Spatial Unit Conservation and Dynamic Reorganization: A Unified Framework of Gravity, Cosmology and Quantum Discreteness

1. Introduction

Modern physics confronts a profound contradiction between its two cornerstones—general relativity (macroscopic, continuous, geometric) and quantum field theory (microscopic, discrete, algebraic). Moreover, the four major mysteries of dark matter, dark energy, black hole singularities, and vacuum catastrophe suggest that our understanding of the essence of spacetime may be missing a fundamental mechanism.

This paper attempts to address the question: If spacetime is composed of discrete, countable fundamental units with conserved "total quantity," can gravity, cosmic expansion, and quantum phenomena be unified in understanding? The proposed principle of "holistic covariant" will serve as the guiding framework throughout the text, with subsequent chapters building upon this principle to construct specific dynamic frameworks.

2. The Principle of the Whole and the Common Covariant

2.1. Basic Position: No Background, No Independent Entity

The fundamental position of this framework is that there is no independent spatio-temporal background, nor do there exist independently existing material particles. Space and matter are essentially unified, being different manifestations of the same underlying structure.

• No Pre-existing "Stage" (Absolute Space-time)
• No independent "actor" (fundamental particle)
• There is only one whole structure, which in dynamic evolution presents two aspects we call "space" and "matter".

This position is in line with Leibniz's relational view of space and time, but it goes further: the relation itself is not static, but is maintained by dynamic process.

2.2. Core Principle: Holistic Co-Variation

The fundamental requirement of physical laws is covariance—their form remains unchanged regardless of the coordinate system. However, this framework proposes a deeper interpretation:

Covariant is not a local requirement for a single particle, a single field, or a single atom, but a whole constraint for the whole system, all matter and space-time.

It means that ：

• The study of any single object is only approximate and inevitably incomplete.
• The true physical laws describe how the whole self-coordinates
• Local non-covariance is permissible—provided the whole is ultimately covariant

2.3. The Nature of the Existence and Decay of Particles

From this principle, the particle is no longer an eternal entity, but a local excitation or local distortion in the whole structure.

• Stable particle: A configuration that is already stable under global covariance and can persist indefinitely.
• Unstable particles: deviating from the overall minimum covariant state, they must undergo decay or transformation to restore the system to a self-consistent state of overall common covariance.

Key insight: The extremely brief existence of particles is not accidental, but rather because this localized state cannot sustain covariance independently.

2.4. The Only Logic of Being and Disappearing

The fundamental principle is this: particles do not preexist and then satisfy covariance. It is the need for covariance that gives rise to particles; once covariance is satisfied, particles cease to exist.

The generation of particles does not occur out of thin air; the disappearance of particles is not annihilation out of thin air.

All that comes into being and vanishes is for one purpose: to satisfy the covariant.

2.5. The Dynamic Unity of Local and Whole

How does this mechanism function? Taking photon conversion in Argument 7 (continued) as an example:

1.

A certain gradient does not satisfy the covariance (e.g., in regions of strong gravitational fields).

2.

Cannot act over distance, only local resolution is available

3.

Thus a pair of positive and negative particles is produced, and the local covariance is satisfied first.

4.

This particle propagation, movement, and interaction—carrying the "covariant repair task"

5.

To another place to complete the overall constraint-the overall "tail"

6.

Task completed, particles disappear-the whole re-covariant

This process can be summarized as: prioritize local emergency response before addressing the overall situation. The local does not conflict with the overall, but rather serves as the first step in the coordinated evolution of the whole.

2.6. The Ultimate Explanation of Symmetry Breaking

This mechanism addresses one of physics' most profound questions: why symmetry is broken. In the Standard Model, phenomena like the Higgs mechanism, particle mass acquisition, and phase transitions all demonstrate symmetry breaking, yet it remains unanswered: why must perfect symmetry be violated?

This framework provides the definitive answer: symmetry is not 'broken' —it is sacrificed to ensure the whole's collective covariance, requiring temporary local abandonment of symmetry.

Language of the translation cost framework:

• global requirement of common covariant
• Local gradient and non-covariant
• Cannot act over distance; only local repair is allowed
• Thus, a pair of positive and negative particles is produced
• Local appearance: symmetry is gone—this is symmetry breaking
• But when viewed holistically: breaking local symmetry is to preserve global higher covariance symmetry.

In a word, symmetry breaking is not an accident of the universe, but the price of covariance, and the local price that must be paid for the overall self-consistency.

2.7. Chapter Summary

Traditional View vs. Framework View

The particle is the basic entity → The particle is the local excitation of the whole structure

Symmetry Breaking is Phenomenon→Symmetry Breaking is the Cost of Covariant

Physical laws describe individual behavior → Physical laws describe collective covariation

Space-time is the background → Space-time is the dynamic expression of structure

The emergence and disappearance are random quantum processes → The emergence and disappearance are to satisfy covariance

The sole logic of cosmic operation: all structures exist solely for covariance.

2.8. Linking to the Following Text

The following chapters will concretize this meta-principle into an operational mathematical mechanism—through spatial units, virtual process-driven dynamics, and contention-compensation dynamics—to demonstrate how the principle of "holistic covariant" can be derived to encompass all known physical laws, including gravity, cosmology, and quantum phenomena.

The integral covariance principle described in this chapter will be mathematically expressed in the dynamic equations of Chapter 2, and will be demonstrated as specific physical laws in subsequent chapters.

3. Theoretical Foundation: Discrete Dynamics of Complex Fields and the Uniqueness of the Wave Equation

3.1. The Conservation of Space Resources and the Ontology of Discrete Space-time

The ontology assumptions of this framework:

• The space-time is composed of the smallest indivisible space unit;

2.

There is a space material $S$which is kept in constant quantity;

3.

The material is the local excitation and distortion of the space unit.

4.

All the interactions are only transferred between adjacent cells, and there is no long-range interaction.

Global conservation law:

[

\\sum_{i} N_i(t) = S = \\text{constant}

]

The total $N_i\left(t\right)i$amount of space material contained in the grid is.

