Fractional-Prime Frequency Division Architecture: Topological Paradigm of Structured Information and Computing Power Regularization

Author: Zhang Suhang, Luoyang, Henan

Abstract

Modern big data systems and generative information fields generally exhibit structural features of nested stratification, self-similar iteration and continuous scaling. Traditional Turing machine computing paradigms and classical information systems mainly process information via full-domain sampling and discrete encoding. When confronted with massive high-dimensional data, this indiscriminate traversal pattern often leads to linear expansion of data volume and computing overhead, gradually creating a practical dilemma of information redundancy and overload of computing resources.

This paper establishes a fractional-prime frequency division architecture for information regularization. Breaking away from the established framework of full-domain traversal in traditional information engineering, this architecture formulates a dual-track topological paradigm: small fractional-primes govern macroscopic topological routing, and large fractional-primes generate microscopic nested details.

The research demonstrates that small fractional-primes feature short-period recursion and naturally form the global topological skeleton of structured information. They support full-domain dimensionality reduction and rapid indexing at low computing cost. Large fractional-primes possess ultra-long recursive cycles and naturally carry fine microscopic details at high dense scales. They can generate massive high-precision information on demand from topological parameters. This dual-track frequency division system replaces enormous sets of raw discrete data with a limited set of topological generation parameters, decoupling the rigid linkage between data volume and computing consumption. It provides an original underlying mathematical support for next-generation information systems, high-dimensional data regularization and lightweight computing.

Keywords: fractional-prime frequency division; structured information; dual-track architecture; topological generation; computing power regularization

1. Introduction: Surpassing the Turing Traversal Paradigm

The foundational underpinnings of modern information science rest on classical statistical information theory and the Turing machine model. Its core logic is built upon statistical characteristics of discrete, random information, with processing schemes centred on point-by-point scanning, full-data storage and matrix traversal.

Such a framework works efficiently for fully disordered random information. However, it diverges from the essential nature of high-value information found in natural systems and modern digital spaces. Fluid evolution fields, celestial mass distributions, repeated gene segments, high-dimensional neural network manifolds and other core datasets commonly feature strong structural properties: self-similar nesting and hierarchical iteration.

Traditional architectures struggle to recognise and exploit the intrinsic topological order embedded in information. This gives rise to a recurring engineering trade-off: finer details demand larger datasets, and larger datasets demand greater computing power and energy consumption.

This paper proposes the fractional-prime frequency division topological architecture. It seeks to transform the conventional “store first, then traverse” workflow into an active generation scheme: define topological skeletons, generate details by parameters, and iterate on demand. It explores the mathematical basis for representation and scheduling of structured information.

Within the fractional-prime framework, fractional-primes are classified into two families of topological primitives by their primitive recursive cycle scales: small fractional-primes for macroscopic topological systems and large fractional-primes for microscopic nested systems.

2. Axioms of the Fractional-Prime Frequency Division Topology: The Dual-Track Division Framework

2.1 Small Fractional-Primes: Global Macroscopic Skeleton and High-Speed Routing

Small fractional-primes have short recursive cycles and rapidly convergent topological structures. In massive information workflows, they bypass intricate microscopic details. Relying on their stable topological configuration alone, they enable coarse global filtering, dimensional regularization and global indexing.

Their core merit lies in governing the global view with low computing overhead. Minimal topological parameters lock the order framework of the information field, alleviating the computing burden imposed by traditional workflows that must traverse the entire dataset to gain global awareness. They constitute the macroscopic scheduling layer of structured information systems.

2.2 Large Fractional-Primes: Microscopic Nested Details and High-Dimensional Generation

Large fractional-primes feature ultra-long recursive cycles, multi-layer nesting and densely scaled native topological traits, matching the fine microscopic architecture of naturally structured information.

Large fractional-primes do not require pre-storage of massive raw data. Using their topological primitives and hierarchical ratio parameters, they can generate microscopic detail information of arbitrary precision and density on demand via controlled recursive iteration.

