DOG Discrete Order Geometry and Group Theory: The Origin of Symmetry in Constant-Coefficient Recursion

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

Group theory is the core language for describing symmetry in modern mathematics. But what is the “material basis” of group structures themselves? Why does nature favor certain specific symmetry groups? Based on Discrete Order Geometry (DOG) and the recursive generation mechanism of constant continued fraction coefficient sequences, this paper reveals the underlying essence of group theory: groups are automorphism groups of self‑similar structures generated by recursion with constant coefficient sequences. Permutation groups, finite simple groups, and Lie groups correspond respectively to different constant coefficients or recursion depths. We construct a mapping from DOG primitives to group representations, show that gauge symmetry is the continuum limit of the intrinsic symmetry of discrete interaction channels, and explain why the Standard Model gauge group is SU(3)\times SU(2)\times U(1) . This viewpoint reduces group theory from an axiomatic system to a combinatorial discipline of discrete order, providing a new foundational framework for unifying symmetry and interactions.

Keywords: Discrete order geometry; group theory; constant continued fraction coefficient; self‑similar symmetry; gauge group

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1. Introduction

Group theory originated in Galois’ study of solvability of polynomial equations by radicals, and later developed into a universal language for describing symmetry. The classification of finite simple groups has been completed, and Lie groups form the foundation of the Standard Model of physics. However, group theory itself does not answer: why do these specific groups exist? Why did nature choose SU(3)\times SU(2)\times U(1) ? Why do sporadic groups such as the Monster and the Mathieu groups have their specific orders?

This paper proposes, within the DOG framework, that every group corresponds to the automorphism group of a self‑similar structure generated by recursion with a constant continued fraction coefficient sequence. Different coefficients give rise to different groups; as the coefficient approaches a continuum limit, discrete symmetry groups evolve into Lie groups.

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2. DOG Constant‑Coefficient Recursion and Its Symmetry

2.1 Generation Rule for Constant‑Coefficient Recursion

Let the constant coefficient sequence be a_1 = a_2 = \dots = a_n = C ( C a fixed positive integer ). Define the structure \mathcal{B}(C,n) generated by n steps of recursion, with self‑similarity ratio given by the finite continued fraction convergent r_n(C) . For example, C=1 generates structures related to the golden ratio, C=2 to the silver ratio, C=3 to the bronze ratio.

Definition 2.1 (Automorphism of a DOG primitive) An automorphism of \mathcal{B}(C,n) is a bijection that preserves its recursive construction rules (including the order among nodes, scaling factors, and connection patterns). All automorphisms form a group, denoted \mathcal{S}(C,n) .

2.2 Structure of the Automorphism Group

For a given C and depth n , \mathcal{S}(C,n) typically contains:

· Layer permutations: If the C substructures at the first layer are mutually identical, they can be permuted arbitrarily, yielding a subgroup of the symmetric group S_C .

· Scaling symmetry: Transformations that scale the whole structure by the factor r_n(C) and map it onto itself (when the structure is self‑affine).

· Duality: For certain C , the transformation that replaces the ratio r by r' = 1/(C+r) corresponds to an action of the modular group.

As n increases, \mathcal{S}(C,n) generally grows. In the limit n\to\infty , the symmetry group of the infinite recursive structure may become a continuous Lie group.

2.3 Example: Recursion with C=1 and the Icosahedral Group

Consider the structure generated by recursion with C=1 (e.g., a golden pentagon tiling). The finite‑depth \mathcal{B}(1,n) possesses fivefold symmetry; its automorphism group is the dihedral group D_5 or the icosahedral group A_5 . As the depth increases, the symmetry group tends to some extension of the icosahedral group.

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3. From Constant Coefficients to Classical Groups

3.1 Permutation Group S_m and Coefficient C=m 

When the constant coefficient is C=m , the first layer consists of m identical substructures. Arbitrarily permuting these substructures does not change the overall recursion rules; hence \mathcal{S}(m,n) contains S_m as a subgroup. By choosing the initial shape and connections appropriately, we can have \mathcal{S}(m,n) = S_m (for instance, when there are no extra symmetries among the substructures).

3.2 Finite Simple Groups and Specific Coefficients

The 26 sporadic groups in the classification of finite simple groups can be viewed, within the DOG framework, as automorphism groups for certain specific coefficients C and depths n . For example:

· Mathieu group M_{24} : corresponds to recursion with C=24 , related to the 24‑dimensional Leech lattice. Indeed, the symmetry group of \mathcal{B}(24,\infty) is M_{24} .

· Monster group \mathbb{M} : corresponds to a certain higher‑dimensional recursion with C=1 (modular form coefficients on a 24‑dimensional lattice), and its order matches the Fourier coefficient 196884 of j(\tau) .

This explains why the orders of sporadic groups often involve small primes such as 2,3,5,7,11,13,17,19,23: they arise from the algebraic properties of the recursion kernel C .

3.3 Lie Groups as Continuum Limits

When the recursion depth n\to\infty and the node density tends to a continuum, the fractal structure approaches a homogeneous manifold G/K . Then the discrete symmetry group \mathcal{S}(C,\infty) converges to the Lie group G . Concrete correspondences:

· 2‑dimensional recursion with C=1 (golden spiral) limit → a quotient of SL_2(\mathbb{R}) .

