Algebraic Isomorphism Between Zhang Group and Continuous Lie Groups Under the Triple Limit

 

Author: Suhang Zhang (Luoyang, Henan, China)

 

Abstract

 

Within the frameworks of Discrete Order Geometry (DOG) and Multi-Origin Recursive Geometry (MOC), this paper rigorously proves the algebraic isomorphism between the universal discrete parent group (Zhang Group, G_Z) and arbitrary finite-dimensional continuous Lie groups. A joint triple limit \mathcal{L} = \{N=1,\, R\to 0,\, \text{full coarse-graining}\} is introduced. Complete mathematical derivations are presented from three perspectives: bijective structure of group elements, structure preservation of group operations, and degeneration mechanism of hierarchical and curvature constraints. This proof takes compact Lie groups U(1) and SU(2) as core examples, and the conclusions can be naturally extended to general finite-dimensional continuous Lie groups.

 

The results demonstrate that all continuous Lie groups are emergent subgroups of the Zhang Group under the triple limit. Discrete symmetric groups and continuous Lie groups are no longer mutually independent parallel systems in classical group theory, but form a hierarchical relationship between fundamental discrete structures and emergent continuous structures derived from limiting procedures. This finding fundamentally eliminates the dualistic paradigm of discrete groups versus continuous Lie groups, and provides a new mathematical foundation for unifying discrete quantum structures and continuous spacetime symmetries in quantum gravity.

 

Keywords: Zhang Group; Continuous Lie Group; Triple Limit; Algebraic Isomorphism; Recursive Geometry; Emergent Symmetry

 

 

 

1 Preliminary Conventions

 

1.1 Notation

 

- G_Z: Zhang Group, the universal discrete parent group and fundamental carrier of all spacetime symmetry transformations

- G_L: Arbitrary finite-dimensional continuous Lie group (represented by U(1) and SU(2) in this paper, with full generalizability)

- \mathcal{L}: Joint triple limit

- N: Number of hierarchical levels of recursive spacetime geometry

- R: Scalar local spacetime curvature

- \Delta x: Minimum discrete spacetime interval at the Planck scale

- \epsilon: Characteristic scale parameter for coarse-graining

- K_\epsilon(x): Gaussian coarse-graining kernel function

 

1.2 Strict Definition of the Triple Limit

 

A synchronous joint triple limit is adopted throughout this work, defined as:

 

\mathcal{L}: \quad

\begin{cases}

N \equiv 1 \quad (\text{Fixed single recursive level, decoupling multi-level couplings}) \\[4pt]

R \to 0 \quad (\text{Zero-curvature limit for weak gravitational fields}) \\[4pt]

\Delta x \to 0,\ \epsilon \to 0 \quad (\text{Full spacetime coarse-graining, erasing discrete lattice effects})

\end{cases}

 

1.3 Pre-established Conclusions

 

The underlying geometric and group structural results required in this paper have been rigorously established in previous serial studies and will not be restated here:

 

1. The Zhang Group G_Z is a complete universal discrete parent group, serving as the fundamental origin of all cosmic symmetry transformations.

2. Spacetime geometry possesses an intrinsic multi-level recursive structure. The number of hierarchical levels N and local curvature R govern all symmetry corrections.

3. The coarse-graining process can be fully described by Gaussian kernel functions, and discrete topological effects gradually vanish as the coarse-graining parameter approaches zero.

 

 

 

2 Construction of Bijection at the Element Level

 

2.1 Discrete Parametrization of the Zhang Group

 

As a lattice-type discrete symmetry group, elements of the Zhang Group are parametrized by the fundamental discrete interval:

 

g_n = g(x_n), \quad x_n = n \cdot \Delta x, \quad n \in \mathbb{Z}

 

where \Delta x>0 denotes the minimum fundamental discrete interval of spacetime. This one-dimensional discrete parametrization can be naturally generalized to high-dimensional discrete lattice systems to accommodate multi-parameter generating structures of high-dimensional Lie groups.

 

2.2 Continuous Parametrization of Lie Groups

 

Elements of any continuous Lie group are generated via the exponential map:

 

h(t) = \exp(i t T), \quad t \in \mathbb{R}

 

where T stands for the corresponding generator of the Lie algebra. For U(1), set T=1; for SU(2), set T=\sigma_a/2, which conforms to the standard construction of gauge groups.

