MOC Information Topology: The Theory of High-Low Dimensional Inverse Topological Information Conservation

Author: Zhang Suhang

Abstract

Traditional geometry, fractal theory and dimensionality reduction theories have long been plagued by inherent cognitive bias: mappings between high and low dimensions are inherently unidirectional and irreversible. Changes in dimensionality inevitably incur topological structural loss and information degradation. Existing projection, compression and univariate fractal construction methods are all premised on the assumption that cross-dimensional transformation is lossy. They fail to account for the objective law of bidirectionally reversible, structure-preserving growth between two-dimensional planar structures and three-dimensional spatial structures in nature.

Inspired by the natural inverse topological relationship between 2D leaf veins and 3D root systems of plants, this paper extracts the underlying universal logic of cross-dimensional evolution and formally establishes the complete theoretical framework of Information Topology. The core macroscopic conclusion: high-dimensional systems and low-dimensional systems admit strictly inverse topological transformation relations. During regular continuous dimensional evolution, all topological information of the system is permanently conserved; dimensional variation cannot annihilate, distort or irreversibly damage topological structures. Dimensional transformation merely redistributes the spatial form of topological flux, while preserving the inherent topological properties of connectivity, hierarchy and redundancy.

This paper further clarifies that classical information theory conflates "metric information" and "structural information". Metric information undergoes flattening or redistribution during dimensional transformation, whereas structural information remains conserved. Structural invariance constitutes a conservation law more fundamental than metric conservation. Based on the geometric foundation of multi-origin curvature geometry, this paper qualitatively interprets the closed structure of high-low dimensional inverse transformation, and defines the core boundary distinguishing this theory from classical fractals and traditional lossy dimensionality reduction systems. It overturns the traditional paradigm that cross-dimensional transformation must cause information loss, and provides a novel fundamental macroscopic theory for fractal topology, complex networks and spatial information systems.

Keywords: Information Topology; high-low dimensional inversion; conservation of topological information; multi-origin curvature; fractal topology; flux redistribution; structural conservation; metric information; structural information

1 Introduction

Dimensional transformation is a fundamental problem shared by geometry, network theory, biological morphology and engineering topology. A fixed mindset has long prevailed: mapping from high to low dimension is information compression that necessarily discards local features; mapping from low to high dimension is structural expansion that inevitably introduces extraneous disordered branches. No symmetric, invertible closed relation exists between the two types of transformation.

Nevertheless, natural topological growth presents contrary evidence: two-dimensional leaf veins can fully extend into three-dimensional root systems, and three-dimensional root systems can also be completely contracted back to the original two-dimensional structure. The two processes are mutual inverses, with connectivity and hierarchical topology preserved throughout. This phenomenon cannot be explained by Euclidean rigid projection, classical fractal theory or various lossy dimensionality reduction algorithms, revealing a fundamental gap in existing theories: a universal, qualitatively complete macroscopic theory for bidirectionally reversible cross-dimensional information conservation is absent.

More fundamentally, classical information theory identifies information with measurable probability distributions, flux or coordinates. It therefore misinterprets the spatial redistribution of metric information during dimensional transformation as the loss of information itself. This paper asserts: information is structure, not quantity. Conservation of structure constitutes the cornerstone proposition of this work.

Drawing inspiration from natural prototypes, this paper breaks free from the shackles of concrete geometric details within a single dimension. It constructs the system of Information Topology at the macroscopic level and establishes two core guiding principles: first, bidirectionally reversible continuous transformations can be constructed between arbitrary high-dimensional and low-dimensional topological systems; second, regular cross-dimensional evolution satisfies global conservation of topological information. Built upon the geometry of multi-origin curvature and grounded in the macroscopic postulate that steady-state topology tends toward optimal information efficiency, this complete theoretical framework characterizes the evolutionary laws of dimensional change in both natural and artificial topological systems.

