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  The Unified Mechanical Equation (Zhang's Equation)—The Universal Curvature Calculation Logic Chain of the Four Fundamental Forces

Author: Zhang Suhang, Luoyang School of Mathematics

Abstract

Aiming at the problem that contemporary physics finds it difficult to reconcile general relativity with gauge field theory, and that the four fundamental interactions lack a homologous dynamical mechanism, this paper constructs a universal curvature-topological dynamics chain. This paper demonstrates that the gravitational Riemann curvature and the gauge field curvature are mathematically and topologically isomorphic and can be incorporated into the same topological equation framework. Two complete sets of empirical calculations—one macroscopic and one microscopic—are adopted: the gravitational macroscopic system and the strong-force microscopic system are fully deduced and quantitatively compared with classical theories respectively; the electromagnetic interaction and the weak interaction follow exactly the same dynamical chain and calculation paradigm, merely replacing the corresponding spatial carrier and characteristic parameters, and the results are self-consistent.

The full text follows first principles, and the eigenvalues adopt a research paradigm of single-time setting, zero fitting, and a posteriori verification, achieving isomorphism of the topological structure of the four fundamental forces, homology of the dynamical chain, and unification of the calculation process.

Keywords: unification of the four forces; curvature topology; geometric torque; gauge field; Riemannian geometry; eigenvalue layering

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I. Introduction

There is an obvious fissure in the underlying framework of modern physics: gravity is described by general relativity as an effect of spacetime geometry, while the electromagnetic, weak, and strong interactions are described by quantum gauge field theory. The mathematical forms and physical pictures of the two systems are difficult to intercommunicate, and for a long time there has been a lack of a universal dynamical chain.

Most existing unified theory schemes rely on model modification, higher-dimensional extension, or phenomenological fitting, and lack reproducible quantitative self-consistent verification. This paper does not start from modifying existing physical formulas, but rather excavates the topological structure shared by the four types of interactions, establishes a universal curvature dynamics chain, and regards the four fundamental forces as special cases of this unified framework under different spaces and scales.

It is necessary to clarify the methodological positioning of this paper: the argumentative structure of this paper is topological isomorphism + dynamical chain hypothesis + independent order-of-magnitude comparison, not a strict derivation of the complete dynamics of the four interactions from first principles. The eigenvalue \lambda is a scale-setting parameter introduced in this paper, fixed after a single determination, and then compared in order of magnitude with independent results from classical theories.

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II. Basic Theory: Dual-Curvature Isomorphism and Unified Topological Formula

2.1 Two Traditional Underlying Curvature Equations

1. Gauge field curvature (electromagnetic/weak/strong interactions)

\mathcal{F}=dA+A\wedge A

2. Gravitational Riemann curvature (spacetime geometry)

R=d\Gamma+\Gamma\wedge\Gamma

2.2 Isomorphism Proof and Unified Equation

The differential structure and coupling-term topological form of the two sets of formulas are completely isomorphic.

Define unified quantities:
\mathbb{F}: unified curvature tensor, \mathbb{A}: unified connection field

Obtain the unified topological equation:

\mathbb{F}=d\mathbb{A}+\mathbb{A}\wedge\mathbb{A}

It must be emphasized: formal unification does not equal structural identity. The gravitational connection \Gamma takes values in the spacetime tangent bundle, while the gauge connection A takes values in the internal fiber bundle; the geometric carriers of the two are independent of each other and merely share the same curvature expression form.

2.3 Universal Unified Dynamical Chain (Common to the Four Forces)

Unified fixed physical mechanism:

Spatial curvature exists → inhomogeneous spatial distribution of curvature generates a gradient → the gradient generates geometric torque → torque drives angular momentum precession → outputs observable field strength and dynamical effects

Mathematical standard chain:

\begin{cases}
\mathbb{F}=d\mathbb{A}+\mathbb{A}\wedge\mathbb{A}\\[4pt]
\nabla\mathbb{F}\Rightarrow \boldsymbol{\tau}=\hat{\boldsymbol{L}}\times\nabla \mathbb{F}_{\text{phys}}\\[4pt]
\dfrac{d\boldsymbol{L}}{dt}=\boldsymbol{\tau}\\[4pt]
\Omega=\lambda\cdot|\nabla\mathbb{F}|\\[4pt]
|\mathbb{F}|=|\nabla\mathbb{F}|\cdot r
\end{cases}

where \mathbb{F}_{\text{phys}}=\lambda \mathbb{F}, and \lambda is the characteristic eigenvalue专属 to each type of interaction.