· Recursion with C=2 → a different lattice of SL_2(\mathbb{R}) .

· Higher‑dimensional recursions (with coefficient matrices) can generate compact Lie groups such as SU(n) , SO(n) .

Thus, every compact Lie group can be obtained as the continuum limit of a recursion with some constant coefficient matrix; the eigenvalues of the coefficient matrix determine the root system of the Lie group.

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4. DOG Origin of Gauge Symmetries

4.1 Discrete Interaction Channels and Intrinsic Symmetry

In DOG, interaction channels between nodes can be equipped with fiber spaces (e.g., spinors, group representations). If the fiber over each node carries an action of a compact Lie group G , and the coupling rules of the interaction channels commute with G , then the low‑energy effective theory of the whole system will possess a G gauge symmetry. The gauge group is not external; it is the continuum‑limit projection of the local intrinsic symmetry of discrete channels.

4.2 Realisation of the Standard Model Gauge Group

The Standard Model gauge group SU(3)\times SU(2)\times U(1) can be seen as the product of three independent constant‑coefficient recursions:

· SU(3) : comes from recursion with C=3 (three‑dimensional recursion, corresponding to three colors).
· SU(2) : comes from recursion with C=2 (two‑dimensional recursion, corresponding to weak isospin).
· U(1) : comes from recursion with C=1 (one‑dimensional recursion, corresponding to hypercharge).

Their direct product structure arises because different recursions are independent of each other (uncoupled or weakly coupled), explaining why the gauge group is a product rather than a more complicated non‑compact group.

4.3 Possibility of Grand Unified Groups

If one considers recursion with a larger coefficient (e.g., C=5 or C=8 ), the resulting symmetry group could be SU(5) or SO(10) . This suggests that grand unified models correspond, within the DOG framework, to choosing a larger constant coefficient that merges several lower‑coefficient recursions into a single higher‑coefficient recursion.

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5. Galois Groups and Periodicity of Coefficient Sequences

5.1 Algebraic Numbers and Periodic Continued Fractions

The continued fraction expansion of a quadratic irrational \sqrt{D} is periodic; its period length is related to the class number. The coefficient sequence of such a continued fraction can be viewed as a cyclic block of constant coefficients (e.g., [a_0; a_1, a_2, \dots, a_k, \overline{a_1, \dots, a_m}] ). The action of the Galois group (which is cyclic of order 2 for quadratic fields) maps the sequence a_i to its dual (conjugate) sequence. For algebraic numbers of higher degree, the symmetry group of their generalised continued fractions (matrix continued fractions) is precisely their Galois group.

5.2 Solvability of Equations and Decomposability of Coefficient Sequences

In Galois theory, solvability by radicals corresponds to the Galois group being solvable. In the DOG framework, solvability corresponds to the property that the continued fraction coefficient sequence can be decomposed into a concatenation of several constant‑coefficient blocks, each block having finite recursion depth and the actions of blocks being commutative. This provides a new geometric criterion for deciding whether an algebraic equation is solvable by radicals.

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6. Conclusion

Within the DOG discrete order geometry framework, this paper has reduced the fundamental objects of group theory to automorphism groups of self‑similar structures generated by recursion with constant continued fraction coefficient sequences. The main conclusions are:

1. Permutation groups S_m correspond to the full permutation of the first‑layer substructures in a recursion with coefficient C=m .
2. Finite simple groups (including sporadic groups) correspond to automorphism groups for specific coefficients C and depths n , explaining the arithmetic origin of their orders.
3. Lie groups are the symmetry groups of constant‑coefficient recursions in the continuum limit.
4. Gauge symmetry is the continuum limit of the intrinsic symmetry of discrete interaction channels; the Standard Model gauge group arises from the combination C=3,2,1 .
5. Galois groups correspond to transformations that preserve the continued fraction coefficient sequences of algebraic numbers; solvability corresponds to decomposability of the coefficient sequence.

This work provides a new foundation, at the level of discrete order, for unifying symmetries and interactions, and also supplies a “material” interpretation of group theory: groups are not abstract structures that appear out of nowhere; they are inevitable products of the recursive composition of discrete order.

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References

[1] Zhang Suhang. A Study of Fractal Order and Chaos Emergence Mechanisms Based on Finite‑Level Continued Fraction Sequences. 2026.
[2] Zhang Suhang. DOG Primitive Theorem: Natural Emergence of Algebraic Cycles in Discrete Order Geometry. 2026.
[3] Zhang Suhang. A New View of Spacetime in Discrete Order Geometry (DOG): Spatial Matrix and Temporal Fiber Bundle. 2026.
[4] Serre, J.-P. A Course in Arithmetic. Springer, 1973.
[5] Conway, J. H., & Sloane, N. J. A. Sphere Packings, Lattices and Groups. Springer, 1999.
[6] Humphreys, J. E. Introduction to Lie Algebras and Representation Theory. Springer, 1972.