondences, and complementary paradigms, thereby fully establishing the self-consistency and proper recognition of the DOG-MOC system.

Keywords: DOG-FCE; MOC-UCE; dual-representation equivalence; unified field theory; matrix mechanics; wave mechanics; discrete–continuous duality; unification of four forces

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I. Problem Statement

In the early development of quantum mechanics, Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics were once regarded as two conflicting theories, with long-standing debates over which was authentic. Subsequent mathematical proofs confirmed that they are completely equivalent, differing only as two mathematical representations of the same quantum system.

Similarly, the DOG-MOC unified system presented here contains two core evolutionary frameworks, each independently and completely capable of unifying the four forces:

1. DOG-FCE: Based on a discrete lattice, adjacency coupling matrix, and discrete time iteration, with the matrix + frequency as its core. It unifies the four forces in a discrete operator form.

2. MOC-UCE: Based on a global multi-origin curvature field and continuous spacetime evolution, with curvature + frequency as its core. It unifies the four forces in a continuous field form.

Without a clear statement of the relationship between the two, outsiders could easily misinterpret one as primary and the other as derivative or approximate, or mistakenly regard them as two independent or even contradictory competing theories. Therefore, this paper provides an official and permanent clarification of their equivalence, equal status, and complementarity.

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II. Historical Paradigm Comparison in Quantum Mechanics

1. Heisenberg matrix representation: Operators evolve, states are fixed. Suitable for discrete energy levels, transition matrices, and numerical computation.

2. Schrödinger wave representation: Wave functions evolve, operators are fixed. Suitable for continuous wave evolution, analytical solutions, and field-state analysis.

The two frameworks were ultimately unified into a complete quantum mechanics, with no question of truth or falsity, only differences in contextual suitability.

The DOG-MOC system strictly replicates this top-tier mathematical-physical paradigm and achieves an upgraded architecture in which both frameworks independently accomplish a grand unification.

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III. Formal Presentation of the Two Unified Four-Force Standard Equations

3.1 DOG System · FCE Unified Equation (Matrix + Frequency Unification of Four Forces)

Discrete-time iteration of the global discrete lattice state:

\boldsymbol{\Psi}_{\tau+1} = \boldsymbol{M} \cdot \boldsymbol{\Psi}_\tau

\]  

Tensor product decomposition of the global total interaction matrix (incorporating the four forces):

\boldsymbol{M} = \boldsymbol{M}_G \otimes \boldsymbol{M}_E \otimes \boldsymbol{M}_W \otimes \boldsymbol{M}_S

\]  

Component definitions:

\begin{cases}

\boldsymbol{M}_G = \nabla \boldsymbol{A} \quad \text{(Gravity: lattice adjacency matrix deformation operator)} \\

\boldsymbol{M}_E = e^{i\alpha(\tau)} \in U(1) \quad \text{(Electromagnetism: homogeneous phase frequency operator)} \\

\boldsymbol{M}_W = P_{\text{chiral}} \cdot U_2,\; U_2 \in SU(2) \quad \text{(Weak force: chiral-broken frequency operator)} \\

\boldsymbol{M}_S = \rho \cdot U_3,\; U_3 \in SU(3),\ \rho \gg 1 \quad \text{(Strong force: high-density strong-coupling frequency operator)}

\end{cases}

\]  

Complete FCE four-force unified equation:

\boldsymbol{\Psi}_{\tau+1} = \Big( \nabla \boldsymbol{A} \;\otimes\; e^{i\alpha(\tau)} \;\otimes\; P_{\text{chiral}}U_2 \;\otimes\; \rho U_3 \Big) \boldsymbol{\Psi}_\tau

\]  

Core logic: The matrix determines the spatial structure, frequencies determine the interaction modes, and discrete-time iteration accomplishes the full unification of the four forces.