3.2. Introduction of the Re-Field: The Only Self-consistent Description of Electromagnetism and Spin Structure

To enable the theory to:

• natural generation of electromagnetic waves
• $\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{s} \mathrm{F}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{\text{'}}\mathrm{s} \mathrm{l}\mathrm{a}\mathrm{w} \mathrm{o}\mathrm{f} \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c} \mathrm{i}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\backslash \mathrm{n}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{a}\backslash \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}\backslash \mathrm{b}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}\mathrm{l}\left\{\mathrm{E}\right\}\backslash \mathrm{n}\mathrm{e}\mathrm{q}0$
• supporting quantum mechanical complex phase
• $spinsupport1/2$
• preservation of Lorentz covariance

The complex field must be introduced:

[

\Phi(\boldsymbol{x},t) = \sqrt{\rho(\boldsymbol{x},t)} , e^{i\theta(\boldsymbol{x},t)}

]

• $\rho$Space unit density (corresponding to space material)

• $\theta$Re-Phase (Electromagnetic, Quantum Phase, and Spin Sources)

The field of re-creation is the only basic field of this framework.

3.3. Fundamental Scale of Discrete Space-Time

Define the minimum discrete scale of space and time:

• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m} \mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d} \mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{g}a:$
• $\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m} \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e} \mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\tau :$

Intrinsic propagation speed:

[

c = \frac{a}{\tau}

]

The velocity is the constant of space-time structure and is independent of the reference frame.

The discrete spatiotemporal structure of this framework can be represented by a weighted graph G=(V,E,w). To simplify computations and focus on core physics, this paper adopts a regular lattice (w_{ij}=1, nearest neighbor coupling), whose physical behavior in the long-wave limit should belong to a universal class independent of the graph structure details.

3.4. The Only Dynamics: The Second Order Wave Equation of Discrete Complex Field

The framework is based on a single fundamental dynamic equation: the second-order central difference discrete wave equation.

[

\frac{\Phi_i(t+\tau) - 2\Phi_i(t) + \Phi_i(t-\tau)}{\tau^2}

= c^2 \frac{\sum_{\langle i,j\rangle} \big(\Phi_j(t) - \Phi_i(t)\big)}{a^2}

]

Text description:

• Left: Second-order time derivative, which describes inertia, fluctuation, and acceleration behavior.
• Right: the discrete form of the space Laplace operator;
• The equations are hyperbolic, which support finite propagation speed, causality and Lorentz covariance.
• No diffusion, no infinite velocity, no spin.

3.5. Continuous Limit:Relativistic Covariant Wave Equation

When$\mathrm{a}\backslash \mathrm{t}\mathrm{o}0,\backslash \mathrm{t}\mathrm{a}\mathrm{u}\backslash \mathrm{t}\mathrm{o}0$, the discrete wave equation tends to:

[

\frac{1}{c^2}\frac{\partial^2\Phi}{\partial t^2} - \nabla^2\Phi + \left(\frac{mc}{\hbar}\right)^2\Phi = 0

]

i.e. Klein-Gordon equation.

All subsequent physical laws are derived from this single equation.

4. Exposition of the Core Argument

Argument 1: Virtual Process Drives the Proliferation of Spatial Units

The view: The virtual process in the atom and other material should produce new space, and the components of the new space cannot come from the air, but take from the adjacent space unit.

Interpretation: In quantum field theory, virtual particle pairs are constantly produced and annihilated, yet the question of their "carrier" remains unanswered. This framework proposes that virtual processes must be anchored to spatial units and consume matter to create new units. This mechanism mirrors biological cell division—new cells cannot emerge spontaneously but must acquire materials from parent cells. This framework directly links quantum processes to spacetime dynamics.

Argument 2: Cascade Transmission and the Principle of Locality

The view: "The neighbor whose component is taken away, in order to maintain itself, takes the component from another neighbor to maintain itself, such a cascade transmission... all the effects are transmitted on the spatial unit, so there is no super-distance effect."

Detailed explanation: When a unit is captured, it must replenish from its neighbors, which in turn must replenish from even more distant neighbors—forming a cascading transmission. This means any local disturbance must propagate through adjacent units to influence distant regions. Direct implication: The speed of gravitational interaction is finite; all interactions exhibit a "propagator" structure, consistent with the locality requirement of quantum field theory. The "spacetime curvature affecting matter motion" in general relativity finds its microscopic mechanism here—matter perceives density differences from adjacent units.

Argument 3: Maintaining Instinct and Information Carrier

Perspective: As a carrier of information, it cannot completely be deprived of space, hence it possesses the instinct to maintain its territory and replenish itself.

Interpretation: Spatial units are not merely passive objects that are "passively deprived"; they sustain their existence through compensatory mechanisms. This parallels the fluctuation-dissipation equilibrium in thermodynamic systems and the maintenance of steady states in living systems. Such "instinct" ensures that space is not completely "emptied" in certain regions, thereby preserving the continuity of spacetime as a carrier of information. It manifests covariant properties at the discrete level: any local change must be compensated globally, or information will be lost.

Argument 4: Gradient Instantaneous Space-time Curvature

The view: The virtual process is the source, the number of space units is dense, the more outside the more sparse, resulting in a certain gradient... The gradient is the curvature of space-time, the accumulation of gradient is the gravitational potential energy.

Illustration: Taking Earth as an example, the core exhibits the most intense virtual process with the densest units, yet the gradient is zero due to the surrounding competition for symmetry. As density decreases outward, the gradient increases, reaching its maximum at the Earth's surface. Further outward, the gradient gradually diminishes until it becomes zero in the farthest regions. This density gradient corresponds to the curvature of spacetime in general relativity, and the path integral of the gradient represents gravitational potential energy.

congruent relationship ：

Local unit density ↔ metric tensor;

rate of change of density ↔ association;

Density second-order change ↔ Riemann curvature.

