ppflib - Physics-Prime Factorization Library

A comprehensive Rust library implementing the Physics-Prime Factorization (PPF) mathematical framework for exploring quantum mechanics through number theory.

Overview

The PPF framework extends classical number theory by treating -1 as a special "Sign Prime", creating multiple valid factorizations for integers and enabling a number-theoretic model of quantum phenomena. This library provides a complete implementation with algebraic, topological, and geometric realizations.

Key Features

• Factorization State Spaces: Complete enumeration of valid P-factorizations S(n)
• Quantum Collapse: Modeling (-a) × (-b) = ab transitions
• Sign Prime: The unique negative prime (-1) enabling quantum behavior
• Algebraic Structures: State space ideals, P-prime spectrum, homological algebra
• Topological Analysis: Simplicial complexes, Galois groups, Betti numbers
• Geometric Realizations: IOT metric, tautochrone operators, Hilbert spaces

Quick Start

Add this to your Cargo.toml:

[dependencies]
ppflib = "0.1.0"

Basic Usage

use ppflib::prelude::*;

fn main() -> Result<(), FactorizationError> {
    // Create factorization state spaces
    let classical = FactorizationStateSpace::new(6)?;     // S(6) - classical state
    let quantum = FactorizationStateSpace::new(-6)?;    // S(-6) - quantum state

    println!("S(6) has {} factorizations", classical.size());
    println!("S(-6) has {} factorizations", quantum.size());

    // Demonstrate quantum collapse: (-2) × (-3) = 6
    let s_neg2 = FactorizationStateSpace::new(-2)?;
    let s_neg3 = FactorizationStateSpace::new(-3)?;
    let result = StateSpaceOperations::multiply(&s_neg2, &s_neg3)?;

    if result.collapse_info.collapsed {
        println!("✓ Quantum collapse: (-2) × (-3) → 6");
        println!("Collapse ratio: {:.2}", result.collapse_info.collapse_ratio);
    }

    Ok(())
}

Core Concepts

Factorization State Spaces

Every integer n has a state space S(n) containing all valid P-factorizations:

use ppflib::prelude::*;

let s6 = FactorizationStateSpace::new(6)?;
for factorization in s6.factorizations() {
    println!("{}", factorization); // "2 × 3", "-2 × -3"
}

// Negative integers create quantum state spaces
let s_neg6 = FactorizationStateSpace::new(-6)?;
for factorization in s_neg6.factorizations() {
    println!("{}", factorization); // "-1 × 2 × 3", "-2 × 3", "2 × -3"
}

Quantum Collapse

When two quantum states multiply, they can collapse to classical states:

let s_neg2 = FactorizationStateSpace::new(-2)?;  // Quantum state
let s_neg3 = FactorizationStateSpace::new(-3)?;  // Quantum state
let result = StateSpaceOperations::multiply(&s_neg2, &s_neg3)?;

// Result is S(6) - a classical state!
assert!(result.collapse_info.collapsed);
assert_eq!(result.result_space.value(), 6);

Examples

1. Basic Factorization and Quantum Collapse

See the complete quantum collapse demonstration:

cargo run --example 01_basic_factorization

This example shows:

• Creating classical vs quantum state spaces
• Observing quantum collapse during multiplication
• Analyzing the Sign Prime (-1)
• Understanding factorization properties

2. Algebraic Structures

Explore the rich algebraic framework:

cargo run --example 02_algebraic_structures

Features demonstrated:

• State space ideals IS(n)
• P-prime spectrum analysis
• Multiplication algebra
• Homological structures

3. Topological Analysis

Dive into topological properties:

cargo run --example 03_topological_analysis

Includes:

• Factorization simplicial complexes K(n)
• Galois groups Gal_P(n) ≅ (ℤ₂)^(k-1)
• Betti numbers and Euler characteristics
• Toroidal topology verification (χ = 0)

4. Geometric Structures

Experience the geometric realization:

cargo run --example 04_geometric_structures

Showcases:

• IOT (Involuted Oblate Toroidal) metric with critical ratio r/R = 1/30
• Tautochrone operators and geodesics
• PPF Hilbert space with complexity-based inner product
• Quantum operators and state evolution

