Homologous Generation Theory of Mathematical Objects (GKT)

——A Meta-Theory on the Generative Structure of Mathematics

Author: Zhang Suhang

Affiliation: Heluo Mathematical School

Date: May 2026

Abstract

This paper announces the birth of a new mathematical cognitive paradigm: the Homologous Generation Theory.

The theory asserts that the ultimate reason why mathematical objects can be expanded, equivalently transformed, and transplanted across domains lies not in technical conditions such as topological convergence, algebraic closure, or analytic regularity, but in the fact that they share the same set of Generative Kernels—a collection of irreducible, complete, and uniquely representable primitive units.

This paper makes three contributions:

First, it proposes the Axiom of Kernel Existence and the Criterion for Homology, elevating the question of "why mathematical objects are related" from a technical level to an ontological level;

Second, it proves that prime factorization and orthogonal basis expansion share the same universal property in the sense of category theory, thereby revealing the structural isomorphism between discrete mathematics and continuous mathematics at the generative level;

Third, it delineates the applicable boundaries of the Homologous Generation Theory, specifying its validity in separable Hilbert spaces and Banach spaces with Schauder bases, while candidly acknowledging its limitations.

This paper does not provide new algorithms, but it offers the underlying logic for all algorithms; it does not prove new theorems, but it makes cross-domain transplantation of theorems predictable and explainable. This represents a Copernican turn from "relational mathematics" to "generative mathematics."

Keywords: Homologous Generation Theory; Generative Kernel; Universal Property; Structural Unity; Meta-Mathematics

1 Introduction: The Forgotten Problem of Genesis

1.1 The Separation of "Use" and "Inquiry" in Mathematics

There exists a clear dividing line in the history of mathematics:

· Classical period: Mathematics was operational—Babylonians solved equations, Egyptians measured areas. They used mathematics but did not ask why it worked.

· Greek period: Mathematics became inquisitive—Euclid asked "how is proof possible," giving birth to axiomatization.

· Modern period: Mathematics has once again become operational—we skillfully use Fourier transforms, Hilbert spaces, and wavelet analysis, yet rarely ask a more fundamental question:

"Why are these expansions possible?"

The traditional answer is: because functions satisfy conditions of convergence, smoothness, integrability, and completeness.

But this answer is technical, not principled. It answers "under what conditions can it be achieved," but never "why can it be achieved."

1.2 The Core Question of This Paper

This paper raises a long-neglected question:

"What is the origin of expansion, transformation, and equivalence?"

In other words:

· Why can an arbitrary function establish an exact correspondence with a set of trigonometric functions?

· Why are discrete prime factorization and continuous series expansion so similar in logical structure?

· Why is cross-domain transplantation between different mathematical branches always feasible?

This paper's answer is:

Because they are homologous.

Homology means sharing the same set of generative kernels, the same generative rules, and the same ontological origin.

1.3 The Positioning of Homologous Generation Theory

The Homologous Generation Theory is not a supplement to any branch of mathematics, but a meta-interpretation of all mathematics. It does not replace group theory, analysis, or topology, but provides them with a common generative foundation.

Just as Galois told posterity: "Do not look only at equations; look at their symmetry groups"—the Homologous Generation Theory tells posterity: "Do not look only at objects; look at their generative kernels."

2 The Prototype: The Meta-Mathematical Significance of the Fundamental Theorem of Arithmetic

2.1 The Threefold Structure of the Fundamental Theorem of Arithmetic

The cornerstone of number theory—the Fundamental Theorem of Arithmetic—is not merely a theorem, but a prototype.

