The Reconstruction of Applied Science from the Perspective of FCFG: From Prime-Fraction Substrates to a Geometric Revolution in Industry and Physics

Author: Suhang Zhang

Luoyang School of Mathematics

Abstract

The research and development paradigms of traditional engineering technology, physical modelling, artificial intelligence and life science have long relied on empirical trial and error, massive computing iterations and discrete numerical simulation. Lacking a unified underlying geometric generative logic, they suffer from inherent bottlenecks such as high cost, blind iteration and suboptimal structures. Based on the core framework of Fractal Continued-Fraction Geometry (FCFG) and its bidirectional evolutionary law of prime fractions, this paper establishes a cross-disciplinary universal paradigm for structure generation. FCFG demonstrates that all complex nested fractal structures can be decomposed into the superposition, resonance and coupling of primordial substrate fractals with prime fractions 1/p. This theoretical breakthrough provides new underlying geometric generation rules for topological display design, graphics rendering acceleration, inverse design of metamaterials, communication array coding, tissue engineering, neural network pruning, turbulent flow modelling and other frontier fields. This paper defines a new division paradigm: theory generates structures, and engineering achieves implementation. It promotes applied science from the paradigm of "simulation and approximation driven by computing power" to "intrinsic generation driven by mathematical genes", offering a unified fundamental theoretical support for the next generation of cross-domain industrial technological innovation.

Keywords: FCFG; bidirectional evolution; prime fractions; primordial nested fractals; metamaterials; graphics rendering; 6G communication; artificial intelligence architecture; turbulence modelling

1. Introduction: Paradigm Bottlenecks in Applied Science and Geometric Breakthroughs

Cutting-edge engineering fields share a common underlying dilemma: complex microstructures, multi-scale systems and nonlinear dynamical systems cannot generate optimal configurations via unified mathematical logic. Researchers can only approximate feasible solutions through exhaustive computing and repeated experimental trial and error. Chip micro-nano structures, optical metamaterials, fluid vortex systems, biological branching networks and deep neural networks are all constrained by this empirically oriented, discrete and resource-intensive traditional research model.

Fractal Continued-Fraction Geometry (FCFG) formulates a universal axiom of bidirectional evolution: integers grow outward to construct topological skeletons, while prime fractions collapse inward to fill fine details. Among them, primordial nested fractals corresponding to prime fractions are indecomposable fundamental units for all self-similar structures. All complex composite structures arise from the multi-scale superposition and resonance of prime-fraction primitives.

This theory thoroughly reshapes the generative logic of engineering structures. Instead of blindly searching for optimal configurations via trial and error, humans can directly generate naturally optimal topological forms following fundamental mathematical rules. The theoretical work covers structural tracing, mechanism definition and paradigm construction. Parameter tuning, process adaptation, simulation verification and practical engineering implementation are left to industrial technology systems, forming a clear and efficient new division of labour for research and innovation.

2. Microelectronics and Information Industry: Eliminating Computing Redundancy and Realizing Intrinsic Geometric Optimality

2.1 Topological Reconfiguration of Novel Displays

Current pixel arrangements for OLED and MicroLED are engineering compromises devised to compensate for material defects, with inherent issues including moiré patterns, uneven brightness and visual loss. Under the FCFG framework, subpixel arrays arranged on primordial nested scales such as 1/3 and 1/5 can produce perfectly self-similar, interference-free topological configurations. Moiré artifacts are eliminated geometrically at the source, and uniform visual performance across the whole panel is achieved, breaking the bottlenecks of conventional display layouts.

2.2 Real-Time Generative Rendering in Computer Graphics

Traditional graphics rendering depends on massive texture storage and frequent GPU iterations for anti-aliasing, which incurs heavy computational costs and imposes an upper bound on detail. Following the recursive evolutionary rules of continued fractions, edge textures and natural fractal fine features of complex scenes can be generated in real time via prime-fraction primitive operators. Pre-made texture libraries are no longer required, and infinite natural details can be produced at low computational cost, fundamentally reshaping the underlying algorithms for real-time rendering, game engines and 3D modelling.

