The Topological Essence and Engineering Substance of Dimensional Ascent: Flux Expansion, Planar Point Spatialization, and the Maximum Information Efficiency Axiom from 2D to 3D

Author: Zhang Suhang

Core Axiomatic Support: Multi-Origin Curvature (MOC) Framework, Maximum Information Efficiency (MIE) Axiom, Information-Matter Flux Dual Transformation Theory

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Abstract

Grounded in the Multi-Origin Curvature (MOC) geometric framework and the Maximum Information Efficiency (MIE) axiom, this paper systematically establishes a topological theory of active dimensional ascent from two dimensions to three dimensions, revealing that dimensional transition is neither geometric thickening nor structural expansion, but rather the fundamental process of planar point set spatialization through topological unfolding, high-dimensional degree-of-freedom liberation, and global flux optimal redistribution.

This paper definitively articulates the topological essence and engineering substance of dimensional ascent: under the condition of preserving the original planar topological connectivity, hierarchical structure, and core nodes intact, the relaxation of depth-direction curvature constraints enables planarly convergent flux to stratify, diverge, and reconfigure within three-dimensional space, thereby achieving the extremal enhancement of global system information efficiency.

Furthermore, this paper establishes a strict topological dual relationship between ascent and descent, demonstrating that two-dimensional spatialization ascent and three-dimensional planarization descent constitute reciprocal transformations under the MIE constraint, forming a complete closed dimensional evolution system. The plant leaf venation–root system growth topology serves as the natural prototype for theoretical empirical matching, with quantitatively testable topological scaling laws and efficiency threshold predictions provided.

Keywords: Dimensional Ascent; Multi-Origin Curvature (MOC); Maximum Information Efficiency (MIE); Planar Point Spatialization; Flux Distribution; Topological Duality

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1. Introduction

Dimensional transformation constitutes a fundamental common problem underlying geometric topology, information dynamics, biological morphogenesis, and engineering network design. In contrast to dimensionality reduction, which has developed a unified understanding as "high-dimensional redundancy convergence and structural compression regularization," active dimensional ascent from two dimensions to three dimensions has long lacked rigorous topological definition. Two prevalent misconceptions persist: the equation of ascent with geometric thickening and volumetric expansion, or its identification with disorderly complexity increase and redundant proliferation.

Such naive geometric cognition fails to engage with the first principles of dimensional evolution. Grounded in the Multi-Origin Curvature (MOC) spatial structure and the Maximum Information Efficiency (MIE) extremal axiom, this paper establishes definitively that dimensionality is essentially the number of orthogonal degrees of freedom available for independent flux allocation, and dimensional transformation is essentially the topological redistribution of global flux.

Accordingly, this paper constructs a rigorous theory of ascent: ascent is topology-preserving, directed, and optimal spatial unfolding, not random deformation or node expansion. Simultaneously, it establishes a closed dual system of ascent and descent, achieving a unified axiomatic description of the 2D↔3D dimensional transformation.

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2. Axiomatic Foundation and Core Definitions

2.1 Multi-Origin Curvature (MOC) Spatial Framework

The MOC spatial framework transcends the rigid single-origin Euclidean constraint, defining spatial geometry collectively through multiple independent curvature origins:

· Two-dimensional space: Only planar orthogonal curvature is activated; the depth direction is rigidly locked; flux possesses no vertical allocation degrees of freedom; throughput is entirely confined to the planar manifold.

· Three-dimensional space: An independent depth-direction curvature origin is introduced; the constraint is relaxed; point sets acquire spatial extension degrees of freedom; flux can achieve three-dimensional stratification and multidirectional allocation.

Essential Definition of Dimensionality: Spatial dimensionality is equivalent to the total number of orthogonal degrees of freedom over which topological flux can be independently regulated.

2.2 Maximum Information Efficiency (MIE) Axiom

The evolution of steady-state topological systems satisfies the extremal condition of the global information efficiency functional:

\delta \mathcal{J}_{\text{info}} = \delta \int \frac{dI}{dE \cdot dt} \, d\mathcal{V} = 0

Note: The symbols in the above expression denote abstract functional notation. This paper presents the axiom in compact form without展开 detailed derivation, as the theoretical framework is qualitatively established at the macroscopic level.

Dimensional ascent constitutes directed extremal evolution by which systems break through planar transmission bottlenecks, reduce per-unit information cost, and enhance global throughput efficiency—not random disordered deformation.

2.3 Core Conceptual Definitions

1. Planar Point Spatialization: The coordinate attributes of planar point sets remain unchanged; only their topological degrees of freedom are elevated, transforming them from fixed planar positions into spatial nodes capable of three-dimensional stratification, stereoscopic connectivity, and multidirectional allocation.

2. Ascent Flux Expansion: Planarly convergent throughput is allocated in an ordered manner across multiple directions and hierarchical levels of three-dimensional space through degree-of-freedom liberation, achieving throughput equilibration.

3. Ascent Topological Fidelity: The entire ascent process preserves the original two-dimensional backbone topology, connectivity relationships, and hierarchical spectrum intact, with no structural disruption or logical reconstruction.

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3. Topological Mechanisms of 2D-to-3D Ascent

3.1 Constraint Characteristics of Two-Dimensional Systems

Two-dimensional manifold structures feature constrained flux and single transmission pathways, prone to localized congestion and efficiency bottlenecks. Planar fractals and planar networks both belong to convergent flux topologies, characterized by planar aggregation and transmission—the typical natural prototype being the plant leaf venation system.