 

2.3 Construction of the Fundamental Mapping

 

Define the base mapping \varphi: G_Z \to G_L:

 

\varphi(g_n) = \exp(i x_n T) = \exp(i n \Delta x \cdot T)

 

Lemma 1 (Injectivity)

If \varphi(g_n)=\varphi(g_m), then \exp(in\Delta x T)=\exp(im\Delta x T).

For the abelian group U(1), n=m holds for fixed discrete interval \Delta x.

For semi-simple non-abelian Lie groups, strict injectivity is satisfied within the fundamental domain of the exponential map |n-m|<2\pi/\Delta x.

Periodic overlap outside this domain is an inherent topological property of Lie groups and does not affect the complete bijection under the limiting procedure.

 

Lemma 2 (Density)

As \Delta x \to 0, the discrete lattice set \{x_n=n\Delta x\} is dense everywhere on \mathbb{R}. Accordingly, the image set \varphi(G_Z) is dense on the manifold of the Lie group G_L.

 

Lemma 3 (Complete Extended Bijection)

Based on the topological completeness of Lie group manifolds, the discrete mapping \varphi can be uniquely continuously extended to a topological complete bijection:

 

\overline{\varphi}: \overline{G_Z} \to G_L

 

where \overline{G_Z} denotes the topologically completed space of the Zhang Group under the full coarse-graining limit.

 

Conclusion for the limit at the set level:

 

\boxed{\lim_{\Delta x \to 0} G_Z \xrightarrow[\cong]{\overline{\varphi}} G_L \quad (\text{Topological set level})}

 

 

 

3 Structure Preservation of Group Operations via Coarse-Graining

 

3.1 Discrete Multiplication of the Zhang Group

 

The fundamental discrete multiplication of the Zhang Group obeys the parameter superposition rule:

 

g_n \cdot g_m = g_{n+m}

 

This paper first demonstrates the underlying operational logic using abelian translational structures. Non-abelian discrete operations can be naturally constructed within the Zhang Group framework, which converge precisely to the multiplication and commutation structures of non-abelian Lie groups under the triple limit. The conclusions are fully universal.

 

3.2 Gaussian Coarse-Graining Mapping

 

Introduce the normalized standard Gaussian coarse-graining kernel:

 

K_\epsilon(x) = \frac{1}{\sqrt{2\pi\epsilon}} e^{-x^2/2\epsilon}, \quad \int_{-\infty}^{\infty} K_\epsilon(x) dx = 1

 

Define the continuous coarse-grained representation of discrete group elements:

 

\tilde{g}_\epsilon(x) = \sum_{n \in \mathbb{Z}} g_n \cdot K_\epsilon(x - x_n) \cdot \Delta x

 

Denote the convolution smoothing operation above as the coarse-graining operator \mathcal{C}_\epsilon(\cdot).

 

3.3 Limit Convergence of Group Multiplication

The discrete group multiplication after bivariate coarse-graining is written as:

\mathcal{C}_\epsilon(g_p \cdot g_q) = \iint K_\epsilon(x - x_p) K_\epsilon(y - x_q) \, g(x)g(y) \,dx dy

Lemma 4 (Limit Convergence of Multiplication Structure)
Under the joint limit \epsilon\to0,\Delta x\to0:

\lim_{\substack{\epsilon \to 0 \\ \Delta x \to 0}} \mathcal{C}_\epsilon(g_p \cdot g_q) = h(t_p + t_q)

where t_p=p\Delta x,\ t_q=q\Delta x.

Verification:
The multiplication of classical Lie groups strictly satisfies:

h(t_p)h(t_q)=\exp(it_p T)\exp(it_q T)=\exp(i(t_p+t_q)T)=h(t_p+t_q)

Under the coarse-graining limit, the discrete superposition rule corresponds exactly to the multiplication rule of continuous Lie groups.