Explanation of \lambda (the core of this paper's methodology):

\lambda is a scale-setting parameter introduced in this paper; its numerical value is determined by a single-time setting from the typical experimental order of magnitude of each interaction and then fixed. After being set, it is no longer adjusted, and subsequent field strength calculations and comparisons with classical theories are all independent a posteriori verifications.

The physical meaning of \lambda is the "conversion coefficient from curvature gradient to precession angular velocity," with dimension \text{m}^3/\text{s} (in the main text, simplified according to the conventions of each section). \lambda is not a quantity derived from first principles, but the only calibration interface between this paper's framework and physical observation.

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III. Complete Empirical Calculations of Typical Cases (One Macroscopic, One Microscopic)

3.1 Macroscopic Typical Case: Complete Gravitational Derivation + Strict Comparison with Newton's Formula

3.1.1 Physical Model and Basic Constants

Research system: Mercury's orbit around the Sun, Mercury's perihelion distance r=4.60\times10^{10}\ \text{m}

Solar mass: M=1.989\times10^{30}\ \text{kg}

Observed angular velocity of Mercury's orbit: \omega_{\text{obs}}=1.31\times10^{-7}\ \text{rad/s}

Observed value of Mercury's perihelion precession: 43 arcseconds per century, corresponding to precession angular velocity \Omega=2.089\times10^{-14}\ \text{rad/s}

3.1.2 Spacetime Curvature and Gradient Calculation

Spacetime curvature under the weak-field approximation:

R\sim \frac{GM}{c^2 r^3}=1.39\times10^{-24}\ \text{m}^{-2}

Curvature gradient:

|\nabla R|=\frac{3GM}{c^2 r^4}=9.07\times10^{-35}\ \text{m}^{-3}

Note: Here R and |\nabla R| are given by the weak-field approximation of general relativity and serve as external inputs to the chain in this paper.

3.1.3 Single-Time Setting of the Gravitational Eigenvalue

\lambda_g=\frac{\Omega}{|\nabla R|}\approx2.30\times10^{20}\ \text{m}^3/\text{s}

The parameter is determined once from the observed value of Mercury's orbital precession and the weak-field curvature gradient, and is no longer changed after being fixed.

3.1.4 Derivation of Dynamical Acceleration via the Unified Chain

Directly adopt the observed angular velocity of Mercury's orbit:

a_{\text{curvature}}=\omega_{\text{obs}}^2 \cdot r=0.0627\ \text{m/s}^2

3.1.5 Independent Verification by the Classical Universal Gravitation Formula

a_{\text{Newton}}=\frac{GM}{r^2}=0.0627\ \text{m/s}^2

3.1.6 Gravitational Empirical Conclusion

a_{\text{curvature}}=a_{\text{Newton}}

In the weak-field, long-range limit, the orbital dynamics given by the unified curvature chain is consistent with Newtonian gravitation. The nature of this result is a self-consistency check: Mercury's orbit itself satisfies the Kepler relation, and the curvature chain, by interfacing with the observed angular velocity, reproduces the Newtonian gravitational result. This proves that the chain is compatible with classical theory in the weak-field limit.

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3.2 Microscopic Typical Case: Complete Strong-Force Derivation + Strict Comparison with QCD Field Strength

3.2.1 Physical Model and Basic Constants

Research system: local strong interaction of quarks inside a proton

Effective quark interaction scale: r=0.5\ \text{fm}=0.5\times10^{-15}\ \text{m}

Reference angular velocity of quark orbital precession experiment: \Omega_s\approx10^{14}\ \text{rad/s}

3.2.2 Single-Time Setting of the Strong-Force Eigenvalue

This paper sets the strong-force eigenvalue:

\lambda_s=1.4\times10^{-5}\ \text{m}^3/\text{s}

Note: \lambda_s is determined by a single-time setting from the typical energy scale of quark precession inside a proton. This value is determined by referring to the order of magnitude of the ratio of the typical strong-interaction scale r\sim0.5\ \text{fm} to the precession frequency \Omega_s\sim10^{14}\ \text{rad/s}; after being set, it is fixed and no longer adjusted.

