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="x3 * sin(x)" --symbol=x theory --json symbolic integrate --expr="exp(-x2)" --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

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:

• 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:

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