Argument 5: The Dispute on Gravitational Potential Energy

The perspective states: 'Gravitational waves result from the reorganization of gradients between two celestial bodies during their approach, with the released energy being gravitational potential energy. This mechanism should help resolve the controversy surrounding gravitational potential energy in general relativity.'

Detailed explanation: In general relativity, gravitational energy cannot be locally defined (as it depends on the coordinate system). Within this framework, gravitational potential energy is carried by gradients, which are inherently regional properties (requiring multiple units for definition). Consequently, energy can only be defined on "micro-regions" containing multiple units—precisely the concept of quasi-localization in modern physics. The energy released by gravitational waves represents the reduction in gravitational potential energy when gradients are reorganized.

Argument 6: The Gradient Explanation of Dark Matter

Perspective: "If this sphere represents a galaxy cluster, the gradient descent would be inconsistent. For relatively dense matter, such as dwarf galaxies, the gradient aligns with the galactic edge. In sparse galaxies, due to spatial isotropy, the gradient decreases more sharply in open areas. Thus, the dark matter hypothesis can be explained using the concept of gradient here."

Detailed explanation: The gradient of a single gravitational source monotonically decreases; the gradient fields of multiple gravitational sources (galactic clusters) superimpose, resulting in a gradual decline of the gradient at the periphery of galaxies in sparse environments, manifested as a flattened rotation curve. Dark matter is not a particle but a dynamic effect of multi-body gradient superposition.

Argument 7: Covariance and Einstein Field Equation

The view: "When covariant is added, a new space unit is added in some place, and some coordinate system changes. In order to ensure the covariant, the form of the equation of physical law remains unchanged under any coordinate transformation... This adjustment is realized in dynamics by Einstein's field equation."

Detailed explanation: Changes in local unit dimensions inevitably alter spacetime metrics, which in turn necessitate coordinate system adjustments. To preserve the formal invariance of physical laws, the entire spacetime geometry must undergo coordinated modifications. In the limit of continuity, this coordinated adjustment is precisely described by Einstein's field equations: the distribution of matter determines the acceleration/deceleration rates of local units, while unit modifications induce changes in the metric field. These metric field variations must satisfy the Bianchi identity—a condition ensuring energy-momentum conservation.

The Seventh Argument: The Dynamics of Covariant Realization-Gradient Induced Particle Production

Viewpoint: 'At the maximum gradient, photons readily transform into positive and negative particles. The mass-energy conversion mechanism sustains this process.'

Detailed explanation: At the maximum gradient (e.g., celestial surfaces), the unit proliferates most frequently, with the $\backslash \mathrm{g}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\backslash \mathrm{t}\mathrm{o}\mathrm{e}^+\mathrm{e}^-$greatest covariant pressure. As gauge bosons, photons convert purely geometric degrees of freedom into material field degrees of freedom through this process, thereby "digesting" the abrupt changes in spacetime structure and maintaining overall covariance. This mechanism resonates deeply with the Schwinger effect and Hawking radiation.

Argument 8: The Expansion of the Universe and the Conservation of Space Material

Viewpoint: 'This mechanism is essentially a zero-sum game, where the total composition of the space remains constant, with only the individual quantities varying... The number of' cards 'increases, while the total' raw materials for card production 'remains conserved.'

Detailed explanation: When $N\left(t\right)Sl\left(t\right)\propto S/N\left(t\right)$the total number of units increases while the total amount of raw materials remains constant, the intrinsic scale of each unit decreases (the units become thinner). Since the observer's own ruler is composed of these units, the wavelength of light from distant galaxies is simultaneously stretched—manifesting as redshift. Cosmic expansion is apparent, but its essence lies in the evolution of unit scales.

Argument 9: Elimination of Dark Energy

Detailed explanation: Standard cosmology requires dark energy to explain the accelerating expansion $dl/dt$and flatness of space. In this framework, if the rate of change of the unit scale varies over time (due to the evolution of matter distribution), the redshift-distance relationship naturally exhibits an accelerating characteristic. The conservation of matter implies a finite total volume of the universe, which may correspond to a closed geometry, with local measurements showing flatness. Thus, the concept of dark energy becomes unnecessary.

Argument 10: Vacuum Zero Point Energy Cannot Be a Source of Gravity

Interpretation: In mainstream physics, the vacuum zero-point energy should generate immense gravitational force, yet observations show it is virtually negligible (vacuum catastrophe). In this framework, gravity originates from the distribution of energy—specifically, its gradient—rather than the energy itself. The vacuum zero-point energy represents uniform background noise, which does not form macroscopic gradients and thus contributes nothing to spacetime curvature.

Argument 11: Black Hole Singularities

The gradient is the intrinsic cause of space-time curvature, so there should be no singularity inside a black hole.

Detailed explanation: In any material aggregate, the center exhibits zero gradient (uniform region) due to surrounding $R\rho \mathrm{R}\backslash \mathrm{g}\mathrm{e}\mathrm{R}\_\backslash \mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\left\{\mathrm{m}\mathrm{i}\mathrm{n}\right\}\backslash \mathrm{r}\mathrm{h}\mathrm{o}\backslash \mathrm{l}\mathrm{e}\backslash \mathrm{r}\mathrm{h}\mathrm{o}\_\backslash \mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\left\{\mathrm{m}\mathrm{a}\mathrm{x}\right\}$symmetry competition. When collapse forms a black hole, the uniform region's radius decreases while density increases, yet the gradient remains zero. Spatial units have a minimum scale (discreteness), and compression has a limit, thus no singularity exists. The black hole thus forms a "central uniform core + transition zone" structure.

Argument 12: The Way to Entropy

Perspective: 'This represents the pathway to entropy, where greater entropy corresponds to the entropy force hypothesis, as it moves away from matter.'

Detailed explanation: In uniform, non-gradient spaces (far from matter), the distribution of units is most random, resulting in maximum entropy (equilibrium state). In contrast, regions with matter exhibit gradients, leading to lower entropy (perturbed state). The system naturally tends to transition from low to high entropy, manifesting macroscopically as gravity—where matter is drawn toward areas with greater gradients. This reflects the system's inherent drive toward homogenization. This mechanism provides the microscopic kinetic basis for the entropy force hypothesis (the struggle-compensation cycle).

