# Cold Review Formula Extraction Plan

## 📋 Executive Summary

This document outlines the systematic approach for extracting standalone mathematical formulas from new modules for independent cold review. The goal is to create a pure mathematical foundation that can be verified separately from implementation details.

---

## 🎯 Objectives

1. **Isolate Pure Mathematical Content** - Remove implementation-specific code
2. **Create Review-Ready Formulas** - Standardized documentation format
3. **Enable Cross-Domain Validation** - Map formulas to problem domains
4. **Facilitate Collaborative Review** - Allow mathematicians to verify independently

---

## 📁 Directory Structure

```
src/formulas/
├── README.md                    # Formula catalog overview
├── extraction_plan.md           # This document
├── sluq_dynamics/
│   ├── stress_accumulation.md
│   ├── state_transition.md
│   └── convergence_formula.md
├── constraint_satisfaction/
│   ├── basis_verification.md
│   ├── order_computation.md
│   └── exact_order.md
├── optimization/
│   ├── search_algorithms.md
│   ├── pruning_strategies.md
│   └── convergence_metrics.md
└── cross_validation/
    ├── consistency_measures.md
    ├── domain_metrics.md
    └── validation_scores.md
|     └── validation_scores.md
| ```

### **Phase 2: Extraction (Day 2)**

#### **Step 2.1: Pure Formula Isolation**
For each identified function:

1. **Extract Mathematical Expression**
   ```rust
   // a_{t+1} = a_t * 0.9375 + 0.1 * |e_t| + 0.05 * Δ_t + 0.02 * m_t
   ```

2. **Document Implementation Mapping**
   ```rust
   // decay = current >> 4  →  a_t * 0.9375 (fixed-point approximation)
   // error_term = 0.1 * abs(error)  →  λ₁|e_t|
   ```

#### **Step 2.2: Create Formula Documents**

Each formula document follows this template:

```markdown
# [Formula Name]

## Mathematical Expression
```
[LaTeX-style mathematical notation]
```

## Computational Implementation
```rust
[Code snippet showing implementation]
```

## Domain Applicability
- [x] Additive Basis Theory
- [ ] Constraint Satisfaction  
- [x] Optimization Problems

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

### **Phase 3: Validation (Day 3)**

#### **Step 3.1: Cross-Reference with Literature**
- Compare with existing mathematical literature
- Verify against known results
- Check for consistency with established theory

#### **Step 3.2: Computational Testing**
- Unit tests for each formula
- Edge case validation
- Performance benchmarking

---

## 🔍 Extraction Process

### **Phase 1: Identification (Day 1)**

#### **Step 1.1: Locate Mathematical Functions**
```bash
# Search for mathematical functions in source code
find src/ -name "*.rs" -exec grep -l "formula\|equation\|constraint\|optimization" {} \;

# Extract function signatures with documentation
grep -r "pub fn\|fn " src/ --include="*.rs" | \
  grep -E "(solve|compute|calculate|optimize|verify)" > math_functions.txt
```

#### **Step 1.2: Categorize Formulas**
Create categories based on mathematical domain:
- **State Dynamics** - SLUQ state transitions
- **Constraint Logic** - Basis and order verification
- **Optimization** - Search and convergence
- **Validation** - Cross-domain metrics
---

## 📊 Formula Categories & Priority

### **High Priority (Days 1-2)**
1. **SLUQ Stress Accumulation** - Core state dynamics
2. **Basis Order Verification** - Erdős #336 core problem
3. **Constraint Satisfaction** - General problem solving
4. **Search Optimization** - Computational efficiency

### **Medium Priority (Days 3-4)**
1. **State Transition Logic** - System behavior
2. **Convergence Analysis** - Fixed-point detection
3. **Pruning Strategies** - Search space reduction
4. **Validation Metrics** - Result verification

---

## 🎯 Success Metrics

| **Metric** | **Target** | **Validation** |
|------------|------------|----------------|
| Formulas Extracted | 15-20 | Count of documented formulas |
| Mathematical Correctness | 100% | Peer review approval |
| Computational Accuracy | 99%+ | Unit test pass rate |
| Cross-Domain Coverage | 3+ domains | Problem types solved |

---

## 🚀 Next Steps

1. **Execute Phase 1** - Identify all mathematical functions
2. **Create extraction scripts** - Automate pure formula isolation
3. **Begin documentation** - Start with highest priority formulas
4. **Setup validation framework** - Enable cross-domain testing

This extraction plan ensures your mathematical innovations can be independently verified while maintaining connection to computational implementation.
```