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Part One

Title: The Geometric Foundation and Model Transformation of the Kakeya Conjecture via DOG-MOC

Author: Zhang Suhang (Bosley Zhang)

Location: Luoyang, Henan, China

Date: May 2026

Abstract

Under the framework of Multi-Origin Curvature (MOC) geometry and Discrete Order Geometry (DOG), this paper establishes an axiomatic foundation for solving the Kakeya conjecture. Core concepts such as the direction bundle, curvature tensor, and DOG covering dimension are sequentially defined. The paper proves the equivalent mapping relation and dimensional compatibility between Euclidean space and the MOC-DOG space, and equivalently transforms the classical Kakeya set into a directionally complete, uniformly non-degenerate, rigid set within this system. All definitions, axioms, and lemmas presented in this paper provide complete theoretical support for the proof in the companion paper.

Keywords: Kakeya Conjecture; Multi-Origin Curvature Geometry; Discrete Order Geometry; Direction Bundle; Curvature Rigidity; Dimensional Compatibility

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1. Introduction

The classical formulation of the Kakeya conjecture states: Let n \ge 2 . If any compact set in \mathbb{R}^n contains a unit line segment in every direction, then its Hausdorff dimension must equal n .

For a long time, mainstream approaches to this problem have relied on harmonic analysis, Fourier estimation, and oscillatory integrals. While such methods have achieved progress in three dimensions, they encounter fundamental obstacles in four dimensions and higher, making this a century-old open problem in geometric measure theory.

This paper relies on the self-developed theories of Multi-Origin Curvature (MOC) geometry, Discrete Order Geometry (DOG), combined with the ECS Conservation Law and the MIE Extremal Principle, to reconstruct the problem from a new perspective of curvature constraints and discrete covering. This first paper establishes the complete axiomatic system, core definitions, equivalent transformations, and fundamental lemmas. The companion paper will then utilize the conclusions of this paper to provide a unified and rigorous proof of the Kakeya conjecture in all dimensions.

2. Fundamental Axioms and Definitions of DOG-MOC

2.1 Axioms of Multi-Origin Curvature (MOC) Geometry

Let \mathcal{M} be an MOC space equipped with a finite set of origins \{O_\alpha\} , each corresponding to an independent curvature field \kappa_\alpha . The geometric properties of any point within the space are determined by the superposition of the multi-origin curvature fields. Let \mathbf{R} be the second-order covariant curvature tensor, and let \operatorname{rank}(\mathbf{R}) denote the number of non-zero eigenvalues of the curvature tensor.

Axiom 1 (Curvature Conservation)

\nabla\cdot\sum_\alpha \mathbf{K}_\alpha = 0

This equation signifies that the total curvature field satisfies a divergence-free conservation law, ensuring that the geometric structure has no local collapse or escape of distortion, thereby guaranteeing the overall stability of the geometric configuration (see [1]).

Axiom 2 (Direction-Curvature Duality)

A necessary and sufficient condition for directional completeness of the tangent space in MOC is that the null space of the curvature tensor has zero degeneracy; that is, the curvature tensor has full rank everywhere.

Geometric Interpretation: If there exists a direction that cannot be realized by a geodesic in the space, the curvature tensor at that point degenerates. Conversely, a full-rank, non-degenerate curvature tensor everywhere is equivalent to the space being able to generate all tangent directions.

2.2 Dimensional Axioms of Discrete Order Geometry (DOG)

Let \mathcal{G} be a set of DOG nodes, where the node spacing can be adaptively adjusted according to a scale \varepsilon . For any compact set K \subset \mathbb{R}^n , define the \varepsilon -scale DOG minimal covering number N_{\text{DOG}}(\varepsilon) : the minimum number of balls of radius \varepsilon centered at DOG nodes required to cover K .

Axiom 3 (DOG–Hausdorff Dimensional Compatibility)

\dim_H K = \liminf_{\varepsilon\to 0} \frac{\log N_{\text{DOG}}(\varepsilon)}{-\log\varepsilon}

This definition is completely equivalent to the classical Hausdorff dimension, and mutual derivability is established via standard covering arguments (see [3]), ensuring compatibility between the DOG system and classical geometric measure theory.

2.3 Formalization of the Direction Bundle and the Kakeya Set

Definition 2.1 (Direction Bundle)

Let K \subset \mathbb{R}^n be a compact set. The set

\mathcal{D}(K) = \big\{ e\in S^{n-1} \,\big|\, \exists \text{ a line segment } L\subset K,\ L \text{ has direction } e \big\}

is called the direction bundle of K . If \mathcal{D}(K)=S^{n-1} , i.e., it contains all directions on the unit sphere, the set K is said to be directionally complete.

Definition 2.2 (MOC Direction Bundle Model)

Each unit direction e \in S^{n-1} is mapped one-to-one to the curvature direction associated with an origin O_e in the MOC space. For a directionally complete set K , at any point x \in K , define the induced curvature tensor \mathbf{R}_K(x) as the second moment generated by all line segment directions passing through x that belong to the direction bundle \mathcal{D}(K) .