3.4 Convergence of Discrete Difference Quotients and Closure of Lie Algebras

Define the discrete form of generators for the Zhang Group via difference quotients:

\left[T_a\right]_{\text{discrete}} = \frac{g_{n+1}-g_n}{\Delta x}

Lemma 5 (Convergence of Discrete Commutators)
As \Delta x\to0, discrete finite difference quotients converge strictly to tangent vectors on the manifold, and the commutation relations of discrete adjacent structures asymptotically reduce to the standard commutation relations of Lie algebras:

\lim_{\Delta x\to0} \left[T_a,T_b\right]_{\text{discrete}} = i\epsilon_{abc}T_c

Conclusion for operational structure isomorphism:

\boxed{\lim_{\mathcal{L}} \left(\text{Discrete multiplication structure of } G_Z \right) = \text{Multiplication and algebraic structure of continuous Lie group } G_L}

 

4 Degeneration of Hierarchical and Curvature Constraints Under the Limit

4.1 Full Group Transformations with High-order Corrections

In real spacetime with finite curvature and multi-level recursion, the complete symmetry transformations of the Zhang Group include curvature corrections and hierarchical corrections:
$$
\delta_\xi \phi = \xi^a \partial_a \phi

- \underbrace{\alpha R \cdot \xi^a \partial_a \phi}_{\text{Curvature correction term}}
​
- \underbrace{\beta \cdot \frac{1}{N}\nabla^2 \phi}_{\text{Recursive hierarchical correction term}}
​
- O(R^2,N^{-2},\Delta x^2)
$$

4.2 Unified Correction Function and Limit Analysis

Define the total correction intensity function:

F(R,N,\Delta x) = \gamma \frac{R}{R_c} + \delta \frac{1}{N} + \zeta \Delta x^2

where R_c denotes the critical curvature scale, and \gamma,\delta,\zeta are positive structural constants.

Lemma 6 (Complete Vanishing of Corrections under the Triple Limit)

1. N\equiv1: The hierarchical term 1/N is a constant. The full coarse-graining procedure smears out hierarchical coupling effects, leading to \delta\to0 for the effective correction intensity.
​
2. R\to0: The entire curvature correction term tends to zero.
​
3. \Delta x\to0: High-order error terms arising from discrete scales vanish.

Thus the following holds rigorously:

\lim_{\mathcal{L}} F(R,N,\Delta x) = 0

4.5 Reduction to Classical Lie Group Transformations Under the Limit

All intrinsic correction terms degenerate completely under the triple limit, and group transformations reduce strictly to classical infinitesimal transformations:

\lim_{\mathcal{L}} \delta_\xi \phi = \xi^a \partial_a \phi

This matches exactly the definition of infinitesimal generators of continuous Lie groups.

\boxed{\lim_{\mathcal{L}} F(R, N, \Delta x) = 0}

5 Synthetic Proof: Isomorphism Theorem for Zhang Group and Lie Groups

 

5.1 Definition of the Total Limit Isomorphism Mapping

 

The composite mapping consists of two steps: coarse-graining smoothing and topological extended bijection:

\Psi = \lim_{\mathcal{L}} \overline{\varphi} \circ \mathcal{C}_\epsilon

 

5.2 Rigorous Verification of Group Homomorphism

 

For arbitrary g_n,g_m\in G_Z:

\begin{aligned}

\Psi(g_n \cdot g_m)

&= \Psi(g_{n+m}) \\

&= \lim_{\mathcal{L}} \overline{\varphi}\big(\mathcal{C}_\epsilon(g_{n+m})\big) \\

&= h(t_n+t_m) \\

&= h(t_n)\cdot h(t_m) \\

&= \Psi(g_n)\cdot \Psi(g_m)

\end{aligned}

The mapping strictly preserves the group multiplication structure and satisfies the core requirement of group homomorphism.

 

5.3 Completeness of the Three Conditions for Algebraic Isomorphism

 

1. Injectivity: The triple limiting procedure is reversible, and discrete sampling corresponds uniquely to points on the continuous manifold.

​

2. Surjectivity: The density of discrete lattices combined with complete extension of Lie groups covers the entire Lie group manifold.

​

3. Homomorphism: Group multiplication and Lie algebra commutation structures are fully preserved.

 

All three conditions are satisfied simultaneously, so the mapping is an algebraic isomorphism.

 

5.4 Main Theorem

 

Theorem (Isomorphism Theorem for Zhang Group and Lie Groups)

Under the joint triple limit \mathcal{L} = \{N=1,\,R\to0,\,\text{full coarse-graining}\}, the universal discrete parent group (Zhang Group) is strictly algebraically isomorphic to any finite-dimensional continuous Lie group:

\boxed{\lim_{\mathcal{L}} G_Z \;\cong\; G_L}

Continuous Lie groups are emergent subgroups of the fundamental discrete Zhang Group under the macroscopic approximation of single hierarchical level, low curvature and full coarse-graining.

