Research-Stack/6-Documentation/docs/research/FORMULA_EXTRACTION_TEMPLATE.md

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

[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

• Additive Basis Theory
• Constraint Satisfaction
• Optimization Problems
• Graph Theory
• Number Theory
• Signal Processing
• Other: _________

5. Validation Status

• Mathematically verified
• 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

## 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

## 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