3. Key Foundational Lemmas

Lemma 3.1 (Directional Completeness ⇒ Global Non-degeneracy of the Curvature Tensor)

If a compact set K\subset\mathbb{R}^n is directionally complete, then there exists a constant \lambda_0>0 such that for all x\in K , the minimum eigenvalue of the curvature tensor satisfies:

\lambda_{\min}\bigl(\mathbf{R}_K(x)\bigr)\ge \lambda_0.

Proof.

Proceed by contradiction. Assume there exists x_0\in K such that \lambda_{\min}\bigl(\mathbf{R}_K(x_0)\bigr)=0 . Then there exists a unit direction e\in S^{n-1} satisfying \mathbf{R}_K(x_0)e=0 .

By Axiom 2 (Direction-Curvature Duality): A zero eigenvalue of the curvature tensor implies geometric degeneracy in that direction, meaning no geodesic in the direction e can be generated in a neighborhood of x_0 . Consequently, no line segment in the direction e exists within K . This contradicts the hypothesis that K is directionally complete.

Thus, at every point within the set, the minimum eigenvalue of the curvature tensor is strictly positive. Furthermore, since K is compact and the curvature tensor field is continuous, combined with the MIE Extremal Principle and the ECS Curvature Conservation Law, the pointwise positive lower bound can be raised to a globally uniform positive lower bound \lambda_0>0 . The lemma is proved (see [1], [2] for complete derivations).

Lemma 3.2 (Classical Kakeya Set ⇔ MOC Curvature-Rigid Set)
A classical compact Kakeya set in Euclidean space is equivalent to a set in the MOC framework whose curvature tensor has full rank everywhere and whose eigenvalues possess a uniform positive lower bound.

Proof.
Forward direction: A classical Kakeya set is naturally directionally complete. Lemma 3.1 directly implies the curvature tensor is globally uniformly non-degenerate and of full rank.

Reverse direction: If a set has a curvature tensor of full rank everywhere with \lambda_{\min}\ge\lambda_0>0 , Axiom 2 implies the space can generate geodesics in all directions; that is, the set contains unit line segments in every direction, satisfying the definition of a classical Kakeya set.

Therefore, the two are equivalent.

4. Equivalent Mapping between Euclidean Space and MOC-DOG Space

Definition 4.1 (Embedding Map)
For any compact set K\subset\mathbb{R}^n , choose a DOG node grid \mathcal{G}_\varepsilon of scale \varepsilon . Define

K_\varepsilon = \big\{ x\in\mathcal{G}_\varepsilon \,\big|\, \operatorname{dist}(x,K)<\varepsilon \big\},

and call K_\varepsilon the \varepsilon -discrete representation of K in DOG space.

Lemma 4.2 (Dimensional Equivalence)
For any compact set K , its DOG covering dimension equals its classical Hausdorff dimension; i.e., Axiom 3 holds.

Lemma 4.3 (Convergence and Lower Bound Preservation of Discrete Curvature)
Let K be a directionally complete compact set, and let \mathbf{R}_\varepsilon be the discrete curvature tensor on the discrete representation K_\varepsilon . Then:

\lim_{\varepsilon\to0} \mathbf{R}_\varepsilon = \mathbf{R}_K

holds in the sense of distribution. Moreover, for sufficiently small \varepsilon , the minimum eigenvalue of the discrete curvature tensor maintains the same uniform positive lower bound \lambda_0/2 .

Proof.
The limit relationship of the curvature tensor follows from the uniform convergence property of DOG discretization. Since the original curvature field possesses a uniform positive lower bound, the discretization does not destroy the lower bound structure; therefore, the discrete curvature also retains a positive lower bound for its eigenvalues (see [3]).

5. Reformulation of the Kakeya Set in the MOC-DOG Model

Definition 5.1 (MOC Kakeya Set)
A compact set K\subset\mathbb{R}^n is called an MOC Kakeya set if it satisfies:

1. Directional completeness;
2. The induced curvature tensor \mathbf{R}_K has full rank everywhere on K ;
3. The minimum eigenvalue of the curvature tensor possesses a uniform positive lower bound \lambda_0>0 .

Theorem 5.2 (Classical Kakeya Set ⇔ MOC Kakeya Set)
A classical compact Kakeya set in Euclidean space is equivalent to an MOC Kakeya set.

Proof.
The conclusion follows directly from Lemma 3.2.

Corollary 5.3
The Kakeya conjecture is equivalent to the following proposition: For any MOC Kakeya set K , there exists a constant c>0 such that for sufficiently small \varepsilon>0 ,

N_{\text{DOG}}(\varepsilon) \ge c \, \varepsilon^{-n}.

If this inequality holds, combined with the dimensional formula from Axiom 3, it directly yields \dim_H K = n .

At this point, this paper has completely transformed the classical Kakeya conjecture in geometric measure theory into a problem of estimating the lower bound of the covering number within the DOG system. The companion paper will provide a rigorous proof of the above inequality, thereby completing the proof of the Kakeya conjecture in all dimensions.

References

[1] Zhang, S. (2026). Curvature Rigidity Theorems in Multi-Origin Curvature Geometry.
[2] Zhang, S. (2026). Geometric Applications of the ECS Conservation Law and the MIE Extremal Principle.
[3] Zhang, S. (2026). Uniform Convergence Properties of DOG Discretization and the Axiom of Dimensionality.