5. Comprehensive Demo

See everything working together:

cargo run --example 05_comprehensive_demo

Mathematical Framework

The Sign Prime

PPF introduces -1 as a "Sign Prime", creating quantum behavior in number theory:

• Classical states: S(n) for n > 0 (limited factorizations)
• Quantum states: S(-n) for n > 0 (expanded factorizations via sign prime)
• Quantum collapse: (-a) × (-b) = ab (quantum → classical transition)

IOT Geometry

The IOT (Involuted Oblate Toroidal) metric provides geometric realization:

• Critical ratio: r/R = 1/30 (derived from variational principle)
• Toroidal embedding of factorization spaces
• Geodesics and parallel transport
• Connection to quantum field theory

Topological Structure

Factorization spaces have rich topological properties:

• Galois groups: Gal_P(n) ≅ (ℤ₂)^(k-1) for k distinct primes
• Euler characteristic: χ = 0 (toroidal topology)
• Betti numbers capture dimensional structure
• Homological algebra connections

API Documentation

Core Module

• [FactorizationStateSpace]: The fundamental state space S(n)
• [PFactorization]: Individual P-factorizations
• [StateSpaceOperations]: Operations between state spaces
• [CollapseInfo]: Information about quantum collapse events

Algebra Module

• [StateSpaceIdeal]: Ideals IS(n) in the state space ring
• [PPrimeSpectrum]: The spectrum of P-primes
• [MultiplicationAlgebra]: Multiplicative structure
• [HomologyGroups]: Homological computations

Topology Module

• [FactorizationSimplex]: Simplicial complex K(n)
• [PPFGaloisGroup]: Galois group Gal_P(n)
• [BettiNumberComputer]: Topological invariants
• [ComplexStatistics]: Complex analysis tools

Geometry Module

• [IOTMetric]: The IOT metric tensor
• [TautochroneOperator]: Geodesic and transport operators
• [PPFHilbertSpace]: Quantum Hilbert space realization
• [QuantumState]: Quantum states and operators

Research Applications

Machine Learning & NLP

PPF provides novel approaches to ML challenges:

• Non-differentiable optimization: Use discrete factorization dynamics
• Latent space topology: IOT-based embeddings with natural toroidal structure
• Quantum-inspired algorithms: Leverage collapse dynamics for search/optimization
• Semantic embeddings: Map linguistic structures to factorization spaces

See ideas.md for detailed ML/NLP integration strategies.

GPU Acceleration

Optimized for high-performance computing:

• WGPU compute shaders: Pure Rust GPU kernels avoiding FFI overhead
• Bit-parallel operations: Exploit factorization structure for parallelism
• Lookup table optimizations: Precomputed prime tables and factorizations
• Workgroup efficiency: Optimized for modern GPU architectures

See ideas2.md for GPU optimization details.

Contributing

Contributions are welcome! Please see our contributing guidelines.

Areas for Contribution

• Performance optimization: Faster factorization algorithms
• Documentation: More examples and tutorials
• GPU kernels: WGPU compute shader implementations
• ML integrations: TensorFlow/PyTorch bindings
• Visualization: Tools for exploring state spaces

Mathematical Background

The PPF framework is based on research in:

• Number Theory: Prime factorization and algebraic structures
• Algebraic Topology: Simplicial complexes and homology
• Differential Geometry: Riemannian metrics and geodesics
• Quantum Mechanics: Hilbert spaces and operator theory
• Group Theory: Galois groups and symmetries

For detailed mathematical exposition, see the papers in /docs.

Performance

ppflib is designed for performance:

• Memory efficient: Lazy evaluation and caching strategies
• Parallel ready: Thread-safe operations where applicable
• GPU optimized: WGPU compute shader support planned
• Asymptotic bounds: Theoretical complexity analysis available

Benchmarks coming soon with criterion integration.

License

Licensed under either of

• Apache License, Version 2.0 (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
• MIT license (LICENSE-MIT or http://opensource.org/licenses/MIT)

at your option.

Citation

If you use ppflib in academic research, please cite:

@software{ppflib,
  title = {ppflib: Physics-Prime Factorization Library},
  author = {PPF Research Team},
  year = {2024},
  url = {https://github.com/your-org/ppflib},
  version = {0.1.0}
}

Explore the quantum structure hidden in numbers with ppflib! 🔢⚛️