Any integer N can be uniquely decomposed as:

N = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}

Its structure contains three layers of meaning:

Layer Content Meta-Mathematical Implication

Atomic Layer Primes p_i are indivisible Existence of irreducible generators

Generative Layer All integers are generated by primes Atoms possess completeness

Unique Layer Decomposition is unique Representation has determinacy

2.2 As a Prototype of "Generative Structure"

The greatness of the Fundamental Theorem of Arithmetic lies not only in its description of a fact, but in its establishment of a worldview:

Complex object = Basic atoms + Combination rules

This worldview permeates all of mathematics:

· In number theory: composite numbers = primes × primes

· In algebra: polynomials = irreducible polynomials × irreducible polynomials

· In analysis: functions = linear combinations of basis functions

· In linear algebra: vectors = linear combinations of basis vectors

· In quantum mechanics: state vectors = superpositions of eigenstates

The core insight of the Homologous Generation Theory is:

All of the above structures are not "analogies" but "isomorphisms."

They differ only in combination rules—multiplication, addition, integral transforms—but the underlying logic of "atoms + generation" is entirely consistent.

3 The Axiomatic System of the Homologous Generation Theory

This paper establishes three basic axioms. They do not replace the ZFC axiomatic system, but serve as meta-axioms that provide a unified "generative" interpretation for all branches of mathematics.

Axiom I: The Axiom of Kernel Existence

Any separable, complete mathematical space \mathcal{S} necessarily possesses a set of generative kernels \mathcal{K} = \{ \kappa_i \}_{i \in I} satisfying:

1. Irreducibility: \kappa_i cannot be expressed as a generative combination of other elements in the space;

2. Completeness: \forall x \in \mathcal{S} , x can be generated by \mathcal{K} ;

3. Uniqueness: The representation of x under a given generative rule is unique.

Axiom II: The Axiom of Homology

Two mathematical objects x, y are said to be homologous if and only if they belong to the same space \mathcal{S} and share the same set of generative kernels \mathcal{K} .

Homology is an equivalence relation—it partitions all objects in a space into mutually disjoint homology classes. Objects within the same homology class share the same generative kernel; objects in different homology classes have different kernels.

Axiom III: The Axiom of Generation

All mathematical objects within the same space are generated by the same set of generative kernels.

Formal representation:

\forall x \in \mathcal{S}, \quad x = \sum_i c_i \kappa_i, \quad \{\kappa_i\} = \mathcal{K}(\mathcal{S})

where c_i are generative coefficients and \kappa_i are kernel atoms.

Meaning: The kernel determines the structure of the space; all objects are manifestations of kernel combinations. Objects differ only in their combination coefficients and combination rules.

3.1 On the Self-Consistency of the Axiomatic System

This axiomatic system possesses intrinsic self-consistency:

· Axiom I defines the "existence of atoms";

· Axiom II defines "homologous relations";

· Axiom III defines the "mode of generation."

Together, they form a complete generative ontological闭环.

4 Core Concepts and the Isomorphism Theorem

4.1 Generative Kernel

Definition: The generative kernel \mathcal{K}(\mathcal{S}) of a space \mathcal{S} is the minimal set of generators satisfying Axiom I.

Note: Kernels are not unique—the same space can have different kernels (e.g., Fourier bases and wavelet bases). However, between any two kernels there must exist a reversible generative transformation. This property ensures the objectivity of homology determination.

4.2 Prime Kernels and Function Kernels

· In the integer space \mathbb{Z} : \mathcal{K}(\mathbb{Z}) = \{ \text{all primes} \} 

· In L^2[0,2\pi] : \mathcal{K}(L^2) = \{ \cos nx, \sin nx \mid n \in \mathbb{N} \} 

The latter is called "function primes"—their status in function spaces is entirely equivalent to that of primes in integer spaces.

4.3 The Isomorphism Theorem

Theorem: The structure of integer factorization and the structure of orthogonal function expansion are isomorphic in the sense of category theory.

The specific correspondences are:

Integer Space Function Space Isomorphic Mapping

Integer N Function f Object correspondence

Prime p Basis function \phi_n Atom correspondence

Prime factorization Series expansion Generative rule correspondence

Uniqueness of factorization Uniqueness of expansion Uniqueness correspondence

Equivalent deformation Representation transformation Transformation rule correspondence

Proof sketch: Both are Abelian generative categories, and their initial objects share the same universal property, namely: any object can be obtained from the initial object through a unique generative sequence. The identity of the universal property is sufficient for structural isomorphism.

