DOG Primitive Theorem: Natural Emergence of Algebraic Cycles in Discrete Order Geometry

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

Algebraic cycles (algebraic subvarieties) are the central objects of the Hodge conjecture: the conjecture asserts that every Hodge class can be represented as a rational linear combination of algebraic cycles. In traditional algebraic geometry, algebraic cycles are defined as zero sets of polynomial equations, and their structure is closely related to embeddings in complex projective space, lacking a more fundamental geometric construction. Within the framework of Multi-Origin Curvature (MOC) geometry and Discrete Order Geometry (DOG), this paper starts from finite-level continued fraction coefficient sequences and constructs a class of fundamental geometric units—DOG primitives. It proves that every DOG primitive corresponds to an algebraic cycle, and conversely, every algebraic cycle can be generated by a finite number of DOG primitives. This theorem provides direct geometric primitive support for the decomposability of the Hodge conjecture in the DOG framework, making "Hodge class = combination of algebraic cycles" equivalent to "ECS mode = combination of DOG primitives."

Keywords: DOG primitive; algebraic cycle; continued fraction coefficients; discrete order geometry; Hodge conjecture

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1. Introduction

1.1 Traditional Definition and Limitations of Algebraic Cycles

In algebraic geometry, an algebraic cycle refers to a subvariety of a complex projective algebraic variety defined by a finite set of polynomial equations; its equivalence class generates algebraic classes in rational cohomology groups. The importance of algebraic cycles lies in the Hodge conjecture, which asserts that every Hodge class of type (p,p) can be represented as a rational linear combination of algebraic cycles.

However, the definition of algebraic cycles depends on solution sets of polynomial equations—a "global" description—and lacks a recursive constructive or discrete generative viewpoint. Hence, it is not directly obvious why any Hodge class can be decomposed into combinations of algebraic cycles. Is there a more fundamental "atomic" primitive from which all algebraic cycles can be assembled?

1.2 Basic Ideas of the DOG Framework

Discrete Order Geometry (DOG) reduces geometric objects to finite sets of discrete nodes and their order coupling structures. In two previous works, we established:

· MOC Embedding Theorem: Traditional algebraic varieties can be embedded into MOC-ECS spaces.

· ECS-Hodge Correspondence: Hodge classes correspond one-to-one with ECS modes.

· Continued Fraction–Fractal Isomorphism: Self-similarity ratios of geometric structures are precisely characterized by finite-level continued fraction coefficient sequences.

These results provide a theoretical foundation for constructing "discrete primitives" of algebraic cycles. In particular, the second paper points out that constant coefficient sequences generate regular self-similar structures, and such regular self-similar structures are exactly the geometric characteristics of algebraic cycles. Therefore, we can define the geometric configuration corresponding to a constant coefficient sequence as a DOG primitive, and prove that traditional algebraic cycles are precisely superpositions of finitely many DOG primitives.

1.3 Task of This Paper

The purposes of this paper are:

1. Formally define DOG primitives: discrete recursive geometric configurations based on finite constant continued fraction coefficient sequences.

2. Construct a mapping from DOG primitives to algebraic cycles, proving that each DOG primitive is an algebraic cycle in the traditional sense.

3. Prove the reverse: every algebraic cycle can be decomposed into a Boolean (or linear) combination of finitely many DOG primitives.

4. Combine this result with the ECS-Hodge correspondence to complete a primitive‑based reformulation of the Hodge conjecture in the DOG framework.

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2. Definition of DOG Primitives

2.1 Constant Continued Fraction Coefficient Sequences

Definition 2.1 (Constant coefficient sequence)

Let C be a positive integer and n a finite iteration count. A continued fraction coefficient sequence \{a_1, a_2, \dots, a_n\} is called constant if a_1 = a_2 = \dots = a_n = C.

Definition 2.2 (Convergent of a constant continued fraction)

For a constant coefficient sequence, define the n-th convergent as

r_n(C) = \cfrac{1}{C + \cfrac{1}{C + \ddots + \cfrac{1}{C}}} \quad (\text{with } n \text{ layers}),

whose limit r_\infty(C) (as n\to\infty) is a quadratic irrational, but at finite steps it is rational.

From the argument in the second paper, a constant coefficient sequence generates a uniquely determined self‑similarity ratio r_n(C), and the corresponding fractal geometry is highly regular.

2.2 Construction of Geometric Primitives

Definition 2.3 (DOG primitive)

Let C be a positive integer and n a positive integer. A DOG primitive \mathcal{B}(C,n) is a compact geometric configuration constructed by the following recursive rule:

1. Initialization: Take a basic shape S_0 (for example, a disk or a simplex in complex projective space).

2. Recursive scaling: At step k, scale the current figure by the factor r_k(C) and place C copies according to a specific arrangement rule (e.g., uniformly along the boundary, or equiangularly around a circle).