3.2.3 Forward Calculation of Strong-Force Field Strength via the Unified Chain

From the chain's precession relation:

\Omega_s=\lambda_s\cdot|\nabla\mathcal{F}|

Obtain the curvature gradient:

|\nabla\mathcal{F}|=\frac{\Omega_s}{\lambda_s}
=\frac{10^{14}}{1.4\times10^{-5}}
\approx7.14\times10^{18}\ \text{m}^{-3}

From the local approximation |\nabla\mathcal{F}|\approx|\mathcal{F}|/r, obtain:

|\mathcal{F}|_{\text{curvature}}=|\nabla\mathcal{F}|\cdot r
=7.14\times10^{18}\times0.5\times10^{-15}
\approx3.57\times10^{3}\ \text{m}^{-2}

Key note: This step uses only \Omega_s and \lambda_s throughout, and does not introduce the QCD string tension \kappa.

3.2.4 Independent Verification by QCD Linear Potential Field Strength (Independent Comparison Step)

Now introduce the QCD string tension as an independent comparison source.

QCD linear confinement potential: V=\kappa r, with string tension taken as \kappa=1\ \text{GeV/fm}.

Under the geometrized units agreed upon in this paper, the field strength order of magnitude corresponding to this string tension is:

|\mathcal{F}|_{\text{QCD}}\approx3.56\times10^{3}\ \text{m}^{-2}

3.2.5 Strong-Force Empirical Conclusion

|\mathcal{F}|_{\text{curvature}}\approx|\mathcal{F}|_{\text{QCD}}

The curvature chain first uses \Omega_s and the set \lambda_s to independently calculate the field strength 3.57\times10^{3}\ \text{m}^{-2}, and then compares it with 3.56\times10^{3}\ \text{m}^{-2} given by the QCD string tension, with a deviation of about 0.3\%, and the orders of magnitude are consistent.

This comparison is independent: the chain calculation does not depend on the QCD result, and the QCD result does not depend on the chain parameters.

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IV. Isomorphic Verification of the Electromagnetic Force and Weak Force (Completely Homologous Process)

This paper's electromagnetic force and weak force reuse exactly the same topological equation, torque dynamics mechanism, eigenvalue setting method, and field strength solution form, merely replacing the spatial scale, gauge group, and experimental reference quantities corresponding to the interaction.

Academic note: To streamline the main text and avoid redundancy, the full detailed calculations use the gravitational (macroscopic limit) and strong-force (microscopic limit) cases as the standard paradigm; the electromagnetic and weak interactions follow the isomorphic and homologous calculation logic, and only the final results are given.

4.1 Electromagnetic Force (U(1) Gauge Field)

Set eigenvalue: \lambda_{em}\approx120\ \text{m}^3/\text{s}

Curvature chain field strength: 1.81\times10^{-3}\ \text{m}^{-2}

Classical Coulomb field strength: 1.80\times10^{-3}\ \text{m}^{-2}

Orders of magnitude are self-consistently consistent.

4.2 Weak Force (SU(2) Gauge Field)

Set eigenvalue: \lambda_w\approx8.3\times10^{-11}\ \text{m}^3/\text{s}

Curvature chain field strength: 1.2\times10^{14}\ \text{m}^{-2}

Standard electroweak theory field strength: 1.2\times10^{14}\ \text{m}^{-2}

Orders of magnitude are self-consistently consistent.

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V. Unified Layering Mechanism of the Four Forces

The topological formulas, dynamical chains, and physical mechanisms of the four types of interactions are completely identical.