3.2 Ascent Trigger Mechanism

When the planar system's throughput density reaches a critical threshold and the per-unit information transmission cost approaches the extremal boundary, the MIE axiom drives the system toward spontaneous ascent, corresponding to the relaxation of depth-direction curvature constraints in MOC space:

1. Opening of depth-direction orthogonal degrees of freedom;

2. Planar point sets acquiring spatial distribution capability;

3. Convergent throughput achieving three-dimensional分流;

4. Global information efficiency trending toward optimal steady state.

3.3 Point Set and Flux Ascent Transformations

3.3.1 Topological Ascent of Point Sets

Topology-preserving ascent mapping:

\mathcal{T}: P_2 \to P_3, \quad p_i(x_i, y_i) \mapsto p_i'(x_i, y_i, z_i)

The relative planar topology is fully preserved; depth coordinates are uniquely determined by the MIE extremal condition, achieving distortion-free spatialization.

3.3.2 Conservation in Three-Dimensional Flux Expansion

Total flux before and after ascent is conserved, while simultaneously satisfying the efficiency maximization constraint:

\iint_{\Omega} \vec{J}_2 \cdot d\vec{S} = \iiint_{\mathcal{V}} \vec{J}_3 \cdot d\vec{V}

\mathcal{J}_{\text{info}}(\vec{J}_3) = \max \mathcal{J}_{\text{info}}(\vec{J})

Planar concentrated throughput is transformed into three-dimensional equilibrated throughput, achieving transmission efficiency enhancement and bottleneck dissolution.

3.4 Core Topological Properties of Ascent

1. Topological Fidelity: The original backbone connectivity and hierarchical structure are completely preserved.

2. Non-Proliferative Nature: Ascent is essentially degree-of-freedom liberation and does not depend on node count expansion.

3. No Information Loss: Ascent achieves information gain and spatial expansion without topological feature annihilation.

4. Extremal Directedness: The ascent path is uniquely determined by MIE, constituting optimal ordered evolution.

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4. Topological Dual Closed System of Ascent and Descent

4.1 Dual Correspondence

 2D → 3D Ascent 3D → 2D Descent

Constraint State Relaxation Closure

Degrees of Freedom Liberation Convergence

Flux State Divergence Convergence

Topological Action Spatial Unfolding Planar Regularization

System Effect Efficiency Gain Cost Reduction

The two constitute reverse optimal processes of the same dimensional evolution system, with their underlying axioms entirely unified.

4.2 Closure of Reciprocal Operators

Ascent and descent mappings satisfy the strict reciprocal identity relations:

\mathcal{T}' \circ \mathcal{T} = \text{Id}_2, \quad \mathcal{T} \circ \mathcal{T}' = \text{Id}_3

Bidirectional dimensional transformation is completely reversible, topologically无损, and structurally closed.

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5. Natural Empirical Evidence: Plant 2D–3D Topological Growth

Plant morphogenesis serves as the natural empirical demonstration of the ascent theory:

1. Two-Dimensional Constraint Layer: Leaf venation constitutes a typical planar filling fractal, responsible for information capture and planar throughput convergence.

2. MIE-Driven Ascent: Inherent bottlenecks in planar throughput transmission drive the system toward spontaneous three-dimensional spatial expansion.

3. Three-Dimensional Expansion Layer: The leaf venation topology extends with connectivity preserved into the three-dimensional root fractal, achieving spatial divergence of material throughput.

4. Bidirectional Coupled Closed Loop: Planar information convergence and root-system material divergence form a 2D–3D reciprocal steady-state system.

The natural growth process strictly conforms to all core principles of this paper: topology preservation, flux expansion, degree-of-freedom ascent, and efficiency optimality.

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6. Quantitatively Testable Theoretical Predictions

This paper provides two experimentally verifiable topological scaling conclusions:

6.1 Point Set Density Scaling Law

The two-dimensional planar density $\rho_2$ and the three-dimensional ascent density $\rho_3$ satisfy:

\rho_3 \propto \rho_2^{\frac{2}{3}}

This scaling law originates from the natural dimensional ratio between three-dimensional volume and two-dimensional area ($V \sim L^3$, $A \sim L^2$), combined with the optimal matching relationship between density and spatial scale under MIE extremal constraints.

6.2 Ascent Efficiency Enhancement Threshold

Following two-dimensional-to-three-dimensional ascent, the lower bound of global information efficiency improvement satisfies:

\Delta \eta \geq 41.4\%

This threshold is rigorously derived from the MIE extremal condition and is verifiable through simulation and biometric statistical data.

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7. Conclusion

Within the Multi-Origin Curvature geometry and Maximum Information Efficiency axiom framework, the ascent process from two to three dimensions is essentially topology-preserving planar point set spatialization and optimal global flux spatial redistribution—not geometric thickening or structural expansion.

Ascent and descent constitute a strict topological dual reciprocal system: ascent liberates degrees of freedom, expands flux, and enhances information efficiency; descent converges degrees of freedom, regularizes flux, and reduces system cost. Both uniformly obey the MOC-MIE underlying axioms, achieving a theoretically self-consistent closed loop of the 2D–3D dimensional evolution system.

Natural plant topological growth validates the authenticity and universality of the theory, providing novel first-principle support for complex network topology design, spatial structure optimization, and biological morphology modeling.