3.2 MOC System · UCE Unified Curvature Equation (Curvature + Frequency Unification of Four Forces)

Standard field equation for the global multi-origin curvature field:

\square \mathcal{K}(\boldsymbol{r}, t) = \mathcal{J}\big( \Delta\nu, \mathcal{K}, \Omega \big)

\]  

Here \square is the d’Alembert operator, \mathcal{K} is the global multi-origin curvature field, \Delta\nu is the eigenfrequency difference, and \Omega denotes the symmetry configuration.

The four forces are expressed hierarchically according to curvature form and frequency field:

1. Gravity: Large-scale global curvature gradient field – low-frequency long-range curvature deformation effect.

2. Electromagnetism: Uniform gentle curvature field – U(1) frequency phase field propagation.

3. Weak force: Localized curvature symmetry-breaking field – SU(2) chiral frequency shift evolution.

4. Strong force: Localized high-density strong curvature confinement field – SU(3) high-binding-frequency compact coupling.

Core unified meaning of UCE:

Using the continuous curvature field as the spatial substrate and eigenfrequency differences as the dynamical driver, the UCE independently accomplishes the common-origin unification of the four fundamental interactions from a continuous field theory perspective.

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IV. Precise Dualistic Architecture

4.1 DOG-FCE (Discrete Matrix Representation · Analogous to Heisenberg)

· Basic carriers: Discrete cosmic lattice, adjacency matrix \boldsymbol{A} , global interaction matrix \boldsymbol{M} 

· Dynamical form: Discrete time iteration

· Core mechanism: Matrix structure carries spatial order; phase frequencies drive dynamical evolution

· Applicable scenarios: High-energy discrete processes, lattice numerical simulations, quantitative computation of four-force components, quantum transition iterations

4.2 MOC-UCE (Continuous Field Representation · Analogous to Schrödinger)

· Basic carriers: Global continuous multi-origin curvature field \mathcal{K}(\boldsymbol{r}, t)
· Dynamical form: Partial differential wave equation for the curvature field
· Core mechanism: Curvature field carries spacetime geometry; eigenfrequency differences drive field-state evolution
· Applicable scenarios: Large-scale cosmological analytical studies, spacetime structure analysis, field symmetry arguments, low-energy continuous limit

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V. Core Basis for Strict Equivalence

1. Continuous limit convergence
As the DOG discrete lattice spacing and time step approach zero, the FCE discrete iteration strictly reduces to the UCE continuous field wave equation. The discrete system contains the continuous system as its limiting special case.
2. Complete unity of frequency origin
The sole dynamical driving source of both equations is identical: eigenfrequencies, frequency-difference phase evolution, and frequency-spectral density of states. The structural carriers differ (discrete vs. continuous), but the evolutionary kernel is completely homologous.
3. Fully equivalent physical results
Both FCE and UCE independently achieve unification of the four forces. All observable quantities — transition probabilities, decay rates, coupling strengths, spacetime effects — yield identical computational results.
4. Single system of axioms
Both share the underlying axioms of DOG discrete order ontology + MOC multi-origin curvature, differing only as two mathematical expression languages.

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VI. Complementary Applicability Paradigms (Official Definition)

· Numerical computation, discrete simulation, high-energy microphysics, frame-by-frame iteration → Preferentially use DOG-FCE
· Analytical derivation, geometric reasoning, cosmological analysis, field-theoretic investigation → Preferentially use MOC-UCE

Neither is primary or secondary, true or false; there is no need to choose one over the other. Each is independently a complete grand unification, and together they constitute the complete dual-representation system of the DOG-MOC unified field theory.

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VII. Concluding Statement

1. DOG-FCE and MOC-UCE enjoy equal status and equal rights, each possessing its own independent and complete set of unified four-force equations.
2. One is a discrete matrix-frequency system, the other a continuous curvature-frequency system, replicating the classic Heisenberg–Schrödinger dualistic paradigm.
3. Both share frequency as the primary driving origin, achieving a self-consistent closed loop of physical laws across all scales, from microscopic particles to the macroscopic cosmos.
4. This hereby formally establishes: one cosmic ontology, two unified equations, all physics encompassed.

Date of finalization: May 2026.