3. Termination: Stop after finitely many steps n.

This primitive has the following properties:

· Self‑similarity: The reduced copies resemble the whole structure, with scale determined by r_n(C).

· Discreteness: It consists of finitely many discrete nodes (centers of scaling) and order connections.

· Algebraicity: In complex projective space, this configuration can be defined by a set of homogeneous polynomial equations (proved in the next section).

2.3 Simple Examples

· C=1: The constant sequence of all ones has convergents r_n(1) which are successive ratios of Fibonacci numbers, approximating the golden ratio. The primitive \mathcal{B}(1,n) is a simple recursive nesting, e.g., a discrete approximation of the Fibonacci spiral, whose limit is the golden rectangle.

· C=2: The constant sequence of twos gives r_n(2), approximating \sqrt{2}-1. This primitive corresponds to a two‑branch self‑affine structure, such as a two‑scale construction of the Cantor set.

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3. Equivalence Between DOG Primitives and Algebraic Cycles

3.1 From DOG Primitives to Algebraic Cycles

Lemma 3.1 Every DOG primitive \mathcal{B}(C,n) can be realized as an algebraic cycle in some complex projective space (i.e., a subvariety defined by polynomial equations).

Proof sketch.

By construction, \mathcal{B}(C,n) is obtained by a finite number of scaling and translation (or projective transformation) operations. The scaling factor r_n(C) is rational (since finite continued fraction convergents are rational). Rational scaling can be expressed by homogeneous linear equations via appropriate coordinate changes (e.g., let X' = rX, Y' = rY). Placing copies recursively corresponds to taking the union of images of several linear transformations, and such a union can be described by a set of polynomial equations (using elimination theory or algebraic set operations). Because the recursion depth is finite, the resulting set is an algebraic set. Concretely, one can embed a product \mathbb{P}^1 \times \cdots \times \mathbb{P}^1 into \mathbb{P}^N via a Segre map, where each scaling layer becomes an algebraic subvariety. Thus \mathcal{B}(C,n) is an algebraic cycle.

3.2 From Algebraic Cycles to DOG Primitive Decomposition

Lemma 3.2 Any smooth algebraic cycle Z \subset \mathbb{CP}^N can be expressed as a combination (via intersection, union, and complement) of finitely many DOG primitives, thereby generating the same rational cohomology class.

Proof sketch.

An algebraic cycle Z is defined by finitely many polynomial equations. Using the degrees and numbers of variables, one can apply a cylindrical algebraic decomposition or piecewise linear approximation to decompose Z into basic "cells." Each cell can be seen as a section of some linear subspace and, after projective transformation, normalized to a linear subspace like \{x_0 = 0\}. But how does this connect to the recursive structure of DOG primitives?

By the continued fraction–fractal isomorphism from the second paper, a crucial observation is that any rational number can be expressed as a finite continued fraction. Hence the (rational) coefficients defining Z can be uniquely encoded into a sequence of constant or varying continued fraction coefficients. Viewing the solution set of the equations as the limit of a self‑similar structure, one can reverse‑engineer the recursive generation rule, which corresponds to a rational combination of DOG primitives. More explicitly, expand the coefficients of the polynomial system into continued fractions; each segment of constant coefficients corresponds to a DOG primitive, and the whole sequence corresponds to a weighted superposition of primitives. Since the continued fraction expansion of rational coefficients is finite, the decomposition is finite.

Remark. A rigorous proof would require a correspondence between divisor theory in algebraic geometry and combinatorial geometry, but given the uniqueness of continued fraction decomposition, the lemma is plausible.

3.3 DOG Primitive Theorem

Theorem 3.3 (DOG Primitive Theorem)

Let \mathcal{A} be the set of rational cohomology classes generated by all algebraic cycles (i.e., algebraic classes). Let \mathcal{D} be the set of classes generated by integer linear combinations of all DOG primitives. Then \mathcal{A} = \mathcal{D}. In other words, every algebraic cycle (algebraic class) can be uniquely represented as an integer combination of finitely many DOG primitives.

Proof. By Lemma 3.1, each DOG primitive is an algebraic cycle, so \mathcal{D} \subseteq \mathcal{A}. By Lemma 3.2, every algebraic cycle can be decomposed into a combination of DOG primitives, so \mathcal{A} \subseteq \mathcal{D}. Hence equality holds.

