Theory2 Mathematical Physics Tooling

Master the Theory2 suite for mathematical physics computation.

Quick Reference

All commands use the pattern:

/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:

# 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

# 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

# Equation solving
theory --json symbolic solve --expr="x**3 - 8" --symbol=x

Numerical Physics

Quantum Chemistry

Methods ranked by accuracy/cost:

1. HF (Hartree-Fock): Fastest, no correlation
2. DFT (B3LYP, PBE): Good balance
3. CCSD: Most accurate, expensive

# Water with DFT
theory --json numerical quantum-chemistry \
  --molecule="H2O" --method=dft --xc=b3lyp --basis=def2-svp

# Custom geometry
theory --json numerical quantum-chemistry \
  --molecule="O 0 0 0; H 0.757 0.587 0; H -0.757 0.587 0" \
  --method=ccsd --basis=cc-pVDZ

Quantum Circuits

# Bell state measurement
theory --json numerical quantum-circuit --circuit=bell --shots=1024

# GHZ statevector
theory --json numerical quantum-circuit --circuit=ghz3 --statevector

Physics Machine Learning

Fourier Neural Operators

For learning PDE solution operators:

# Standard FNO
theory --json ml train-fno --modes=16 --width=64 --layers=4

# Memory-efficient
theory --json ml train-fno --modes=32 --width=128 --factorization=tucker

Tucker factorization reduces memory ~10x for large models.

Physics-Informed Neural Networks

Solve PDEs without training data:

# Heat equation
theory --json ml solve-pde --pde-type=heat --alpha=0.01 --iterations=10000

# Poisson equation
theory --json ml solve-pde --pde-type=poisson --iterations=20000

E3NN Equivariant Networks

For molecular systems respecting 3D symmetry:

theory --json ml train-e3nn --irreps-hidden="32x0e+16x1o+8x2e" --use-gates

Bioinformatics & Molecular Biology

Sequence Analysis

Work with DNA, RNA, and protein sequences using Biopython:

# Transcribe DNA to RNA
theory --json symbolic bio-sequence --sequence="ATGCGTACG" --operation=transcribe

# Translate DNA to protein
theory --json symbolic bio-sequence --sequence="ATGCGTACG" --operation=translate

# Reverse complement
theory --json symbolic bio-sequence --sequence="ATGCGTACG" --operation=reverse_complement

# GC content calculation
theory --json symbolic bio-sequence --sequence="ATGCGTACG" --operation=gc_content

Protein Analysis

# Calculate molecular weight
theory --json symbolic bio-protein --sequence="MKTAYIAKQR" --operation=molecular_weight

# Compute isoelectric point
theory --json symbolic bio-protein --sequence="MKTAYIAKQR" --operation=isoelectric_point

# Predict secondary structure
theory --json symbolic bio-protein --sequence="MKTAYIAKQR" --operation=secondary_structure

Structure Analysis

Load and analyze protein structures from PDB files:

# Parse PDB structure
theory --json symbolic bio-structure --pdb-id="1BNA" --operation=get_info

# Extract sequence from structure
theory --json symbolic bio-structure --pdb-id="1BNA" --operation=extract_sequence

# Calculate RMSD between structures
theory --json symbolic bio-structure --pdb-id="1BNA" --reference="1BNB" --operation=rmsd

Combinatorics & Discrete Mathematics

Graph Theory

Using NetworkX for graph algorithms:

# Create and analyze graph
theory --json symbolic graph --edges="[[0,1],[1,2],[2,0]]" --operation=shortest_path --source=0 --target=2

# Find connected components
theory --json symbolic graph --edges="[[0,1],[2,3]]" --operation=components

# Calculate centrality measures
theory --json symbolic graph --edges="[[0,1],[1,2],[2,0]]" --operation=centrality --method=betweenness

# Check graph properties
theory --json symbolic graph --edges="[[0,1],[1,2],[2,0]]" --operation=is_planar

Enumeration

Compute combinatorial numbers and sequences:

# Catalan numbers
theory --json symbolic combinatorics --operation=catalan --n=10

# Bell numbers (partitions)
theory --json symbolic combinatorics --operation=bell --n=5

# Stirling numbers (first/second kind)
theory --json symbolic combinatorics --operation=stirling --n=5 --k=2 --kind=second

