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

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

Numerical Physics

Quantum Chemistry

Methods ranked by accuracy/cost:

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

bash

# 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

bash

# 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:

bash

# 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:

bash

# 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:

bash

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:

bash

# 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

bash

# 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:

bash

# 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:

bash

# 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:

bash

# 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:

bash

# 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:

bash

# 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

bash

# 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:

Part→Whole: Analyze components individually
Whole→Part: Use overall structure to inform details
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:

bash

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

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

Search AI Tools

theory2-physics

Install this agent skill to your Project

SKILL.md

Theory2 Mathematical Physics Tooling

Quick Reference

Module Selection Guide

Symbolic Mathematics

Lie Algebra Computations

Expression Operations

Numerical Physics

Quantum Chemistry

Quantum Circuits

Physics Machine Learning

Fourier Neural Operators

Physics-Informed Neural Networks

E3NN Equivariant Networks

Bioinformatics & Molecular Biology

Sequence Analysis

Protein Analysis

Structure Analysis

Combinatorics & Discrete Mathematics

Graph Theory

Enumeration

Optimization Problems

Theorem Proving

RobustLeanProver (Recommended)

Tactic Tiers (Auto Mode)

Problem Type Detection

Proof Caching

Searching & Saving Proofs

Scientific Validation Workflow

Hermeneutic Circle Methodology

Prior Knowledge Integration

Multi-Method Verification

Documentation

MCP Tools

Agents

Best Practices