The Homologous Generation Theory of Mathematical Objects

Author: Zhang Suhang, Luoyang

Abstract

This paper proposes and systematically establishes the Homologous Generation Theory of Mathematical Objects, revealing the unified essence of function fitting, series expansion, linear representation, representation transformation, and algebraic equivalence in mathematics: any mathematical objects that can be fitted, decomposed, or mutually represented must necessarily share a set of indivisible atomic units and obey the same generative structure. This common origin is homology, which is the fundamental reason why mathematical objects can approximate and transform into one another. Conditions such as convergence, smoothness, and integrability are merely technical sufficient conditions.

Taking the Fundamental Theorem of Arithmetic as a prototype, this paper proves that the integer space and function space are globally isomorphic in the categorical sense, introduces the concept of function primes, and incorporates prime decomposition and function basis expansion into a unified framework, thereby unifying the underlying logic of discrete and continuous mathematics. This theory directly explains classical structures such as Taylor expansions, Fourier transforms, and the equivalence of matrix mechanics and wave mechanics, and can be extended to functional analysis, quantum mechanics, information theory, and other fields, forming a unified program that spans branches of mathematics.

Keywords: Homologous generation; Fittability; Function primes; Global isomorphism; Unification of mathematical structures

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I. Introduction

In mathematics and theoretical physics, a widespread and profound phenomenon has long remained in a state of "being used without being questioned":

Any sufficiently regular complex function can almost always be infinitely approximated or exactly fitted by a set of simple functions.

Traditional theory attributes this to technical conditions such as convergence, completeness, and smoothness, answering the question of "whether fitting can be achieved." Yet it has never answered:

Why is fitting possible?

Why can mathematical objects of different forms be transformed, represented, and rendered equivalent to one another?

The core contribution of this paper is:

To reduce "fittability" from a methodological question to an ontological one, and to provide a unified answer:

Mathematical objects can be fitted, expanded, and equivalently substituted—

not because of clever techniques,

but because they are homologous.

Homology means sharing the same set of basic atoms, the same generative structure, and the same underlying space.

This paper formalizes this idea into an axiomatic system and proves:

Integers and primes, functions and basis functions, share a completely isomorphic generative relationship;

The necessary and sufficient condition for fittability is homology.

This theory is not merely a supplement to existing formulas but a reinterpretation of the overall structure of mathematics, representing an original framework at the meta-mathematical level.

II. The Prototype of Mathematical Structure: The Fundamental Theorem of Arithmetic and the Atomic Generation View

The cornerstone of number theory, the Fundamental Theorem of Arithmetic, states:

Any integer greater than 1 can be uniquely decomposed into a product of prime powers:

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

and the decomposition is unique.

Here, primes act as indivisible basic atoms:

1. Irreducible: cannot be further decomposed into products of smaller units.

2. Complete: all integers are generated from them.

3. Unique: each integer corresponds to a unique combination of atoms.

All integer operations—reduction, common denominators, equivalence, approximation—are essentially recombinations of these prime atoms.

This structure is not exclusive to number theory but is the underlying pattern of all mathematics.

III. Axiomatic System of the Homologous Generation Theory

This paper establishes the entire theoretical system on three axioms, which are self-consistent, complete, and capable of deriving all conclusions.

Axiom 1: The Axiom of Basic Atoms

Any complete, closed, and self-consistent mathematical space necessarily possesses a set of basic atomic units that are irreducible, linearly independent, complete in generation, and unique in representation. Every object in the space is generated by these atoms, without exception.

Axiom 2: The Axiom of Homologous Generation

All objects within the same space share the same set of basic atoms and obey the same generative rules. This property of sharing a common origin is called homology.

Axiom 3: The Axiom of the Origin of Fittability

A necessary and sufficient condition for two mathematical objects to be exactly fitted, mutually represented, or mutually approximated is that they belong to the same space and are homologous.

In other words:

\text{Homology} \iff \text{Fittability}.

Convergence, smoothness, and integrability are merely technical thresholds for achieving fitting, not the essential reason why fitting is possible.

IV. Core Definitions and the Global Isomorphism Theorem

Definition 1: Basic Atomic Unit

An indivisible, minimal, and complete generative object within a mathematical space.

Definition 2: Homology

Objects within the same space that share the same set of atoms and the same generative rules are said to possess homology.

Definition 3: Function Prime

In a Hilbert function space, an orthonormal, irreducible, complete basis function with unique expansion is called a function prime.

Its status in the function world is entirely equivalent to that of primes in the integer world.

Theorem 1: The Prime–Function Prime Global Isomorphism Theorem

The integer arithmetic space and the space of square-integrable functions are globally isomorphic, with the following correspondences:

· Integers ↔ Functions

· Primes ↔ Function primes (basis functions)

· Prime factorization ↔ Series expansion / orthogonal projection

· Uniqueness of decomposition ↔ Uniqueness of expansion

· Equivalent transformations of integers ↔ Function fitting and representation transformations

Proof sketch: Both are Abelian categories; their generator structures correspond one-to-one; their operations are homomorphic; their decomposition theorems are isomorphic; and their logics of object approximation are identical. Hence, discreteness and continuity share the same underlying structure.

Theorem 2: Fittability is Equivalent to Homology

\text{Fittability} \iff \text{Homology}.

Proof:

· (Necessity) If two objects are fittable, they can be decomposed into the same combination of atoms; hence they are homologous.
· (Sufficiency) If they are homologous, they share atoms and can therefore achieve approximation and fitting through coefficient recombination.

Non-homologous objects cannot establish a stable, unique, and exact fitting relationship; they can only be approximated, not rendered equivalent.

V. Unified Explanation of Classical Structures

1. Taylor Expansion
Takes the power function family \{1, x, x^2, x^3, \dots\} as function primes. Any analytic function is homologically generated from these primes and can therefore be infinitely fitted.

2. Fourier Transform
Takes the trigonometric system as function primes. All square-integrable functions are homologous and can therefore be represented as superpositions of harmonics.

3. Matrix Mechanics and Wave Mechanics
Both arise from the same underlying Hilbert space, using different representations: the former employs discrete atomic bases (e.g., energy eigenstates), the latter continuous position-space bases. Their physical equivalence is precisely a manifestation of homologous generation.

4. Canonical transformations in analytical mechanics, integral transforms, and kernel methods
All are homologous representation transformations: the object remains unchanged; only the choice of atomic basis is altered.

VI. Conclusion

Mathematics is not a collection of fragmented formulas but a holistic structure of homologous generation.

All objects are generated from basic atoms;
All operations are recombinations of atoms;
All fitting, expansion, equivalence, and transformation are merely natural manifestations of homology.

Fittable, because homologous;
Unifiable, because isomorphic;
Understandable, because sharing the same root.

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Theory Title: The Homologous Generation Theory of Mathematical Objects
Version: Final Authoritative Condensed Edition
Date: May 2, 2026