Structural Information Theory: The Dual-Layer Complementary Foundational Framework of Dimension-Preserving Topological Transport and Shannon Information Theory

Author: Zhang Suhang (Heluo School of Mathematics)

Abstract

Shannon information theory established an algebraic measurement system for the quantity of information, resolving the boundary problem of information “quantity” and defining the universal laws of entropy measurement, coding limits, and channel capacity. However, its theoretical framework is entirely detached from spatial topological structures and geometric transport constraints, rendering it incapable of characterizing the structural evolution, topological persistence, or spatial flux distribution of information during dimensional transformations. Based on the MOC (Multi-Origin Curvature) axiomatic system, this paper introduces the core concepts of topological flux and structural information, and establishes a foundational theory of dimension-preserving topological transport under dimensional transformations. We explicitly assert that Shannon information theory and MOC-based structural topology constitute two mutually orthogonal foundational layers of information science—the former as the algebraic quantification layer governing probabilistic measures and bit limits, and the latter as the topological transport layer governing spatial structures, path distributions, and dimensional evolutions. The two are fully complementary, non‑overlapping, and synergistically complete, together forming the complete foundational architecture of information science. Accordingly, this paper formally defines Structural Information Theory as an independent foundational branch of information science.

Keywords: Structural Information Theory; Shannon Information Theory; Topological Flux; Dimension-Preserving Dimensional Transformation; MOC Multi-Origin Geometry; Information Topological Transport

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1 Introduction

1.1 Achievements and Inherent Boundaries of Shannon Information Theory

In 1948, Shannon founded classical information theory based on probability and statistics, establishing information entropy, channel capacity, and coding theorems, and uniformly resolving the quantitative boundaries of measurability, compressibility, and transmissibility of information. It serves as the fundamental “dimensional system” of modern information science.

The essential characteristic of Shannon’s theory is its complete de‑spatialization and de‑structurization: information is abstracted as probabilistic random variables, and channels are purely algebraic abstract channels, devoid of geometric forms, topological connectivity, hierarchical structures, or spatial path attributes.

This feature is both the source of its universality and its theoretical blind spot for over seventy years: Shannon’s system can only answer “how much information exists,” but cannot answer “how information exists, flows, deforms, and persists within spatial structures.”

1.2 The Long‑Missing Topological‑Structural Dimension in Information Science

Contemporary complex information systems—including three‑dimensional IoT, 3D chip interconnects, spatial digital twins, biological network topologies, and field‑based volumetric transmission networks—are all spatially structured information systems. Their core problem is no longer merely the bit‑rate limit, but rather: how information undergoes structural migration, conformal mapping, flux redistribution, and topological persistence between high‑ and low‑dimensional spaces.

Addressing this gap, this paper focuses on the laws of conformal dimension reduction from high to low dimensions; the strictly dual low‑to‑high conformal dimension expansion obeys symmetric flux conservation and inverse transformation rules, together constituting a complete dimensional transformation system. The detailed axiomatic derivation, quantitative models, and engineering applications are left for subsequent series of studies.

1.3 Core Proposition of This Paper

Shannon information theory characterizes the algebraic‑quantitative attributes of information; MOC structural topology theory characterizes the spatial‑structural attributes of information. Together they form a “quantitative‑topological dual orthogonal” foundational architecture for information science, supplementing the geometric foundation long missing from modern information theory.

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2 Essential Boundaries and Applicability Domain of Shannon Information Theory

2.1 Core Contributions of Shannon Theory

Shannon information theory established a complete algebraic system for information quantification and transmission limits:

1. Information entropy:
H = -\sum p_i \log p_i
2. Channel capacity:
C = B \log_2(1 + S/N)
3. Source and channel coding theorems, defining the theoretical limits of lossless compression and reliable transmission.

The entire theory is built entirely on probability space, independent of geometry, topology, dimensions, or structural forms.

2.2 Inherent Structural Deficiencies of Shannon Theory

1. No spatial dimension constraint: information is not tied to a geometric carrier and has no dimensional definition.
2. No topological structure constraint: channels have no nodes, connections, loops, or hierarchies.
3. No morphological evolution capability: cannot describe structural changes during dimensional ascent or descent.
4. No path‑flux mechanism: lacks the flux allocation and redistribution laws of spatial networks.

In summary: Shannon information theory is a purely numerical information theory, not a spatial‑structural information theory.

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3 MOC Structured Dimensional Transformation Axiomatic System (Dimension‑Reduction Trunk)

3.1 Essence of Conformal Dimension Reduction: Flux Redistribution and Structural Convergence

Under the MOC multi‑origin geometry framework, conformal dimension reduction is not information loss; it is defined as:

\text{Dimension Reduction} = \text{Controllable Topological Flux Redistribution} + \text{Ordered Structural Convergence of Spatially Dispersed Information}

The dispersed topological flux in high‑dimensional space is orderly aggregated and redistributed toward low‑dimensional planes during dimensional compression; the flux in the depth dimension decays while the flux borne by the low‑dimensional plane increases, with the overall structural morphology preserved.

3.2 Three Core Axioms (Dimension‑Reduction Trunk + Dual Completeness of Expansion)

Axiom I (Topological Fidelity Invariance):
A conformal dimension‑reducing transformation from high to low dimensions completely preserves all connected components, loop structures, adjacency relations, and hierarchical topologies of the system. Only the metric coordinate parameters of the depth dimension are discarded; all core topological structural information is fully migrated to the low‑dimensional space.

