# Formula Extraction Template

This template standardizes the extraction of mathematical formulas from code for cold review and verification.

---

## Template Structure

### 1. Function Identification
```
File: [path/to/file.ext]
Function: [function_name]
Line: [line_number]
Language: [Rust/Python/Lean/C++ etc.]
```

### 2. Mathematical Expression
```
[LaTeX or mathematical notation]

Example:
a_{t+1} = a_t \cdot e^{-\lambda t} + \sum_{i=1}^{n} w_i \cdot x_i
```

### 3. Computational Implementation
```rust
[code showing implementation]

Example:
pub fn exponential_decay(
    current: Q16_16, 
    decay_rate: Q16_16, 
    input: Q16_16
) -> Q16_16 {
    let decayed = current * exp(-decay_rate);
    decayed + input
}
```

### 4. Domain Applicability
- [x] Additive Basis Theory
- [ ] Constraint Satisfaction  
- [x] Optimization Problems
- [ ] Graph Theory
- [ ] Number Theory
- [ ] Signal Processing
- [ ] Other: _________

### 5. Validation Status
- [x] Mathematically verified
- [x] Computationally tested
- [ ] Cross-domain validated
- [ ] Peer reviewed

### 6. References
- [Academic paper or source]
- [Related documentation]

---

## Quick Reference Guide

### Common Mathematical Patterns

| Pattern | Example | Documentation |
|---------|---------|---------------|
| Recurrence Relations | `a_{n+1} = f(a_n)` | Use subscripts in doc comments |
| Matrix Operations | `C = A × B` | Document dimensions and types |
| Optimization | `minimize f(x)` | Include constraints |
| Signal Processing | `Y = FFT(X)` | Specify window, overlap |
| Statistical | `μ = E[X]` | Define population vs sample |

### Documentation Checklist

- [ ] Function signature documented
- [ ] Mathematical formula included
- [ ] Parameters explained
- [ ] Return value specified
- [ ] Domain tags applied
- [ ] Validation status marked
- [ ] References cited

---

## Usage Examples

### Example 1: DSP Function
```markdown
## FFT Spectral Analysis

### Mathematical Expression
```
X(k) = \sum_{n=0}^{N-1} x(n) \cdot e^{-j\frac{2\pi kn}{N}}
```

### Computational Implementation
[See flac_dsp_node.py:process_flac_chunk]

### Domain Applicability
- [x] Signal Processing
- [ ] Other domains

### Validation Status
- [x] Mathematically verified
- [x] Computationally tested
```

### Example 2: State Machine
```markdown
## SLUQ State Update

### Mathematical Expression
```
a_{t+1} = a_t - (a_t >> r) + \lambda_1|e_t| + \lambda_2\Delta_t + \lambda_3m_t
```

### Computational Implementation
[See SLUQ.lean:stress_accumulation]

### Domain Applicability
- [x] Optimization Problems
- [x] Constraint Satisfaction
- [ ] Other domains

### Validation Status
- [x] Mathematically verified
- [ ] Cross-domain validated
```