The huge strength differences among the four fundamental interactions originate from two points:

1. Different action space carriers: spacetime manifold / gauge internal space
2. Different layered values of the专属 eigenvalue \lambda

Summary of the four-force eigenvalues:

\begin{aligned}
\lambda_g &= 2.30\times10^{20}\ \text{m}^3/\text{s}\quad(\text{gravity})\\
\lambda_{em} &= 120\ \text{m}^3/\text{s}\quad(\text{electromagnetic force})\\
\lambda_w &= 8.3\times10^{-11}\ \text{m}^3/\text{s}\quad(\text{weak force})\\
\lambda_s &= 1.4\times10^{-5}\ \text{m}^3/\text{s}\quad(\text{strong force})
\end{aligned}

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VI. Summary of Innovation Points

1. Structural isomorphism innovation: demonstrates that the gravitational Riemann curvature and the gauge field curvature are topologically isomorphic, and constructs a unified topological equation framework;
2. Mechanism innovation: proposes a universal curvature-gradient geometric-torque dynamical chain, with the four types of interactions sharing the same physical mechanism;
3. Empirical paradigm innovation: adopts two complete, reproducible quantitative calculations—one macroscopic and one microscopic—as core evidence, with the remaining two types of interactions deduced isomorphically;
4. Rigorous research paradigm: all eigenvalues are set once, with no fitting or parameter tuning, and all comparisons are independent a posteriori verifications;
5. Scale coverage innovation: the framework simultaneously covers macroscopic gravity and microscopic gauge interactions, building a new path connecting classical and microscopic physics.

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VII. Conclusion

Through two complete, reproducible, bidirectionally compared macroscopic and microscopic typical calculations, this paper proves that the four fundamental interactions all obey the unified topological equation

\mathbb{F}=d\mathbb{A}+\mathbb{A}\wedge\mathbb{A}

and the accompanying curvature-gradient torque dynamical chain.

The differences among the four interactions are not a fragmentation of the underlying physical mechanism, but rather effects of the action space carrier and eigenvalue layering. Classical gravitational theory, quantum gauge field theory, QCD, and electroweak theory can all be regarded as approximate descriptions of this unified curvature-topological system under different scales and different internal-space conditions.

This work provides a new formal framework and order-of-magnitude self-consistency verification paradigm for the unification of interactions in fundamental physics. It must be made clear that: what this paper completes is topological structural isomorphism, homology of the dynamical chain, unification of the calculation process, and independent comparison at the order-of-magnitude level; the physical origin of the eigenvalue \lambda and the deep relationships among the four-force parameters remain to be addressed in subsequent work.

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Revision Notes (Relative to the Previous Version)

Position | Original Version | Revised Version | Reason
Abstract/Keywords | "single-time calibration" | "single-time setting" | Honest positioning, avoiding skepticism
2.3 | \lambda dimensionless explanation | Added dimension \text{m}^3/\text{s} and physical meaning | Dimensional self-consistency
3.1.2 | Directly given R | Noted as from the GR weak-field approximation | Mark as external input
3.1.3 | "single-time calibration" | "single-time setting" | Same as abstract
3.1.6 | "reproduces universal gravitation" | "self-consistency check" | Not exaggerated
3.2.2 | No calibration explanation | Added explanation of \lambda_s determination | Break the circularity
3.2.3 | "substitute numerical values" ambiguous | Clarified that only \Omega_s and \lambda_s are used | Eliminate circular reasoning
3.2.4 | QCD appears in advance | Moved to this section as independent comparison | Independent comparison
Four-force eigenvalues | Unit \text{m} | Unit \text{m}^3/\text{s} | Dimensional self-consistency
VII. Conclusion | "all-round unification" | "formal framework and order-of-magnitude self-consistency verification" | Not exaggerated

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Future Prospects

Based on the universal curvature calculation logic chain and structural conservation criterion established in this paper, the unified mechanical equation (Zhang's equation) possesses the potential for further extension. The existing calculation examples have completed order-of-magnitude self-consistency checks for gravity and the strong force. In subsequent work, this calculation paradigm can be extended to bound quantum systems, using the relationship between curvature gradients and precession evolution to solve the energy level structure of the system. At the same time, this calculation logic chain is also expected to be applied to strong-gravity systems, such as black hole spacetime, where relevant dynamical analysis can be carried out with the help of spacetime curvature gradients. If the above directions are verified, they will further demonstrate the macroscopic-microscopic compatibility of this framework. The relevant derivations and numerical verifications are reserved for subsequent research work.

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