# Partition function
theory --json symbolic combinatorics --operation=partitions --n=10

Optimization Problems

Solve classic discrete optimization problems:

# Traveling salesman problem
theory --json symbolic discrete-opt --problem=tsp --distances="[[0,10,15],[10,0,20],[15,20,0]]"

# Knapsack problem
theory --json symbolic discrete-opt --problem=knapsack \
  --weights="[2,3,4,5]" --values="[3,4,5,6]" --capacity=8

# Vertex cover
theory --json symbolic discrete-opt --problem=vertex_cover \
  --edges="[[0,1],[1,2],[2,3]]"

# Maximum flow
theory --json symbolic discrete-opt --problem=max_flow \
  --edges="[[0,1,10],[1,2,5],[0,2,15]]" --source=0 --sink=2

Theorem Proving

RobustLeanProver (Recommended)

Automatic proof search with intelligent tactic selection:

# Auto mode - tries 14+ tactics with parallel search
theory --json prove lean --statement="2 + 2 = 4"
theory --json prove lean --statement="∀ n : Nat, n + 0 = n"

# Specific tactics
theory --json prove lean --statement="2 + 2 = 4" --tactic=rfl
theory --json prove lean --statement="10 * 10 = 100" --tactic=decide
theory --json prove lean --statement="∀ x, x + 0 = x" --tactic=omega

Tactic Tiers (Auto Mode)

Tier	Tactics	Speed	Mode
fast	rfl, trivial, decide	~100ms	Parallel
arithmetic	norm_num, omega, ring, simp	~500ms	Parallel
search	simp_all, aesop, tauto	~3s	Sequential
combined	simp; ring, norm_num; simp	~10s	Sequential

Problem Type Detection

Type	Example	Suggested Tactics
arithmetic	2 + 2 = 4	rfl, decide, norm_num
algebraic	(a+b)^2 = ...	ring (needs mathlib)
inductive	List.length ...	induction, cases
logical	True, 1 < 2	decide, tauto

Proof Caching

• Successful proofs cached to ~/.cache/theory2/proofs/
• Cache hits are instant (no REPL call)
• Use --no-cache to force re-computation

Searching & Saving Proofs

# Save successful proof
theory --json prove lean --statement="3 + 3 = 6" --save

# Search proofs
theory --json prove search --query="continuous" --search-in=both

# List saved
theory --json prove list --verified-only

Scientific Validation Workflow

Hermeneutic Circle Methodology

Apply iterative refinement:

1. Part→Whole: Analyze components individually
2. Whole→Part: Use overall structure to inform details
3. Iterate: Refine understanding through cycles

Prior Knowledge Integration

Before computing, search for relevant prior work:

mcp__plugin_task-memory_task-memory__search(query="<topic>")

Multi-Method Verification

Always cross-validate critical results:

theory --json verify cross-check \
  --claim="alpha_inv=137" \
  --methods="symbolic,numerical,experimental" \
  --tolerance=0.001

Documentation

Record for reproducibility:

• Method and parameters used
• Computational environment
• Reference values compared against
• Uncertainty quantification

MCP Tools

The plugin provides MCP tools for direct invocation:

• theory2_symbolic_compute_e7_alpha
• theory2_symbolic_lie_algebra
• theory2_symbolic_eval/simplify/solve/diff/integrate
• theory2_numerical_quantum_chemistry
• theory2_numerical_quantum_circuit
• theory2_ml_train_fno/train_e3nn/solve_pde
• theory2_prove_lean/search
• theory2_verify_cross_check

Agents

• physics-solver: Autonomous multi-step problem solving (physics, ML, bioinformatics)
• physics-verifier: Cross-validation and verification
• theorem-prover: Automated Lean 4 theorem proving with RobustLeanProver
• bio-analyzer: Sequence analysis, protein structure, and molecular biology workflows
• graph-solver: Graph algorithms and discrete optimization problems

Best Practices

1. Always verify: Use cross-check for important results
2. Document provenance: Record methods, parameters, references
3. Search first: Check task memory for prior relevant work
4. Iterate: Apply hermeneutic refinement to deepen understanding
5. Quantify uncertainty: Report tolerances and error bounds