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

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.

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🎯 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

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📁 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

   // 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:

# [Formula Name]

## Mathematical Expression

[LaTeX-style mathematical notation]

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

Domain Applicability

• Additive Basis Theory
• Constraint Satisfaction
• Optimization Problems

Validation Status

• Mathematically verified
• 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

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## 🔍 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

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📊 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

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🎯 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

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🚀 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.