2 Fundamental Macroscopic Postulates of the System

2.1 Core Viewpoints of Multi-Origin Curvature Space

Traditional space adopts single-origin rigid metric, which locks the degrees of freedom for high-dimensional extension and can only generate truncated lossy mappings. Information Topology employs multi-origin curvature to construct space: orthogonal degrees of freedom in low-dimensional space are constrained, and flux tends to converge on the plane; multiple independent curvature origins act in high-dimensional space, enabling flux to diverge in stratified three-dimensional forms.

Macroscopic definition: Spatial dimension is the number of independent orthogonal degrees of freedom available for the distribution of topological flux.

2.2 Postulate of Optimal Steady-State Topological Transport

All long-term stable topological systems spontaneously adjust their spatial configuration toward a steady state that maximizes effective information flux per unit energy consumption. This postulate serves solely as a macroscopic constraint to explain the ordered, directional nature of high-low dimensional inverse evolution and excludes subdivided derivations of variational, numerical or biological laws.

2.3 The Essence of Information: Structure Rather Than Quantity

Represented by Shannon’s information theory, conventional information theory equates information with uncertainty-eliminating measures, quantified by probability distributions and bits. This framework holds within single-origin Euclidean space, yet it conflates two concepts of different hierarchical levels: metric information and structural information.

- Metric information: the spatial distribution form of information in concrete coordinates, probabilities and flux volumes. It changes as spatial degrees of freedom contract.

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- Structural information: topological invariants of the system including connectivity, hierarchical genealogy, homology classes and Betti numbers. It remains unchanged under dimensional shifts.

Take the blood circulatory system as an analogy: total blood volume and flow velocity change drastically from the aorta to capillaries (metric alteration), yet the physiological structure and functional topology carried by blood remain intact and undergo precise spatial redistribution. Shannon information theory observes the variation in flux and thus judges information to be lost. Within the framework of Information Topology, however, flux variation merely represents the flattening of metric information, while structural information remains conserved.

We hereby propose a rigorous definition of information: Information is structure, not quantity. Dimensional transformation alters the spatial distribution of metric information, without changing the total amount of structural topological information. The classical conclusion that "cross-dimensional transformation must incur information loss" is a special case of metric distortion under single-origin Euclidean projection and lacks universal geometric validity.

Structural invariance forms a conservation law more fundamental than metric conservation. This does not overthrow classical conservation laws, but extends them to the domain of topological structure.

3 Natural Inspiration: Inverse Prototypes between 2D and 3D

The growth systems of plants intuitively demonstrate the fundamental form of high-low dimensional inversion:

Low-dimensional leaf veins are constrained to the plane, with flux converging directionally. The dimensional-raising process relaxes the curvature constraint along the depth axis, and the original complete topology is preserved as it extends to form high-dimensional parallel root systems. Reversing the operation tightens the constraint, and the diffuse high-dimensional flux is reorganized into a planar regular form, fully recovering the initial low-dimensional planar structure.

During this process, the metric information (area, flux, density) of leaf veins differs drastically from that of root systems, yet their topological structures — branching hierarchy, connectivity and redundancy patterns — are isomorphic. This natural phenomenon visually proves that dimensional rise and fall can be bidirectionally reversible and structurally lossless, and such "structural invariance" itself represents conservation. Generalizing this property to all high-low dimensional topological systems yields the universal theory.

4 Macroscopic Nature of High-Low Dimensional Inverse Transformation

This paper defines a pair of inverse topological mappings: the dimension-raising operator extends low-dimensional structures into high-dimensional ones, while the dimension-lowering operator condenses high-dimensional structures into low-dimensional ones. The two operations form a complete closed inverse relation; successive application restores the original topology.

At the macroscopic level, inverse transformation possesses four conservation properties:

1. Conservation of connectivity: switching between high and low dimensions does not rupture the global connected structure.

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2. Conservation of hierarchy: the primary-secondary genealogy of branches is fully transmitted bidirectionally.

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3. Conservation of redundancy: high-dimensional multi-path fault tolerance can be completely inherited into low-dimensional space.

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4. Conservation of structure: all topological information of the system is preserved without loss. Structural invariance implies conservation.

