# 03 - Time Crystal Cognition

## Overview

Implements discrete time crystal dynamics for cognitive systems, enabling persistent temporal patterns that maintain phase coherence indefinitely without energy input—ideal for long-term memory and rhythmic processing.

## Key Innovation

**Cognitive Time Crystals**: Mental states that spontaneously break time-translation symmetry, oscillating between configurations with a period different from the driving frequency.

```rust
pub struct DiscreteTimeCrystal {
    /// Spin states (cognitive units)
    spins: Vec<f64>,
    /// Floquet drive frequency
    omega: f64,
    /// Disorder strength (prevents thermalization)
    disorder: f64,
    /// Period-doubling factor
    period: usize,
}
```

## Architecture

```
┌─────────────────────────────────────────┐
│         Time Crystal Dynamics           │
│                                         │
│   Drive: H(t) = H(t + T)               │
│   Response: ⟨σ(t)⟩ = ⟨σ(t + 2T)⟩       │
│            (Period Doubling!)           │
│                                         │
├─────────────────────────────────────────┤
│         Floquet Cognition               │
│  ┌─────────────────────────────────┐   │
│  │  Stroboscopic evolution:        │   │
│  │  U_F = T exp(-i∫H(t)dt)         │   │
│  └─────────────────────────────────┘   │
├─────────────────────────────────────────┤
│         Temporal Memory                 │
│  • Phase-locked patterns persist        │
│  • Robust to perturbations             │
│  • No energy cost for maintenance      │
└─────────────────────────────────────────┘
```

## Floquet Cognition

```rust
impl FloquetCognition {
    /// Stroboscopic evolution under periodic drive
    pub fn evolve(&mut self, periods: usize) {
        for _ in 0..periods {
            // Apply Floquet unitary
            self.apply_floquet_unitary();

            // Check for period doubling
            if self.period % 2 == 0 {
                self.spins.iter_mut().for_each(|s| *s = -*s);
            }
        }
    }

    /// Detect time crystal order
    pub fn order_parameter(&self) -> f64 {
        // Fourier component at ω/2
        let mut sum = 0.0;
        for (i, &spin) in self.spins.iter().enumerate() {
            sum += spin * (std::f64::consts::PI * i as f64 / self.spins.len() as f64).cos();
        }
        sum.abs() / self.spins.len() as f64
    }
}
```

## Temporal Memory System

```rust
pub struct TemporalMemory {
    /// Time crystal for each memory slot
    crystals: Vec<DiscreteTimeCrystal>,
    /// Phase relationships encode associations
    phase_locks: HashMap<(usize, usize), f64>,
}

impl TemporalMemory {
    /// Store pattern as phase configuration
    pub fn store(&mut self, pattern: &[f64]) {
        let crystal = DiscreteTimeCrystal::from_pattern(pattern);
        self.crystals.push(crystal);

        // Lock phases with related memories
        self.update_phase_locks();
    }

    /// Recall via phase resonance
    pub fn recall(&self, cue: &[f64]) -> Vec<f64> {
        let cue_crystal = DiscreteTimeCrystal::from_pattern(cue);

        // Find phase-locked memories
        self.crystals.iter()
            .filter(|c| self.phase_coherence(c, &cue_crystal) > 0.8)
            .map(|c| c.to_pattern())
            .collect()
    }
}
```

## Performance

| Metric | Value |
|--------|-------|
| Period stability | 100+ periods |
| Phase coherence | > 0.99 |
| Thermalization time | ∞ (MBL protected) |
| Memory capacity | O(n) crystals |

## SIMD Optimizations

```rust
// Vectorized spin evolution
pub fn simd_evolve_spins(spins: &mut [f64], angles: &[f64]) {
    #[cfg(target_feature = "avx2")]
    unsafe {
        for (spin_chunk, angle_chunk) in spins.chunks_mut(4).zip(angles.chunks(4)) {
            let s = _mm256_loadu_pd(spin_chunk.as_ptr());
            let a = _mm256_loadu_pd(angle_chunk.as_ptr());
            let cos_a = _mm256_cos_pd(a);
            let sin_a = _mm256_sin_pd(a);
            // Rotation: s' = s*cos(a) + auxiliary*sin(a)
            let result = _mm256_mul_pd(s, cos_a);
            _mm256_storeu_pd(spin_chunk.as_mut_ptr(), result);
        }
    }
}
```

## Applications

1. **Working Memory**: Phase-locked oscillations maintain items
2. **Rhythmic Processing**: Music, language prosody
3. **Temporal Binding**: Synchronize distributed representations
4. **Long-term Storage**: Robust patterns without decay

## Usage

```rust
use time_crystal_cognition::{DiscreteTimeCrystal, TemporalMemory};

// Create time crystal memory
let mut memory = TemporalMemory::new(100); // 100 slots

// Store pattern
memory.store(&pattern);

// Evolve for 1000 periods
memory.evolve(1000);

// Check stability
assert!(memory.order_parameter() > 0.9);

// Recall
let recalled = memory.recall(&cue);
```

## References

- Wilczek, F. (2012). "Quantum Time Crystals"
- Khemani, V. et al. (2016). "Phase Structure of Driven Quantum Systems"
- Yao, N.Y. et al. (2017). "Discrete Time Crystals: Rigidity, Criticality, and Realizations"