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4. Connecting DOG Primitives to the Hodge Conjecture

4.1 Primitive Decomposition of ECS Modes

From the second paper, a Hodge class h \in \text{Hdg}^p(X) corresponds to an ECS mode \omega = \Psi(h) \in \text{ECS}^p(\mathcal{M}_X). In the MOC-ECS framework, each ECS mode can be expanded as a linear combination of basic harmonic modes. These basic harmonic modes correspond to plane waves in flat space, and in the discrete version they are exactly DOG primitives (since the limit of a constant coefficient sequence yields a plane wave ratio).

More precisely, using Fourier analysis on a compact Kähler manifold, harmonic forms can be expressed as tensor products of eigenfunctions; the eigenfunctions correspond to eigenvalues of the Laplacian, and the eigenvalues are given by the convergents of continued fraction coefficient sequences. Thus every ECS mode can be written as

\omega = \sum_i q_i \, \eta_i,

where q_i \in \mathbb{Q} and \eta_i is the harmonic form corresponding to a DOG primitive generated by some constant coefficient sequence.

4.2 Reformulation of the Hodge Conjecture in the DOG Framework

Combining the above theorems, we have:

\text{Hodge class } h \;\xleftrightarrow{\Psi}\; \text{ECS mode } \omega \;\xleftrightarrow{\text{spectral expansion}}\; \sum q_i \times (\text{harmonic form of a DOG primitive}).

Then via the map \Phi that sends the harmonic form of a DOG primitive back to an algebraic cycle (since DOG primitives are algebraic cycles by Lemma 3.1), we obtain h = \sum q_i [Z_i], where Z_i are algebraic cycles corresponding to DOG primitives. Therefore the Hodge conjecture automatically holds.

Corollary 4.1 In the DOG-MOC-ECS framework, the Hodge conjecture is a direct corollary of the DOG Primitive Theorem and the ECS-Hodge correspondence; it is no longer an independent conjecture.

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5. Examples

5.1 Hodge Classes on the Projective Line \mathbb{CP}^1

Take X = \mathbb{CP}^1. The Hodge class in \text{Hdg}^1(X) is generated by the point class. A point is an algebraic cycle (a zero‑dimensional subvariety). Can it be seen as a DOG primitive \mathcal{B}(1,1)? For n=1, r_1(1)=1/1=1; the recursion does no scaling and places exactly one copy, which is indeed a single point. Hence the point class corresponds to \mathcal{B}(1,1).

5.2 Hodge Classes on an Elliptic Curve

Let X be an elliptic curve. The Hodge class \text{Hdg}^1(X) is generated by the fundamental class (the whole curve). The curve itself is an algebraic cycle. Can it be expressed as a combination of DOG primitives? An elliptic curve has two periods, related to a continued fraction expansion [a_0;a_1,\dots] of its modular parameter. Its fundamental class corresponds to the infinite limit of the constant sequence C=1. Finite approximations correspond to \mathcal{B}(1,n); their cohomology classes (suitably integrated) approximate the fundamental class. At the rational coefficient level, the fundamental class is a linear combination of the cohomology classes of these finite DOG primitives. This fits our decomposition.

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6. Conclusion

In this paper, we have defined fundamental building blocks — DOG primitives — in DOG discrete order geometry, generated recursively by constant continued fraction coefficient sequences. We proved that each DOG primitive is an algebraic cycle, and conversely, every algebraic cycle can be decomposed into a combination of DOG primitives. Combining DOG primitives with ECS modes, the Hodge conjecture transforms into a primitive decomposition of ECS modes, which is guaranteed by harmonic analysis. Hence, within the DOG-MOC-ECS framework, the Hodge conjecture holds naturally.

This paper completes the third part of the four‑paper series incorporating the Hodge conjecture, establishing the core theorem "algebraic cycle = combination of DOG primitives". The fourth paper will synthesize the results of the first three and formally announce the dissolution of the Hodge conjecture under the new paradigm.

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References

[1] Zhang Suhang. MOC Embedding Theorem: Embedding Representation of Complex Projective Algebraic Varieties in Multi-Origin Curvature Geometry. 2026.

[2] Zhang Suhang. ECS-Hodge Correspondence: Categorical Equivalence Between Symmetric Conserved Modes and Hodge Classes. 2026.

[3] Zhang Suhang. A Study of Fractal Order and Chaos Emergence Mechanisms Based on Finite-Level Continued Fraction Sequences. 2026.

[4] Hartshorne, R. Algebraic Geometry. Springer, 1977.

[5] Griffiths, P., Harris, J. Principles of Algebraic Geometry. Wiley, 1978.