Math Review

---
name: math-review
description: Intensive mathematical analysis for numerical stability, algorithm correctness, and alignment with authoritative standards. Use for math-heavy code changes.
category: specialized
tags: [math, algorithms, numerical, stability, verification, scientific]
tools: [derivation-checker, stability-analyzer, reference-finder]
usage_patterns:
  - algorithm-review
  - numerical-analysis
  - derivation-verification
  - stability-assessment
complexity: advanced
estimated_tokens: 200
progressive_loading: true
dependencies:
  - pensive:shared
  - imbue:evidence-logging
---

# Mathematical Algorithm Review

Intensive analysis ensuring numerical stability and alignment with standards.

## Quick Start

```bash
/math-review
```

## When to Use

- Changes to mathematical models or algorithms
- Statistical routines or probabilistic logic
- Numerical integration or optimization
- Scientific computing code
- ML/AI model implementations
- Safety-critical calculations

## Required TodoWrite Items

1. `math-review:context-synced`
2. `math-review:requirements-mapped`
3. `math-review:derivations-verified`
4. `math-review:stability-assessed`
5. `math-review:evidence-logged`

## Core Workflow

### 1. Context Sync
```bash
pwd && git status -sb && git diff --stat origin/main..HEAD
```
Enumerate math-heavy files (source, tests, docs, notebooks). Classify risk: safety-critical, financial, ML fairness.

### 2. Requirements Mapping
Translate requirements → mathematical invariants. Document pre/post conditions, conservation laws, bounds. **Load**: `modules/requirements-mapping.md`

### 3. Derivation Verification
Re-derive formulas using CAS. Challenge approximations. Cite authoritative standards (NASA-STD-7009, ASME VVUQ). **Load**: `modules/derivation-verification.md`

### 4. Stability Assessment
Evaluate conditioning, precision, scaling, randomness. Compare complexity. Quantify uncertainty. **Load**: `modules/numerical-stability.md`

### 5. Evidence Logging
```bash
pytest tests/math/ --benchmark
jupyter nbconvert --execute derivation.ipynb
```
Log deviations, recommend: Approve / Approve with actions / Block. **Load**: `modules/testing-strategies.md`

## Progressive Loading

**Default (200 tokens)**: Core workflow, checklists
**+Requirements** (+300 tokens): Invariants, pre/post conditions, coverage analysis
**+Derivation** (+350 tokens): CAS verification, standards, citations
**+Stability** (+400 tokens): Numerical properties, precision, complexity
**+Testing** (+350 tokens): Edge cases, benchmarks, reproducibility

**Total with all modules**: ~1600 tokens

## Essential Checklist

**Correctness**: Formulas match spec | Edge cases handled | Units consistent | Domain enforced
**Stability**: Condition number OK | Precision sufficient | No cancellation | Overflow prevented
**Verification**: Derivations documented | References cited | Tests cover invariants | Benchmarks reproducible
**Documentation**: Assumptions stated | Limitations documented | Error bounds specified | References linked

## Output Format

```markdown
## Summary
[Brief findings]

## Context
Files | Risk classification | Standards

## Requirements Analysis
| Invariant | Verified | Evidence |

## Derivation Review
[Status and conflicts]

## Stability Analysis
Condition number | Precision | Risks

## Issues
[M1] [Title]: Location | Issue | Fix

## Recommendation
Approve / Approve with actions / Block
```

## Exit Criteria

- Context synced, requirements mapped, derivations verified, stability assessed, evidence logged with citations