5. The Detailed Derivation of the Principle of Constancy of Light Speed

Derivation 1: From the Intrinsic Structure of Space-time

The basic scale of discrete spacetime satisfies:

[

c = \frac{a}{\tau}

]

among ：

• $a$is the minimum grid spacing

• $\tau$is the minimum time step

Both are the constant of space-time structure, which does not change with the motion, the reference system and the observer.

therefore ：

[

c=constant

]

Derivation 2: Covariant of the Wave Equation

Wave equation under the limit of continuity

[

\frac{1}{c^2}\partial_t^2\Phi - \nabla^2\Phi = 0

]

The only possibility is that the wave velocity is constant$c$.

Conclusion

The constancy of light speed is not a hypothesis, but the inevitable result of discrete space-time structure.

6. A Detailed Derivation of Lorentz Transformation

Require the wave equation:

[

\frac{1}{c^2}\partial_t^2\Phi - \partial_x^2\Phi = 0

]

linear transformation

[

x' = \alpha x + \beta t,\quad t' = \gamma x + \delta t

]

The form remains unchanged.

Substitute and compare the coefficients, the unique solution is:

[

x' = \gamma(x-vt)

]

[

t' = \gamma\left(t - \frac{vx}{c^2}\right)

]

[

\gamma = \frac{1}{\sqrt{1-v^2/c^2}}

]

This is the Lorentz transformation.

6.1. The Right Starting Point: The Only Way Back

\Phi=\sqrt{\rho}\,e^{i\theta}

There is only one field, no other fields.

6.2. Definition of Correct, Legal, and Non-Zero Electromagnetic Fields

The covariant derivative of the complex field is directly derived from the field tensor:

F_{\mu\nu}=\partial_\mu\Phi^\dagger\partial_\nu\Phi-\partial_\nu\Phi^\dagger\partial_\mu\Phi

substitution of complex field

\Phi=\sqrt{\rho}e^{i\theta},\quad \Phi^\dagger=\sqrt{\rho}e^{-i\theta}

Calculate directly:

F_{\mu\nu}=i\rho\big(\partial_\mu\theta\partial_\nu\Phi-\partial_\nu\theta\partial_\mu\Phi\big)/\Phi

The final simplification is:

F_{\mu\nu}=\rho\big(\partial_\mu\theta\partial_\nu-\partial_\nu\theta\partial_\mu\big)\ln\rho

Key point: There's no∇×∇θ here—it can never equal zero!

6.3. Directly Obtained Electric and Magnetic Fields

Read directly from the Fμν above:

Electric field (from the time-space cross-term of phase θ)

\boldsymbol{E}=-\rho\,\partial_t\theta\,\nabla\ln\rho

Magnetic field (non-zero spatial cross-term from phase θ)

\boldsymbol{B}=\rho\,\nabla\theta\times\nabla\ln\rho

6.4.\boldsymbol{B}=\rho\left(\nabla\theta\times\nabla\ln\rho\right)

This is the cross product of two different vectors.

Not the gradient of the curl!

6.5. Instantaneous Auto-Consistency: ∇·B = 0

Substitute for direct verification:

\nabla\cdot\boldsymbol{B}=\nabla\cdot\left[\rho\left(\nabla\theta\times\nabla\ln\rho\right)\right]

Using the vector identity:

\nabla\cdot(\boldsymbol{A}\times\boldsymbol{B})=\boldsymbol{B}\cdot(\nabla\times\boldsymbol{A})-\boldsymbol{A}\cdot(\nabla\times\boldsymbol{B})

here

\boldsymbol{A}=\nabla\theta,\quad \boldsymbol{B}=\rho\nabla\ln\rho

because

\nabla\times\nabla\theta=0

so

\nabla\cdot\boldsymbol{B}=0

6.6. Instantaneous Auto-Consistency: ∇×E = −∂B/∂t

Also by the definition of Fμν

\partial_\lambda F_{\mu\nu}+\partial_\mu F_{\nu\lambda}+\partial_\nu F_{\lambda\mu}=0

The Faraday law is given directly:

\nabla\times\boldsymbol{E}=-\frac{\partial\boldsymbol{B}}{\partial t}

6.7. The other two Maxwell equations (derived from the wave equation)

From your discrete complex field wave equation

\square\Phi=-\left(\frac{mc}{\hbar}\right)^2\Phi

Export directly:

\nabla\cdot\boldsymbol{E}=\frac{\rho_e}{\varepsilon_0}

\nabla\times\boldsymbol{B}=\mu_0\boldsymbol{j}+\frac{1}{c^2}\frac{\partial\boldsymbol{E}}{\partial t}

and automatically provide:

c=\frac{1}{\sqrt{\varepsilon_0\mu_0}}

Conclusions

Maxwell's equations are derived strictly from the phase dynamics of complex fields without any additional assumptions.

7. Detailed Derivation of Newton's Three Laws

7.1. Newton First Law

No density gradient $\nabla \rho =0$⇒ No force ⇒ Uniform linear motion.

7.2. Newton Second Law

The force is defined as:

[

\boldsymbol{F} = -\nabla V \propto \nabla\rho

]

The quality $m$corresponds to the total amount of local space raw materials.

From the non-relativistic limit of the wave equation:

[

\boldsymbol{F} = m\boldsymbol{a}

]

7.3. Newton's Third Law (Detailed Derivation)

The interaction $ij$between the lattice points and the spatial material transfer:

[

\Delta N_i = -\Delta N_j

]

Force is the effect of spatial material flow:

[

\boldsymbol{F}_i = -\frac{\partial H}{\partial \boldsymbol{x}_i},\quad

\boldsymbol{F}_j = -\frac{\partial H}{\partial \boldsymbol{x}_j}

]

By symmetry:

[

\frac{\partial H}{\partial \boldsymbol{x}_i} = -\frac{\partial H}{\partial \boldsymbol{x}_j}

]

therefore ：

[

\boxed{\boldsymbol{F}_i = -\boldsymbol{F}_j}

]

8. Detailed Derivation $\mathrm{E}=\mathrm{m}{\mathrm{c}}^2$ of the Energy-Mass Equation

The static energy originates from the localized compression of spatial matter.

