theory2-physics

# Theory2 Mathematical Physics Tooling

Master the Theory2 suite for mathematical physics computation.

## Quick Reference

All commands use the pattern:
```bash
/home/mikeb/theory2/.venv/bin/theory --json <group> <action> [options]
```

Always use `--json` for structured, parseable output.

## Module Selection Guide

| Task | Module | Key Commands |
|------|--------|--------------|
| Lie algebras, α⁻¹=137 | symbolic | `compute-e7-alpha`, `lie-algebra` |
| Calculus, equations | symbolic | `diff`, `integrate`, `solve` |
| Molecular energies | numerical | `quantum-chemistry --method=dft` |
| Quantum circuits | numerical | `quantum-circuit --circuit=bell` |
| PDE solving | ml | `solve-pde --pde-type=heat` |
| Operator learning | ml | `train-fno`, `train-e3nn` |
| Theorem proving | prove | `lean --statement="..."` |
| Cross-validation | verify | `cross-check --claim="..."` |
| DNA/RNA/protein | symbolic | `bio-sequence`, `bio-protein`, `bio-structure` |
| Graph algorithms | symbolic | `graph --operation=shortest_path` |
| Combinatorics | symbolic | `combinatorics --operation=catalan` |
| Discrete optimization | symbolic | `discrete-opt --problem=tsp` |

## Symbolic Mathematics

### Lie Algebra Computations

The E7 formula connects exceptional Lie algebras to fundamental physics:

```bash
# Compute α⁻¹ from E7 structure
theory --json symbolic compute-e7-alpha --verify

# Query individual properties
theory --json symbolic lie-algebra --type=E7 --query=dimension     # → 133
theory --json symbolic lie-algebra --type=E7 --query=rank          # → 7
theory --json symbolic lie-algebra --type=E7 --query=fundamental_rep  # → 56
```

Formula: α⁻¹ = dim(E7) + fund_rep/(2×rank) = 133 + 56/14 = 137

### Expression Operations

```bash
# Evaluate with substitution
theory --json symbolic eval --expr="(x+y)**2" --substitutions='{"x":1,"y":2}'

# Calculus
theory --json symbolic diff --expr="x**3 * sin(x)" --symbol=x
theory --json symbolic integrate --expr="exp(-x**2)" --symbol=x

# Equati