Their core merit lies in generating massive details from parameters. Topological iteration replaces stacked discrete samples, easing data inflation caused by high-precision content. They constitute the microscopic generation layer of structured information systems.

3. Core Mechanism of the Frequency-Division Architecture: Decoupling Data and Computing Resources

3.1 The Computing Bottleneck of Traditional Architectures

Traditional computing systems operate on indiscriminate traversal: as information volume grows, sampling scope and iteration count rise in tandem, driving sharp growth in computing and storage demand. Computational complexity is tightly coupled to data volume. This is the fundamental mathematical source of excessive reliance on high-end computing hardware and rampant information proliferation.

3.2 Innovation of the Dual-Track Frequency-Division Topology

The fractional-prime frequency division architecture proposed in this paper separates the topology and computing workload for macroscopic routing and microscopic generation:

First, the global control phase invokes small fractional-prime topology. Regardless of the total data scale, short-cycle topology quickly locks the overall structure and completes global classification and indexing, with low computing costs independent of data expansion.

Second, the detail rendering phase selectively invokes large fractional-prime topology. Only when local high-resolution analysis is required, long-cycle recursion is activated to precisely generate dense details in targeted regions, without full-domain computation over the entire massive dataset.

This establishes a new computing rule: computing overhead is determined not simply by the total data volume, but controllably by iteration depth and the selected observation scale.

4. Redundancy Elimination Mechanism for Structured Information under the Fractional-Prime Architecture

Three core topological conclusions are formulated in this work:

1. Massive self-similar structured information can be treated as a topological field formed by superimposed small and large fractional-primes with tuned ratios.

Digital structured information is more than a mechanical pile of discrete samples; it possesses stable skeletons, hierarchical evolution and scale resonance.

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2. One major source of modern information redundancy is scale aliasing introduced by undifferentiated processing.

Traditional workflows do not distinguish macroscopic from microscopic content or short from long cycles. Everything is stored as discrete samples, which easily inflates data volume artificially.

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3. Dual-track frequency-division regularization with small and large fractional-primes transforms information from bulk accumulation to parametric order.

Small fractional-primes organise global order; large fractional-primes carry fine-grained information. Topological order replaces discrete disorder, standardising structured information and reducing its footprint.

5. Distinction between this Paradigm and Conventional Information Systems

5.1 Complementarity with the Shannon Entropy Framework

Shannon information theory centres on statistical entropy and applies to the measurement of disordered, random information. The frequency-division architecture here is built on topological recursion and serves the regularization of ordered, nested structured information. Disordered information belongs to Shannon; ordered structured information belongs to fractional-prime topology. The two systems complement each other and together form the complete theoretical dimension of modern information science.

5.2 Logical Advancement beyond Classical Turing Traversal Computing

A Turing machine performs linear traversal and passive reading and depends on complete raw datasets. This paradigm implements topological layering and active generation, governed by tunable topological parameters. It upgrades the paradigm from “ read all data before computation ” to “ define the structure and then generate content ”, achieving a dimensional shift in computing logic.

5.3 Reversal of Direction Compared with Traditional Fractal Algorithms

Traditional fractal workflows scan data first and fit rules, relying on traversal of source data. This paradigm defines fractional-prime topological primitives and constructs macroscopic skeletons first, and generates data from intrinsic mathematical structures. The research approach shifts from inductive fitting to deductive generation.

6. Conclusion: Topological Evolution of Information Computing Systems

This paper constructs a dual-track frequency division topological architecture based on small and large fractional-primes. Small fractional-primes handle macroscopic topological routing while large fractional-primes produce microscopic details, achieving full-scale topological regularization of structured information.

This architecture fundamentally releases the rigid coupling between data volume and computing load, mitigates the problem of redundant proliferation of structured information, and reduces the computing requirements of large-scale information systems. It provides a novel topological theory and technical paradigm for big data processing, generative modelling, high-dimensional retrieval and lightweight computing infrastructure.