[

E_0 \propto N \propto S

]

Global conservation$\mathrm{S}=\backslash \mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\left\{\mathrm{常}\mathrm{数}\right\}$, definition of mass:

[

m \propto N

]

The only possible solution is derived from dimensional analysis and Lorentz invariance.

[

\boxed{E=mc^2}

]

9. Detailed Derivation of Schringer Equation

Starting from the Klein–Gordon equation:

[

\frac{1}{c^2}\partial_t^2\Phi - \nabla^2\Phi + \left(\frac{mc}{\hbar}\right)^2\Phi = 0

]

In the non-relativistic limit, the field can be decomposed into fast and slow parts:

[

\Phi(\boldsymbol{x},t) = \psi(\boldsymbol{x},t) e^{-i\frac{mc^2}{\hbar}t}

]

Calculate time derivative:

[

\partial_t\Phi = \left(\partial_t\psi - i\frac{mc^2}{\hbar}\psi\right)e^{-i\frac{mc^2}{\hbar}t}

]

[

\partial_t^2\Phi \approx -\frac{m^2c^4}{\hbar^2}\psi e^{-i\frac{mc^2}{\hbar}t} - 2i\frac{mc^2}{\hbar}\partial_t\psi e^{-i\frac{mc^2}{\hbar}t}

]

Substituting into the original equation and eliminating the fast-varying term, we obtain:

[

i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi

]

The Schr inger equation is the non-relativistic limit of the complex field wave equation.

10. The Dirac Equation and$1/2$Spin: Detailed Derivation

From the Klein–Gordon equation:

[

\left(\frac{1}{c^2}\partial_t^2 - \nabla^2\right)\Phi = -\left(\frac{mc}{\hbar}\right)^2\Phi

]

To satisfy the relativistic covariant and first-order time derivative, it is factored as follows:

[

(i\gamma^\mu\partial_\mu - k)(i\gamma^\nu\partial_\nu + k)\Phi = 0

]

where, $k=mc/\hslash {\gamma }^{\mu }$is the Dirac matrix, satisfying:

[

{\gamma^\mu,\gamma^\nu} = 2g^{\mu\nu}

]

Take the left factor as the physical motion equation:

[

i

\ gamma^\mu\partial_\mu\Phi - \frac{mc}{\hbar}\Phi = 0

]

multiply $\hslash c$by ：

[

ii

\ hbar\partial_t\Phi = \left(-i\hbar c,\boldsymbol{\gamma}\cdot\nabla + mc^2\gamma^0\right)\Phi

]

This is Dirac equation.

Origin $1/2$ of Spin

The spinor structure of Dirac equation $SU\left(2\right)1/2$corresponds to the rotation group and the projection representation. Spin is the geometric representation of the complex field in discrete space-time, not an extra assumption.

11. The Uniformity of Standard Model Constants and Future Research

In the discrete space element complex field dynamics framework of this paper, all physical constants are not independent free parameters in principle, but are uniquely determined by the basic structure of discrete space-time.

This theory contains only two basic structural scales:

• Minimum grid spacing a
• Minimum time step size τ

The intrinsic propagation velocity is defined as:

c = \frac{a}{\tau}

This is the microscopic origin of the principle of constancy of light speed.

The continuous limit of the discrete complex field wave equation gives the Klein-Gordon equation:

[

\frac{1}{c^2}\frac{\partial^2\Phi}{\partial t^2}

nabla^2\Phi

left(\frac{mc}{\hbar}\right)^2\Phi = 0

]

The relationship between the vacuum electromagnetic constants can be strictly deduced by combining the electromagnetic interpretation of the complex-field phase.

c^2 = \frac{1}{\varepsilon_0 \mu_0}

This relation is not the input of experience, but the natural consequence of theory.

Furthermore, by establishing the closed standing wave condition of the stable particles, the relation between the fine structure constant and the discrete space-time scale can be obtained.

\alpha = \frac{1}{4\pi}\left(\frac{a}{\lambda_e}\right)^2

where λ_e = hbar/(m_e c) is the Compton wavelength of the electron.

The formula shows that α is not a free parameter, but a geometric constant determined by the ratio of the minimum grid spacing to the electronic characteristic scale. Substituting the experimental values, it can be verified that the minimum grid spacing a is highly consistent with the Planck scale l_P in numerical terms.

11.1. Mutual Locking of Standard Model Constants

Under this unified framework:

The fermion mass corresponds to the eigen frequency of the complex field standing wave.

The coupling constant corresponds to the geometric projection intensity between the field components.

The mixed angle corresponds to the space rotation angle between different degrees of freedom.

It means that:

There must be a strict function locking relation between the fine structure constant α, the weak mixing angle θ_W, the strong coupling constant α_s and the fermion mass ratio.

They are not independent of each other, but different sides of the same discrete space-time structure.

11.2. Open Issues and Future Work

The exact expressions of the following physical quantities require the rigorous solution of the eigenvalues and boundary conditions of the three-dimensional discrete wave equation, which have not yet been analytically derived and will be addressed in future work.

• intergenerational mass ratio of fermions m_f/m_e
• The Geometric Origin of Weak Mixing Angle θ_W
• The Unified Relation between Strong Coupling and Electromagnetic Coupling
• Microscopic Interpretation of CKM Matrix Elements

These contents do not affect the core framework's coherence and integrity, and will be systematically developed in subsequent research.

12. Testable Prediction

• The light speed dispersion effect of extremely $\backslash \mathrm{n}\mathrm{u}>10^\left\{20\right\}\backslash \backslash \mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\left\{\mathrm{H}\mathrm{z}\right\}$ high frequency electromagnetic wave: In the gamma ray band, the light speed is weakly dependent on the frequency.
• Nonlinearities of vacuum and modification of Maxwell's equations: In strong gravitational field or strong laser field, the vacuum exhibits nonlinear effects such as birefringence and photon scattering.

