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The Homologous Generation Theory of Mathematical Objects（GKT）

——A Meta-Theory on the Genesis of Mathematical Structures

Author: Zhang Suhang
Affiliation: Heluo School of Mathematics

Abstract

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

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

The contributions of this paper are threefold:

First, it proposes the Axiom of Kernel Existence and the Criterion of Homology, elevating "fitability" from a computational issue to a question of ontology.

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

Third, it delineates the boundaries of applicability of the Homologous Generation Theory, explicitly affirming its validity in separable Hilbert spaces and Banach spaces with Schauder bases, while candidly acknowledging its limits.

This paper offers no new algorithms, but provides the underlying logic for all algorithms; it proves no new theorems, but makes cross-domain transplantation of theorems an expectable and explainable phenomenon. This is a Copernican turn from relational mathematics to generative mathematics.

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I. Introduction: The Forgotten Genetical Question

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

There is 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 rise 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 convergence, smoothness, integrability, and completeness.

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

1.2 The Core Inquiry of This Paper

This paper raises a long-neglected question:

What is the ultimate origin of "fitability"?

In other words:

Why can two seemingly utterly different mathematical objects—say, an arbitrary function and a family of trigonometric functions—establish a precise correspondence?
Why are discrete integer decomposition and continuous series expansion so similar in logical structure?

The answer given in this paper is:

Because they are homologous.

Homologous means sharing the same Generative Kernel, the same generation rules, and the same genetic origin.

1.3 The Position of the Homologous Generation Theory

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

Just as Galois told later generations: "Do not look only at equations—look at their symmetry groups"—
The Homologous Generation Theory tells later generations: "Do not look only at objects—look at their Generative Kernels."

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II. 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 Meaning
Atomic layer Primes p_i are indivisible There exist irreducible generators
Generative layer All integers are generated by primes Atoms possess completeness
Uniqueness layer The decomposition is unique Representation is deterministic

2.2 The Prototype as a "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 + Composition 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 their composition rules—multiplication, addition, integral transforms—but the underlying logic of "atoms + generation" is entirely consistent.

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III. The Axiomatic System of the Homologous Generation Theory

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

Axiom I: The Axiom of Kernel Existence

For any separable, complete mathematical space \mathcal{S} , there must exist a Generative Kernel \mathcal{K} = \{ \kappa_i \}_{i \in I} satisfying:

1. Irreducibility: \kappa_i cannot be expressed as a generative combination of other elements within the space;
2. Completeness: \forall x \in \mathcal{S} , x can be generated from \mathcal{K} ;
3. Uniqueness: The representation of x under the given generation rules is unique.

Axiom II: The Axiom of Homology

Two mathematical objects x and y are said to be homologous if and only if they belong to the same space \mathcal{S} and share the same Generative Kernel \mathcal{K} .

Homology is an equivalence relation—it partitions the mathematical universe into "families by blood."

Axiom III: The Axiom of the Origin of Fitability

The essential condition under which two mathematical objects can be strictly fitted, mutually approximated, or equivalently transformed is homology.

Technical conditions such as convergence, integrability, and smoothness are sufficient conditions for realizing fitting, but not the essential cause.

Formal formulation:

> x \sim y \quad \Longleftrightarrow \quad \mathrm{Gen}(x) = \mathrm{Gen}(y)
>

where \mathrm{Gen}(x) denotes the Generative Kernel of object x .

3.1 Remarks on the Self-Consistency of the Axiomatic System

This axiomatic system is internally self-consistent:

· Axiom I defines "the existence of atoms";
· Axiom II defines "blood relations";
· Axiom III defines "the possibility of interaction."

Together, they form a complete generative ontological closure.

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IV. 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.

Remark: The kernel is not unique—the same space can have different kernels (e.g., Fourier bases and wavelet bases). However, between any two kernels there must exist an invertible generative transformation. This property guarantees the objectivity of homology determination.

4.2 Prime Kernel and Function Kernel

· 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} \}

We call the latter function primes—their status in function spaces is entirely equivalent to that of primes in the integer space.

4.3 Isomorphism Theorem

Theorem: The structure of integer decomposition 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 Generation rule correspondence
Uniqueness of decomposition Uniqueness of expansion Uniqueness correspondence
Equivalent transformation Representation transformation Transformation rule correspondence

Proof sketch: Both are Abelian generative categories whose initial objects possess the same universal property: every object can be obtained from the initial object through a unique generation sequence. The identity of the universal property is a sufficient condition for structural isomorphism.

4.4 Criterion for Homology Determination

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

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V. Reinterpretation of Classical Structures

5.1 Taylor Expansion

· Kernel: \{1, (x-a), (x-a)^2, \dots\}
· Generation rule: linear combination + limit
· Domain of applicability: space of analytic functions
· Homological explanation: Analytic functions share the power-function kernel, hence they are expandable.

5.2 Fourier Series

· Kernel: \{\cos nx, \sin nx\}
· Generation 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 are expandable.

5.3 The Superposition Principle in Quantum Mechanics

· Kernel: eigenfunction system of the Hamiltonian operator
· Generation 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 Equivalence of Matrix Mechanics and Wave Mechanics

· Matrix mechanics: kernel is energy eigenstates
· Wave mechanics: kernel is position eigenstates
· Root of equivalence: both share the same Hilbert space; only the choice of kernel differs
· Homological explanation: Representation transformations do not alter homology.

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VI. Boundaries, Objections, and Responses

6.1 Boundaries of Applicability

The Homologous Generation Theory presupposes:

1. The space is separable (has a countable dense subset);
2. The space possesses a basis (has a Schauder basis or orthonormal basis).

Cases of non-applicability:

· Banach spaces without a basis (Enflo 1973 counterexample);
· Non-separable Hilbert spaces.

6.2 Responses to Possible Objections

Objection 1: "The existence of a kernel is not guaranteed."

Response: This theory does not claim that "all spaces have a kernel." We acknowledge the existence of Enflo's counterexample and restrict the Homologous Generation Theory to separable spaces that possess a basis. This restriction is not a flaw but a mark of precision. Just as group theory does not deal with non-group structures, the Homologous Generation Theory only deals with spaces that have a generative structure.

Objection 2: "Homology and fitability are tautological."

Response: This criticism was valid in earlier versions. However, this paper has clearly distinguished:

· Homology: defined by Axiom II, belonging to the ontological category;
· Fitability: determined by Axiom III, belonging to the epistemological category.

Their equivalence is not a definitional identity, but a profound discovery within the theory—it reveals the unity between "what an object is" and "how an object can be known."

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

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

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VII. 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 to which branch they "belong."

The Homologous Generation Theory reverses this perspective.

It asks from the inside: What generates this object? Is it homologous with that other object?

The consequences of this shift in perspective are profound:

· Fitability is no longer a technical issue, but a natural manifestation of homology;
· Equivalent transformation is no longer a matter of skill, but a manifestation of homology under different representations;
· Cross-domain transplantation is no longer an accidental discovery, but an inevitable recurrence of homologous structures in different spaces.

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

Fitable, because homologous.
Unifiable, because isomorphic.
Comprehensible, because of common roots.

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 beneath the ground.

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