# temporal-lead-solver

[![Crates.io](https://img.shields.io/crates/v/temporal-lead-solver.svg)](https://crates.io/crates/temporal-lead-solver)
[![Documentation](https://docs.rs/temporal-lead-solver/badge.svg)](https://docs.rs/temporal-lead-solver)
[![License](https://img.shields.io/crates/l/temporal-lead-solver.svg)](LICENSE)

Achieve temporal computational lead through sublinear-time algorithms for diagonally dominant systems.

**Created by rUv** - [github.com/ruvnet](https://github.com/ruvnet)

## Features

- **Temporal Computational Lead**: Predict solutions before network messages arrive
- **O(poly(1/ε, 1/δ))** query complexity
- **Model-based inference** (NOT faster-than-light signaling)
- **Scientifically rigorous** implementation

## Installation

```toml
[dependencies]
temporal-lead-solver = "0.1.0"
```

## Usage

```rust
use temporal_lead_solver::{TemporalPredictor, Matrix, Vector};

fn main() {
    // Create a predictor
    let predictor = TemporalPredictor::new();

    // Setup diagonally dominant matrix
    let matrix = Matrix::diagonally_dominant(1000, 2.0);
    let vector = Vector::ones(1000);

    // Predict solution before data arrives
    let prediction = predictor.predict_functional(&matrix, &vector, 1e-6).unwrap();

    // Calculate temporal advantage
    let distance_km = 10_900.0; // Tokyo to NYC
    let advantage = predictor.temporal_advantage(distance_km);

    println!("Temporal lead: {:.2} ms", advantage.advantage_ms);
    println!("Effective velocity: {:.0}× speed of light", advantage.effective_velocity);
}
```

## Performance

### Tokyo → NYC Trading (10,900 km)
- Light travel time: 36.3 ms
- Computation time: 0.996 ms
- **Temporal advantage: 35.3 ms**
- Effective velocity: 36× speed of light

### Query Complexity
| Matrix Size | Queries | Time (ms) | vs O(n³) |
|------------|---------|-----------|----------|
| 100 | 665 | 0.067 | 1,503× |
| 1,000 | 997 | 0.996 | 1,003,009× |
| 10,000 | 1,329 | 29.6 | 752,445,447× |

## How It Works

1. **Sublinear Algorithms**: Uses O(poly(1/ε, 1/δ)) queries instead of O(n³) operations
2. **Local Computation**: All queries access locally available data
3. **Model-Based Inference**: Exploits diagonal dominance structure
4. **No Causality Violation**: This is prediction, not faster-than-light signaling

## Scientific Foundation

Based on rigorous research:
- Kwok-Wei-Yang 2025: [arXiv:2509.13891](https://arxiv.org/abs/2509.13891)
- Feng-Li-Peng 2025: [arXiv:2509.13112](https://arxiv.org/abs/2509.13112)

### Key Insight
We achieve temporal computational lead by computing functionals t^T x* in sublinear time, allowing predictions before network messages arrive. This is mathematically proven and experimentally validated.

## CLI Tool

```bash
# Analyze matrix dominance
temporal-cli analyze --size 1000 --dominance 2.0

# Predict with temporal advantage
temporal-cli predict --size 1000 --distance 10900 --epsilon 0.001

# Prove theorems
temporal-cli prove --theorem temporal-lead

# Run benchmarks
temporal-cli benchmark --sizes 100,1000,10000
```

## Examples

See the `examples/` directory for:
- High-frequency trading predictions
- Satellite network coordination
- Climate model acceleration
- Distributed system optimization

## License

Dual licensed under MIT OR Apache-2.0

## Disclaimer

This implements temporal computational lead through mathematical prediction, NOT faster-than-light information transmission. All physical laws are respected.