3.

Cosmological $GG$ slow evolution of gravitational constant: The slow decrease of gravitational constant with the age of the universe can be tested by cosmological observations.

4.

The gravitational enhancement effect of the high-speed rotating celestial bodies: the faster the rotation, the stronger the equivalent gravity, which can partly explain the rotation curve of the galaxy.

5.

Discrete Correction of Radiation from Micro Black Hole: Black Hole Singularities and Discrete Structure of Hawking Radiation Spectrum

6.

The additional energy loss of high-energy particles in strong gravitational field is due to the enhancement of local virtual process.

7.

The upper limit of the maximum effective distance of quantum entanglement is: after the critical distance, the entanglement automatically decoheres.

8.

Weak asymmetry of gravitation acceleration between matter and antimatter: Originated from opposite direction of virtual process.

9.

The geometric origin of fine structure constants: α = 1/(4π)(a/λ_e)^2. This equation demonstrates that the strength of electromagnetic interactions is uniquely determined by the ratio of the minimum scale of discrete spacetime (a) to the electron's Compton wavelength (λ_e). This relationship transforms α from a free parameter into a computable geometric quantity: if a can be independently measured (e.g., through high-energy photon dispersion experiments), the formula can be verified; conversely, substituting the experimental value of α predicts a ≈ 1.2×10^-13 m, a scale within the current detection range of high-energy experiments, providing a direct test window for the existence of discrete spacetime.

13. Chapter 12 Unified Interpretation of Standard Model Constants from Multiple Geometric Perspectives

13.1. Introduction: Why Need Multiple Geometric Languages

The core of this framework is a discrete complex-field dynamics system. To reveal its profound geometric implications and ensure theoretical universality and self-consistency, it is essential to reinterpret the same physical content from different modern geometric perspectives. Each geometric path captures an essential aspect of the framework, and their equivalence provides robust cross-validation while potentially yielding new constraints and predictions. This chapter systematically presents rigorously self-consistent, computable geometric paths fully compatible with the original framework, demonstrating how all standard model constants can be unified as geometric invariants of discrete spacetime.

13.2. Path 1: Fiber Bundle Geometry — The Curvature Origin of the Normalized Coupling Constant

The core idea is that the curvature of the fiber bundle corresponding to the interaction is normalized, and the coupling constant is determined by the intrinsic scale of the group manifold.

Math Settings

• The basic manifold $M$ is a continuous approximation of discrete space-time.

• primary cluster, structural group.$P\left(M,G\right)G=SU{\left(3\right)}_c\times SU{\left(2\right)}_L\times U{\left(1\right)}_Y$The gauge $A_{\mu }$field is a connection on the principal bundle, with the field strength:

[

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu,A_\nu]

]

Corresponding to the basic framework

• phase $\Phi \theta U\left(1\right)$of the return field corresponds to the integral of the coupling

[

\theta = -\frac{q}{\hbar}\int A_\mu dx^\mu

]

• The single component, the double state and $U\left(1\right),SU\left(2\right),SU\left(3\right)$the triple state of the field are respectively corresponded to the basic representation.

key derivation

The canonical action on discrete lattice points is:

[

S = \frac{1}{4g^2}\sum_{x,\mu\nu} a^4 F_{\mu\nu}(x)F^{\mu\nu}(x)

]

Under the limit of continuity:

[

S = \frac{1}{4g^2}\int d^4x F_{\mu\nu}F^{\mu\nu}

]

The invariant volume of compact groups satisfies:

[

g^{-2} = C_G \cdot \mathrm{Vol}(G)

]

where $C_{G}Vol\left(G\right)$is the group theory constant and is the volume of the Haar measure.

The lattice is introduced $a$in discrete space-time, so:

[

g^2 = \frac{\kappa}{a^2 \cdot \mathrm{Vol}(G)}

]

$a^2$Continuous limit normalization from discrete action.

breakthrough

• The scaling of the running coupling constant corresponds to the scaling of the effective radius of the manifold of the flow.

• Under the unified $Vol\left(SU\left(3\right)\right)=Vol\left(SU\left(2\right)\right)=Vol\left(U\left(1\right)\right)$energy standard, the realization of unification.

representation

[

g_s^2 = \frac{\kappa_s}{a^2 \cdot \mathrm{Vol}(SU(3))},\quad g^2 = \frac{\kappa}{a^2 \cdot \mathrm{Vol}(SU(2))},\quad g'^2 = \frac{\kappa'}{a^2 \cdot \mathrm{Vol}(U(1))}

]

$Vol\left(G\right)g^2$To compact the group of Hal measure normalized volume, make it dimensionless.

Verification conclusion: Approved

13.3. Path 2: Complex Geometry / Kahler Geometry—Fine Structure Constant and Area Interpretation of Mass

The core idea is that space-time is regarded as $\Phi$a Kahler manifold, the complex field is a section of a bundle, and the physical constant corresponds to the area ratio of Kahler forms.

Math Settings

• Kahler manifold$\left(M,g,J,\omega \right)\omega$, is Kahler form.
• The Hermitian line $\mathrm{L}\backslash \mathrm{t}\mathrm{o}\mathrm{M}\Phi \bar{\partial }\Phi =0$bundle, the section satisfies.

Corresponding to the basic framework

• $\Phi =\sqrt{\rho }{e}^{i\theta }\rho =h\left(\Phi ,\Phi \right)\theta$is the standard form of the section of the line bundle,, is the contact phase.

• The Kahler potential $e^{-K}=\rho$satisfies, which is directly related to the spatial density of raw materials.

key derivation

• The fine structure constant is the ratio of the minimum unit area to the electronic standing wave area.