The core mechanism of transformation lies in a single macroscopic logic: dimensional switching is essentially the global bidirectional redistribution of topological flux. Dimension raising releases degrees of freedom, enabling convergent flux to diverge in three dimensions; dimension lowering tightens degrees of freedom, reorganizing diffuse flux into planar regular forms. The geometric outline changes, but the underlying topological structure and total information remain constant.

5 Core Macroscopic Theorems of Information Topology

5.1 Theorem of High-Low Dimensional Inverse Topological Information Conservation

Under regular continuous transformation constructed in multi-origin curvature space, high-dimensional topological systems and low-dimensional topological systems admit strictly inverse transformations. Topological information is globally conserved throughout bidirectional dimensional evolution. Reciprocal transformation fully recovers the original configuration, with no annihilation or degradation of topological information accompanying dimensional change. The structures (connectivity, hierarchical genealogy, homology classes) are strictly isomorphic before and after transformation, and this structural invariance constitutes a topological conservation law more fundamental than metric conservation.

5.2 Paradigm of Dimensional Transformation (Core Thesis of the Paper)

High-low dimensional inverse transformation \triangleq global bidirectional redistribution of topological flux

High-low dimensional inverse transformation \neq gain/loss of information, structural damage or irreversible reconstruction

The conventional belief that cross-dimensional transformation inevitably loses information is merely a limitation of single-origin Euclidean rigid projection and has no universal geometric meaning. Information Topology revises this one-sided conclusion.

5.3 Two Macroscopic Corollaries

1. Connectivity of high-low dimensional topological systems remains equal under inverse transformation; dimensional switching does not weaken the network connectivity capacity.

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2. Redundant topological parallelism in high dimensions can be fully transmitted to low dimensions via dimensionality reduction, granting ordinary planar networks strong fault tolerance against local disconnection.

6 Theoretical Macroscopic Positioning and Comparison

1. Compared with classical fractal geometry: traditional theories only investigate self-similar structures within a single dimension, without a unified macroscopic framework for bidirectional high-low dimensional inversion and information conservation. This paper fills the gap in cross-dimensional topology.

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2. Compared with conventional dimensionality reduction and projection theories: all existing methods are unidirectional lossy compression with no closed inverse structure. This work constructs a universal reversible lossless macroscopic theory for cross-dimensional topology.

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3. Compared with classical information theory: Shannon information theory addresses metric information, and its rate-distortion conclusion holds only at the metric level. This theory addresses structural information and reveals the underlying law of structural invariance as conservation. The two describe distinct domains and must not be conflated.

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4. Overall definition of the Information Topology system: built upon the geometry of multi-origin curvature, with optimal transport as its evolutionary postulate, high-low dimensional inverse transformation as its core mechanism, and conservation of topological information as its central theorem. It can uniformly interpret all dimensional evolutionary behaviors of natural biological topology and artificial engineering networks, forming an independent and complete fundamental theory.

7 Conclusion

Inspired by natural 2D–3D topological reciprocal growth phenomena, this paper establishes Information Topology, the theory of high-low dimensional inverse topological information conservation, at the macroscopic level, avoiding intricate detailed derivations.

The core macroscopic result: high-dimensional and low-dimensional topological systems possess complete inverse transformation relations. Topological information is globally conserved during regular continuous dimensional evolution. Dimensional rise and fall do not alter inherent topological structures, and merely implement convergent or divergent redistribution of flux across spaces of different degrees of freedom. This fundamentally revises the traditional geometric view that cross-dimensional transformation inevitably loses information and lacks bidirectionally reversible closed transformation.

This paper further establishes that structural invariance also qualifies as conservation. This proposition extends conservation laws from the metric domain to the topological structural domain, offering a new underlying interpretation for the dimensional evolution of complex systems. This theory builds a self-consistent macroscopic topological framework, fills the fundamental theoretical gap in fractal and network dimensional transformation, and provides new principles for analysis, design and optimization of various spatial topological systems.