Axiom II (Topological Flux Conservation):
Throughout the dimensional transformation, the total global topological flux of the system is a strict invariant:

\Phi_{\text{total}}(3D) = \Phi_{\text{total}}(2D)

This equation describes flux conservation for 3D‑to‑2D conformal reduction. The dual expansion transformation satisfies the symmetric conservation relation:

\Phi_{\text{total}}(2D) = \Phi_{\text{total}}(3D)

Dimensional ascent or descent only changes the distribution proportions of flux across different spatial dimensions; the total global topological flux is absolutely conserved.

Axiom III (Lossless Limit Approximability):
Structured dimension reduction can approach the algebraic compression lower bound given by Shannon’s rate‑distortion theory, while, relying on the MOC topological self‑reconfiguration mechanism, it maintains complete structural integrity during extreme dimensional convergence, achieving both numerical approximation to the limit and zero structural loss.

3.3 Essential Differences between Shannon Compression and MOC Structured Dimension Reduction

Aspect Shannon Compression MOC Structured Reduction
Core objective Minimize bit‑coding length Minimize system spatial dimensions
Distortion characteristic Probabilistic lossy compression allowed Topological structure strictly lossless and conformal
Constraints Probability distribution constraints Topological connectivity and flux conservation constraints
Application domain Data bit compression, coding optimization Spatial network topology simplification, structural mapping

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4 Four‑Layer Orthogonal Complementary Structure of the Two Systems

4.1 Overall Complementary Relationship

Shannon information theory governs the quantitative limits of information; Structural Information Theory governs the spatial‑morphological and path limits of information. The two systems are orthogonal, independent, non‑overlapping, and fully cover the entire information system.

4.2 First Layer Complementarity: Numerical Abstraction vs. Spatial Carrier

Shannon’s system is detached from geometry, a purely probabilistic numerical abstraction; the MOC system anchors information to real spatial topology, dimensional structure, and multi‑origin geometric carriers, completing the transition from “pure numbers” to “structured entities.”

4.3 Second Layer Complementarity: Algebraic Compression Lower Bound vs. Topological Conformal Convergence

Shannon provides the algebraic lower bound for data compression; MOC provides a spatial structural convergence scheme that approaches that lower bound, ensuring that at extreme compression the topological structure does not collapse, lose, or deform.

4.4 Third Layer Complementarity: Coding Protocol Specifications vs. Topological Architecture Design

Shannon’s system underpins signal‑layer coding, modulation, and transmission protocols; the MOC system underpins spatial‑layer network topology, traffic scheduling, structural reduction, and system simplification design.

4.5 Fourth Layer Complementarity: Probabilistic‑Topological Bivariate Completeness

The state of a complete information system must be described by two variables:

\text{System State} = \mathcal{F}(\text{Probability Distribution},\ \text{Topological Structure})

Shannon addresses the probabilistic variable; Structural Information Theory addresses the topological variable. Neither theory alone can fully describe complex information systems.

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5 Foundational Positioning of Structural Information Theory

5.1 Disciplinary Definition

Structural Information Theory is the foundational branch of information science that studies the modes of existence, transport paths, dimensional transformations, conformal mappings, and flux evolutions of information within spatial topologies.

The complete foundational architecture of information science is:

\text{Information Science} = \text{Shannon Information Theory (Algebraic Layer)} + \text{Structural Information Theory (Geometric‑Topological Layer)}

5.2 Orthogonality of the Dual‑Layer System

Aspect Shannon Algebraic Layer MOC Geometric‑Topological Layer
Core question How much information can be transmitted? How does information transport spatially?
Basic variables Probability distributions Topological structures
Conservation constraint Probability conservation Topological flux conservation
Theoretical limit Channel capacity limit Conformal dimensional transformation limit

5.3 Engineering Value

Optimal design of modern information systems must simultaneously satisfy:

1. Shannon entropy constraints: optimal coding efficiency and transmission capacity;
2. Topological flux constraints: stable spatial structure, reliable paths, and lossless topology.
Both are indispensable.

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6 Conclusion

1. Shannon information theory completed the algebraic quantification system of information but lacks descriptions of spatial topology and dimensional structure, covering only the numerical side of information science.
2. Based on the MOC multi‑origin geometry, this paper establishes a theory of dimension‑preserving topological transport, revealing the mechanisms of topological fidelity and flux conservation during dimensional reduction, and thereby supplying the missing geometric foundation of information science.
3. The Shannon algebraic system and the MOC topological system form a four‑layer strictly orthogonal complementarity, constituting a truly complete dual‑layer foundational basis for information science.
4. The Structural Information Theory established herein upgrades information from a purely bit‑quantified entity to a unified “quantity‑form” framework.
5. The dual dimensional expansion and reduction transformations together form a complete topological transport system. This paper completes the core dimension‑reduction trunk theory; the accompanying expansion theory will be elaborated in subsequent work, forming a closed loop.

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References

[1] Shannon, C. E. A mathematical theory of communication. Bell System Technical Journal, 1948.
[2] Zhang, S. Dimension reduction as flux redistribution: the engineering essence of point‑set flattening. 2026.
[3] Zhang, S. Foundational axiomatic system of MOC multi‑origin geometry. 2026.
[4] Cover, T. M., Thomas, J. A. Elements of Information Theory. Wiley, 2006.