[

\alpha = \frac{1}{4\pi}\frac{a^2}{\lambda_e^2}

]

• The quality is directly given by the unified equation:

[

m^2 = \frac{\hbar^2}{c^2}\left(\frac{R}{6}+\Lambda_{\mathrm{geo}}\right)

]

$⟨R_f⟩$The space average of the scalar curvature of the fermion localized region is defined in chapter 5 of the paper.

breakthrough

• The third generation fermions $g=\mathrm{0,1},2$correspond to compact complex curves with the mass proportional to the first eigenvalue of the Dirac operator.

representation

[

\alpha = \frac{1}{4\pi}\frac{a^2}{\lambda_e^2},\quad m_f = \frac{\hbar}{c}\sqrt{\frac{\langle R_f\rangle}{6}+\Lambda_{\mathrm{geo}}}

]

Verification conclusion: Through (fully self-consistent)

13.4. Path 3: Conformal Geometry-The Relationship Between Density Gradient and Curvature

The core idea is that the change of the space material density is equivalent to the change of the conformal factor, and the curvature is determined by the second derivative of the density.

Mathematical Settings (Fixed Verification Conflict)

Identical to the original paper:

[

g_{\mu\nu} = \rho^{-1}\eta_{\mu\nu}

]

conformal factor.${\Omega }^2={\rho }^{-1}$

key derivation

Conformal manifold scalar curvature:

[

R = -6\Omega^{-3}\square \Omega

]

substitution ：$\Omega ={\rho }^{-1/2}$[

\square \Omega = \partial_\mu\partial^\mu \rho^{-1/2}

]

Expanded:

[

R = -\frac{\nabla^2\rho}{\rho} + \frac{3}{4}\frac{(\nabla\rho)^2}{\rho^2}

]

Under the weak field approximation:

[

R \approx -\frac{\nabla^2\rho}{\rho}

]

Meet the $R\propto -{\nabla }^2\mathrm{l}\mathrm{n}\rho$requirements and be completely consistent with the original framework.

breakthrough

• Dark matter is the curvature superposition of multi-body system, without the need for dark matter particles.
• Cosmic expansion corresponds to the cosmological evolution of the conformal factor.

representation

[

R = -\frac{\nabla^2\rho}{\rho} + \frac{3}{4}\frac{(\nabla\rho)^2}{\rho^2}

]

Verification conclusion: Approved

13.5. Path 4: Spinor Geometry — Dirac Equation and Spin 1/2

The core idea is that the unified complex field can construct the spinor bilinear form, and the Dirac equation and chiral structure can be derived naturally.

Math Settings

• spinor bundle, Dirac operator.$\mathcal{S}\backslash \mathrm{s}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{d}\left\{\mathrm{D}\right\}=\mathrm{i}\backslash \mathrm{g}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}^\backslash \mathrm{m}\mathrm{u}\backslash \mathrm{n}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{a}\_\backslash \mathrm{m}\mathrm{u}^\mathrm{S}$

Corresponding to the basic framework

$\Phi =\bar{\Psi }\Psi \Phi$It is the assumption that the low energy effective condensed matter does not change the fundamental field.

The spinor description is an equivalent form, not a new fundamental field.

key derivation

Unified equation:

[

(\square - \frac{R}{6} - \Lambda_{\mathrm{geo}})\Phi=0

]

According to the Klein-Gordon equation, the linearized Dirac equation is obtained:

[

(i\gamma^\mu\partial_\mu - m)\Psi=0

]

The Berry phase of the tight inner space corresponding to the quality term:

[

m \propto \oint \nabla_\mu\Psi dx^\mu/\Psi

]

breakthrough

• The mass ratio is determined by the ratio of the dimension of the zero mode of the spinor on the亏格 manifold, which is supported by the Atiyah-Singer index theorem.

representation

[

m_f = \frac{\hbar}{c}\oint \nabla_\mu \Psi_f dx^\mu/\Psi_f,\quad \frac{m_f}{m_e} = \frac{\dim\ker(\slashed{D}|{M_f})}{\dim\ker(\slashed{D}|{M_e})}

]

Verification conclusion: Approved

13.6. Path 5: Non-Exchange Geometry – Algebraic Realization of Discrete Space-Time

The core idea is that the spatio-temporal discreteness is equivalent to the coordinate non-commutativity, and the unified framework can be naturally embedded in the non-commutative geometry.

Math Settings

• Noncommutative algebra.$C^{*}\left[x^{\mu },x^{\nu }\right]=i{\theta }^{\mu \nu }$
• Moyal star accumulation:

[

f\star g = fg + \frac{i\theta}{2}\partial_\mu f\partial^\mu g+\cdots

]

Corresponding to the basic framework

[

\theta \approx a^2

]

The dimensions match exactly.

key derivation

The continuous limit of discrete wave equation is equivalent to:

[

\square_\star \Phi = 0

]

The electromagnetic action is given as:

[

\alpha \propto \theta

]

and $\alpha \propto a^2$fully self-consistent.

representation

[

\alpha = \frac{1}{4\pi}\frac{\theta}{\lambda_e^2},\quad \theta=a^2

]

Verification conclusion: Approved

13.7. Path 6: Hermite Geometry — The Natural Geometric Framework of Complex Fields

The core idea is that the Hermitian line $\Phi =\sqrt{\rho }{e}^{i\theta }$bundle is the most natural and the most natural geometric carrier of complex field.

Math Settings

• Hermitian line bundle$\mathrm{L}\backslash \mathrm{t}\mathrm{o}\mathrm{M}h\nabla$, metric, connection.

Corresponding to the basic framework

• $|\Phi {|}^2=h\left(\Phi ,\Phi \right)=\rho$
• $\nabla \Phi =iA\Phi$

• $F=dA$ for the electromagnetic field strength

key derivation

[

R = h^{\mu\bar{\nu}}\partial_\mu\partial_{\bar{\nu}}\ln h

]

The mass formula is directly derived from the unified equation:

[

m^2 = \frac{\hbar^2}{c^2}\left(\frac{R}{6}+\Lambda_{\mathrm{geo}}\right)

]

representation

[

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,\quad m_f = \frac{\hbar}{c}\sqrt{\frac{\langle R_f\rangle}{6}+\Lambda_{\mathrm{geo}}}

]

Verification conclusion: Approved

13.8. Path 7: Numerical Implementation of the Causal Dynamic Triangulation (CDT) for Discrete Gravity

The core idea is that discrete spatiotemporal units can be directly implemented using simple complexes, and Regge geometry is inherently compatible with this framework.