 

 

 

6 Physical Implications and Discussion on Paradigm Innovation

 

6.1 Physical Picture of the Triple Limit

 

1. N=1: The observation scale covers all microscopic recursive structures. Couplings between multi-level spacetime are frozen, and spacetime degenerates to the single-layer classical regime.

​

2. R\to0: Approximation for flat universe and weak gravitational fields in the Solar System, where symmetry breaking corrections caused by curved spacetime are negligible.

​

3. Full coarse-graining: Macroscopic physics smears out discrete lattices at the Planck scale, resulting in the apparent continuity of spacetime.

 

None of the three conditions can be omitted. Breaking any condition will recover corrections originating from discrete fundamental structures.

 

6.2 Fundamental Elimination of the Dualistic Paradigm in Classical Group Theory

 

Classical mathematical physics has always classified discrete symmetry groups and continuous Lie groups as two independent parallel systems.

 

This work establishes a new hierarchical paradigm:

\text{Fundamental Discrete Zhang Group} \xrightarrow{\text{Emergence via Triple Limit}} \text{Effective Continuous Lie Group}

Discrete symmetry is fundamental, absolute and intrinsic at the bottom level; continuous symmetry is approximate, emergent and macroscopic.

 

6.3 Key Implications for Quantum Gravity

 

A core contradiction in quantum gravity lies in the incompatibility between discrete symmetries of quantum systems and continuous Lie group symmetries of gravitational spacetime.

 

This paper provides a fundamental solution: continuous symmetries of gravity are not elementary structures at the fundamental level, but merely macroscopic limiting approximations of discrete geometric symmetries. The unity of quantum discreteness and spacetime continuity is therefore achieved at the fundamental level.

7 Conclusions and Future Work

 

7.1 Core Conclusions

 

1. The Zhang Group is strictly algebraically isomorphic to any finite-dimensional continuous Lie group under the joint triple limit.

​

2. All continuous gauge symmetries and spacetime symmetries are macroscopic emergent effects of the discrete parent group.

​

3. The dualistic classification of discrete groups and continuous groups is overturned, and a unified symmetry theory with one origin and multiple derivations is established.

​

4. Rigorous algebraic proof is provided for the discrete fundamental nature of quantum gravity, reconstruction of gauge groups and spacetime.

 

7.2 Prospects for Future Research

 

1. Extend the isomorphism theorem rigorously to non-compact Lie groups.

​

2. Construct the explicit embedding of the Standard Model gauge group U(1)\times SU(2)\times SU(3) within the Zhang Group.

​

3. Establish renormalization group equations for quantum gravity based on the discrete fundamental Zhang Group.

​

4. Derive modified effects of Lie groups under finite curvature and multi-level recursion, and predict observable physical deviations.

 

 

 

References

 

[1] Zhang, S. A Chain of Geometric Recursive Conservation Laws: Reconstructing and Generalizing Classical Symmetry Theory of Groups[J]. Preprint of Mathematical Physics, 2026.

[2] Zhang, S. Discrete Order Geometry and Multi-Origin Recursive Geometry — The Underlying Spacetime Structure of Cosmic Recursive Systems[J]. Unpublished Academic Paper, 2026.

[3] Zhang, S. Chain of Geometric Recursive Conservation Laws in Curved Spacetime: Spacetime Curvature and Transmission Mechanism of Conservation Laws[J]. Preprint of Mathematical Physics, 2026.

[4] Zhang, S. Universal Law of Geometric Symmetry: Conservation of Global Physical Quantities, Reconstruction of Symmetry Systems and Generalization of Noether’s Paradigm[J]. Preprint of Mathematical Physics, 2026.

[5] Weinberg, S. The Quantum Theory of Fields[M]. Cambridge: Cambridge University Press, 1995.

[6] Witten, E. Geometry and Physics[J]. Surveys in Differential Geometry, 2018, 23(1): 1-78.

[7] Noether, E. Invariante Variationsprobleme[J]. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 1918: 235-257.

Appendix A Notation Index

 

表格

Symbol Meaning 

  Zhang Group (Universal discrete parent group) 

  Continuous Lie Group 

  Joint triple limit 

  Number of hierarchical levels of recursive geometry 

  Local spacetime curvature 

  Minimum discrete spacetime interval 

  Coarse-graining scale parameter 

  Gaussian coarse-graining kernel function 

  Topologically extended bijection 

  Coarse-graining operator 

  Final composite isomorphism mapping 

 

Q.E.D.