This repository contains formal Metamath proofs for the Reflectology axiom system and its relationship to ZFC set theory.

• axioms.mm - The 40 axioms of Reflectology formally stated in Metamath
• rzfc.mm - Proofs showing that ZFC set theory can be derived from Reflectology axioms
• omega.mm - Extension of set.mm with Reflectology axioms (requires external set.mm)

• CLASSICAL_LOGIC_MAPPING.md - 📖 NEW: Comprehensive analysis showing how each .mm file maps to classical logic, validation roadmap, and honest assessment of proof status
• AXIOMS_MM_VALIDATION_CHECKLIST.md - Detailed validation checklist and status report
• METAMATH_COMPLIANCE.md - Standards compliance documentation
• IMPLEMENTATION_SUMMARY.md - Implementation details and achievements
• QUICKSTART.md - Quick start guide for contributors

All standalone Metamath files in this repository are validated against the official Metamath specification.

This repository uses the official Metamath proof verifier to validate all proofs.

To validate the Metamath files locally:

# Make the script executable (if not already)
chmod +x validate-metamath.sh

# Run validation
./validate-metamath.sh

The script will:

1. Automatically download and build the Metamath verifier if not present
2. Validate all .mm files in the repository
3. Report any errors or warnings
4. Provide a summary of validation results

You can also validate files manually using the Metamath executable:

# Install metamath (if not already installed)
git clone https://github.com/metamath/metamath-exe.git
cd metamath-exe
./build.sh

# Validate a file
./metamath
MM> read "path/to/file.mm"
MM> verify proof *
MM> exit

This repository uses GitHub Actions to automatically validate all Metamath files on every push and pull request. The workflow:

1. Builds the Metamath verifier from source
2. Runs validation on all .mm files
3. Reports any errors or warnings
4. Fails the build if validation errors are found

See .github/workflows/metamath-validation.yml for details.

All files in this repository conform to the Metamath specification and follow best practices from the set.mm project:

• ✅ All constants declared with $c
• ✅ All variables declared with $v
• ✅ Type assignments for all variables with $f
• ✅ Proper axiom syntax: label $a |- statement $.
• ✅ Proper theorem syntax: label $p |- statement $= proof $.
• ✅ Proper inference rule syntax with hypotheses
• ✅ All statements end with $.
• ✅ Comments enclosed in $( ... $)

• ✅ All proofs are complete and explicit
• ✅ Every step justified by axioms or proven theorems
• ✅ Distinct variable conditions properly maintained
• ✅ No circular dependencies
• ✅ All referenced labels exist

This project is validated against standards from:

• metamath/set.mm - The comprehensive Metamath database for set theory and mathematics
• metamath-book - Documentation and tutorials on Metamath
• Metamath Home Page - Official Metamath website with tools and documentation

When contributing changes to .mm files:

1. Run ./validate-metamath.sh locally before committing
2. Ensure all validations pass
3. Fix any errors or warnings reported
4. The CI will automatically validate your changes on push

The axioms are organized into five categories following the Reflectology framework:

1. Configuration Space (Axioms 1-5): Define the fundamental structure Omega
2. Redundancy Reduction (Axioms 6-10): Quotient operations and symmetry
3. Canonical Forms (Axioms 11-14): Loss functions and optimal configurations
4. Evaluate Options (Axioms 15-24): Convergence and goodness criteria
5. Optimize Decision-Making (Axioms 25-40): Dynamical systems and optimization

The file rzfc.mm explores the relationship between the 40 Reflectology axioms and the 10 axioms of Zermelo-Fraenkel set theory with Choice. The file provides:

• ✅ Formal statements of all 10 ZFC axioms in Metamath syntax
• ✅ Detailed informal derivation strategies for each axiom
• ✅ Formal proofs for 2 axioms (Empty Set and a simplified Foundation)
• 📋 Roadmap for completing formal derivations of remaining 8 axioms

Derivation Status:

1. Extensionality ← ax-rf10 (Omega-Bijection) - 📋 Proof planned
2. Empty Set ← ax-rf1 (Initial Emptiness) - ✅ PROVEN
3. Pairing ← ax-rf2, ax-rf3 (Structure, Encapsulation) - 📋 Proof planned
4. Union ← ax-rf5 (Hierarchical Structuring) - 📋 Proof planned
5. Power Set ← ax-rf6 (Redundancy Reduction) - 📋 Proof planned
6. Infinity ← ax-rf15 (Reflective Convergence) - 📋 Proof planned
7. Separation ← ax-rf8 (Symmetry Breaking) - 📋 Proof planned
8. Replacement ← ax-rf39 (Internal Emergence) - 📋 Proof planned
9. Foundation ← ax-rf25 (Gradient Dynamics) - ⚠️ Simplified proof
10. Choice ← ax-rf14 (Canonical Selection) - 📋 Proof planned

For detailed analysis, see CLASSICAL_LOGIC_MAPPING.md which provides:

• Complete mapping of each .mm file to classical propositional and first-order logic
• Comparison with the tauto tactic from GinoGiotto/mm1_tactics
• Comprehensive validation and benchmarking roadmap
• Honest assessment of current proof status vs. aspirational claims

See individual file headers for license information. Most files are either public domain or GNU GPL as indicated in their headers.