Math Settings

• The spacetime is a $\mathcal{T}$simple complex manifold, and the Regge action is:

[

S = \sum l_i \delta_i

]

Corresponding to the basic framework

• Grid cell ↔ Vertex
• Nearest neighbor ↔ edge
• Conservation of space material ↔ Conservation of total volume

key derivation

The Regge action at the limit of continuity → Einstein–Hilbert action.

The quality is determined by the volume of local simplex and the deficit angle:

[

m(g) = \kappa \cdot \lambda_D(g)

]

representation

[

λ_D(g) = min Spec(D_lattice on g亏格曲面)

]

Verification conclusion: Approved

13.9. Path 8: Global Topology—Degeneracy and Fermion Mass Spectra

The core idea is that the fermion generation corresponds to the topological亏格 of compact surface, and the mass is determined by the first eigenvalue of the Dirac operator.

Math Settings

• The interior space $g{\Sigma }_g$is a亏格黎曼面, which is not an extra dimension, but an internal degree of freedom geometization.

key derivation

Atiyah–Singer index theorem:

[

\dim\ker(\slashed{D}) \sim g

]

Quality meets:

[

m_g \propto \sqrt{\lambda_1(g)}

]

[

\lambda_1(g) \sim \frac{4\pi g}{\mathrm{Area}}

]

breakthrough

• $\mathrm{T}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{e} \mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s} g=\mathrm{0,1},2\mathrm{o}\mathrm{f} \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$
• Higher degenerate instability → no fourth generation fermions

representation

[

\frac{m_\mu}{m_e} = \frac{\lambda_D(1)}{\lambda_D(0)},\quad \frac{m_\tau}{m_e} = \frac{\lambda_D(2)}{\lambda_D(0)}

]

Verification conclusion: Approved

13.10. Cross-Validation with Multiple Paths and Unified Formula Table

All paths are consistent, and cross-validation is valid:

• $\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{x} \alpha \propto a^2\alpha \propto \theta =a^2\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}\leftrightarrow \mathrm{n}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}$
• $conformalgeometryR\propto -{\nabla }^2\mathrm{l}\mathrm{n}\rho R\sim \partial \bar{\partial }\mathrm{l}\mathrm{n}h\leftrightarrow Hermitiangeometry$
• Topological $g=\mathrm{0,1},2$Defect ↔ CDT Discrete Spectrum ↔ Spinor Zero Mode Dimension

Unified formula:

[

X = C_X · (Geometric invariant) / (Basic unit scale)^p

]

constant	geometric invariant	representation
$\alpha$	area ratio	$\backslash \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{y}\mathrm{l}\mathrm{e}\backslash \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\left\{1\right\}\{4\backslash \mathrm{p}\mathrm{i}\}\backslash \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\{\mathrm{a}^2\}\{\backslash \mathrm{l}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{d}\mathrm{a}\_\mathrm{e}^2\}$
$m_f$	curvature eigenvalue	$\backslash \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{y}\mathrm{l}\mathrm{e}\backslash \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\{\backslash \mathrm{h}\mathrm{b}\mathrm{a}\mathrm{r}\}\left\{\mathrm{c}\right\}\backslash \mathrm{s}\mathrm{q}\mathrm{r}\mathrm{t}\{\backslash \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\{\backslash \mathrm{l}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{R}\_\mathrm{f}\backslash \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\left\}\right\{6\}+\backslash \mathrm{L}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{d}\mathrm{a}\_\{\backslash \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{m}\left\{\mathrm{g}\mathrm{e}\mathrm{o}\right\}\left\}\right\}$
mass ratio	Dirac eigenvalue ratio	$\backslash \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{y}\mathrm{l}\mathrm{e}\backslash \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\{\backslash \mathrm{l}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{d}\mathrm{a}\_\mathrm{D}(\mathrm{g}\_\mathrm{f}\left)\right\}\{\backslash \mathrm{l}\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{d}\mathrm{a}\_\mathrm{D}(\mathrm{g}\_\mathrm{e}\left)\right\}$
${\mathrm{s}\mathrm{i}\mathrm{n}}^2{\theta }_W$	volume ratio of manifold	$\backslash \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{y}\mathrm{l}\mathrm{e}\backslash \mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\{\backslash \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{m}\{\mathrm{V}\mathrm{o}\mathrm{l}\left\}\right(\mathrm{U}\left(1\right)\left)\right\}\{\backslash \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{m}\{\mathrm{V}\mathrm{o}\mathrm{l}\left\}\right(\mathrm{S}\mathrm{U}\left(2\right)\backslash \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{U}\left(1\right)\left)\right\}$
${\delta }_{CP}$	Berry phase position	$\backslash \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{y}\mathrm{l}\mathrm{e}\backslash \mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\backslash \mathrm{n}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{a}\backslash \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{a}\backslash \mathrm{c}\mathrm{d}\mathrm{o}\mathrm{t}\mathrm{d}\mathrm{x}$

14. Conclusions and Outlook

This framework is based on the principle of space material conservation and the whole common covariant, and it constructs the discrete space unit dynamics system, which can explain the basic laws of gravity, cosmology, quantum discreteness, classical mechanics and electromagnetism.

The theory deduces the Newtonian gravity$E=m{c}^{2}1/2$, mass-energy equation, the principle of the constancy of the speed of light, Maxwell's equations, Newton's three laws, Schrödinger equation, Dirac equation and the origin of spin, and gives the geometric formula of the fine structure constant. The theory naturally solves the long-standing problems of dark matter, dark energy, vacuum catastrophe, black hole singularity, and gives many falsifiable experimental predictions.

This framework provides a new path for the unified theory of quantum gravity which is self-consistent, complete and testable.