2.3 6G Communication and Intelligent Antenna Arrays

Antenna array topologies designed with the spiral fractal evolution logic of FCFG support multi-band compatibility, broadband resonance and low-loss transmission within a compact physical footprint. Meanwhile, recursive sequences of continued fractions can construct self-similar anti-interference coding schemes, suited for harsh transmission environments such as urban canyons, low-altitude space and underwater channels, to greatly boost communication stability and spectral efficiency.

3. Materials Science: Establishing a Mathematically Oriented Structural Customization System

The microstructure design of traditional metamaterials and nano-functional materials relies heavily on blind trial and error, lacking unified design criteria, and suffers from long research cycles and low success rates. FCFG supplies a fractal topological taxonomy for material design: according to target optical, acoustic, electromagnetic or thermodynamic properties, researchers can match primordial fractal substrates based on corresponding prime numbers, and realize precise inverse generation of geometric structures directly from performance requirements.

In new energy devices and quantum sensing, the topological coupling of linear fractals and spiral fractals enables the design of nanoelectrode and sensing structures with maximized specific surface area and highly self-similar charge distribution. It improves charge-discharge efficiency, sensing sensitivity and structural stability from a geometric perspective, opening a new design route for iterative upgrades of advanced functional materials.

4. Life Science and Artificial Intelligence: Decoding Natural Folding and Complex Networks

4.1 Tissue Engineering and Targeted Medicine

Human vascular networks, bronchial trees and glandular systems are naturally evolved optimal fractal nested architectures that conform fully to the bidirectional evolutionary laws of FCFG. Prime-fraction primordial fractals can be adopted to design the pore topology of biological scaffolds, matching the natural scaling rules of cell growth and facilitating cell adhesion and tissue regeneration. Fractal-structured drug carriers can be tailored to fit the hierarchical channels of microvessels in the human body, substantially enhancing the precision of targeted therapy.

4.2 Next-Generation Lightweight AI Architectures

Modern deep learning models suffer from bloated architectures, redundant parameters and severe computing waste. A core inference from FCFG states that all complex self-similar networks can be decomposed into recursive superpositions of a limited set of prime-fraction primitives. Instead of stacking massive parameters, large neural networks can be reconstructed with a small number of native recursive operators, achieving extreme model compression, faster inference and improved generalization, providing a new architectural paradigm for lightweight general artificial intelligence and edge deployment.

5. Classical Physics and Dynamical Systems: Unified Modelling of Multi-Scale Chaos

5.1 Turbulence and Wave Dynamics Modelling

Phenomena such as turbulent vortices, atmospheric perturbations and seismic wave propagation are essentially bidirectional dynamical processes: outward expansion of the spatial skeleton and inward convergence of fine scales. FCFG decomposes fluid evolution into the macroscopic extension governed by integer scales and microscopic nesting governed by prime-fraction scales. This breaks the limitations of conventional single-scale equations and establishes a multi-scale recursive algorithm for turbulence, addressing the long-standing challenge of accurate turbulence simulation.

5.2 Topological Classification of Chaotic Systems

For nonlinear chaotic dynamical systems, the macroscopic skeleton of phase-space evolution is governed by prime-number growth rules, while microscopic convergence characteristics are determined by prime-fraction nesting rules. FCFG provides a universal underlying geometric tool for judging the dimension of chaotic attractors, classifying topological structures and analysing system stability across diverse scenarios.

6. Conclusion: Paradigm Shift — From Computational Simulation to Geometric Generation

The century-old paradigm of conventional applied science is computation by numerical methods and approximation via trial and error, where numerical simulation is used to fit natural structures. The paradigm pioneered by FCFG is mathematical generation and intrinsic matching, where naturally optimal topological structures are directly produced from fundamental mathematical axioms.

The prime-fraction primordial fractal system bridges number theory, fractal geometry and engineering disciplines, forming a universal map of mathematical genes spanning information science, materials, biology, AI and fluid mechanics.

This theory defines the underlying mechanisms, structural rules and application pathways, laying the theoretical foundation for a new generation of applied science. Subsequent technical implementation, process optimization and engineering iteration can be advanced by industrial communities. In the final vision of FCFG: Mathematics defines the principle; industry creates tangible things.

Luoyang School of Mathematics