4.4 The Criterion for Homology Determination

Criterion: x and y are homologous if and only if \mathcal{K}(x) = \mathcal{K}(y) .

5 Reinterpretation of Classical Structures

5.1 Taylor Expansion

· Kernel: \{1, (x-a), (x-a)^2, \dots\} 

· Generative rule: Linear combination + limit

· Domain of applicability: Space of analytic functions

· Homological explanation: Analytic functions share the power function kernel, hence they can be expanded.

5.2 Fourier Series

· Kernel: \{\cos nx, \sin nx\} 

· Generative rule: Linear combination + L^2 convergence

· Domain of applicability: L^2[0,2\pi] 

· Homological explanation: Square-integrable functions share the trigonometric kernel, hence they can be expanded.

5.3 The Superposition Principle in Quantum Mechanics

· Kernel: Eigenfunction system of the Hamiltonian operator

· Generative rule: Linear superposition

· Physical meaning: Eigenstates are the "atomic states" of the physical world

· Homological explanation: All physical states are homologous, hence they can be superposed, interfered, and transformed.

5.4 The Equivalence of Matrix Mechanics and Wave Mechanics

· Matrix mechanics: Kernel is energy eigenstates

· Wave mechanics: Kernel is position eigenstates

· Source of equivalence: Both share the same Hilbert space, differing only in kernel choice

· Homological explanation: Representation transformation does not alter homology.

6 Boundaries, Objections, and Responses

6.1 Applicable Boundaries

The prerequisites for the applicability of the Homologous Generation Theory are:

1. The space is separable (possessing a countable dense subset);

2. The space has a basis (possessing a Schauder basis or orthonormal basis).

Non-applicable cases:

· Banach spaces without a basis (Enflo's 1973 counterexample);

· Non-separable Hilbert spaces.

6.2 Responses to Possible Objections

Objection 1: "The existence of kernels is not necessary."

Response: This theory does not claim that "all spaces have kernels." We acknowledge the existence of Enflo's counterexample and restrict the Homologous Generation Theory to separable spaces with bases. This restriction is not a flaw but a manifestation of precision. Just as group theory does not address non-group structures, the Homologous Generation Theory only addresses structures with generative properties.

Objection 2: "Homology and generation are tautological."

Response: This paper has clearly distinguished:

· Homology: Defined by Axiom II, belonging to the category of relational judgment;

· Generation: Defined by Axiom III, belonging to the category of mode of composition.

Together, they reveal the unity of "what an object is" and "how objects are related."

Objection 3: "This is merely a rephrasing of category theory."

Response: Category theory concerns relations between objects (arrows), while the Homologous Generation Theory concerns the internal generative structure of objects (atoms). The former is a "relational ontology," the latter a "generative ontology." They are complementary, not substitutive.

7 Conclusion: A Manifesto for Generative Mathematics

The long-standing fragmentation of mathematics—arithmetic, algebra, analysis, geometry—stems from a cognitive habit: we always observe mathematical objects from the outside, asking which branch they "belong to."

The Homologous Generation Theory reverses this perspective.

It asks from the inside: What generates this object? Is it homologous to another object?

The consequences of this shift in perspective are profound:

· Expandability is no longer a technical issue, but a natural expression of homology;

· Equivalent transformation is no longer a matter of technique, but a manifestation of homology under different representations;

· Cross-domain transplantation is no longer an accidental discovery, but an inevitable recurrence of homologous structures across different spaces.

The final conclusion of the Homologous Generation Theory can be summarized in three sentences:

Expandable, because homologous.

Unifiable, because isomorphic.

Understandable, because of the same root.

Mathematics is not a loose toolbox, but a great tree growing from a single root system.

The task of this theory is not to cut down this tree, but to reveal its hidden and complete root system buried deep underground.