Galileo Index (GI)

Abstract

The Galileo Index (GI) introduces a practical, empirical framework for evaluating AI model truthfulness through standardized test cases and verifiable metrics. By combining rigorous mathematical validation with transparent blockchain record-keeping on Solana, we provide a reliable measure of AI models' ability to provide accurate information.

1. Introduction

1.1 Motivation

As AI models become increasingly sophisticated, the need for objective truth measurement becomes critical. The Galileo Index addresses this by establishing a standardized testing framework focused on domains where ground truth can be definitively established.

1.2 Core Principles

• Verifiable Ground Truth: Focus on domains with definitive correct answers
• Reproducible Results: Standardized test cases and evaluation methods
• Transparent Scoring: Clear metrics and evaluation criteria
• Immutable Records: Blockchain-based result verification

2. Methodology

2.1 Test Case Categories

• Mathematical Problems (35%)
  • Differential equations
  • Complex analysis
  • Linear algebra
  • Probability theory
• Physical Laws (25%)
  • Classical mechanics
  • Thermodynamics
  • Electromagnetic theory
  • Quantum mechanics
• Logical Reasoning (20%)
  • Formal logic
  • Boolean algebra
  • Set theory
  • Algorithm analysis
• Empirical Validation (20%)
  • Statistical analysis
  • Experimental design
  • Data interpretation
  • Error analysis

2.2 Evaluation Process

1. Test Case Generation
   • Problems with known, verifiable solutions
   • Multiple complexity levels
   • Diverse domain coverage
2. Model Response Collection
   • Standardized input format
   • Controlled testing environment
   • Response validation
3. Answer Validation
   • Automated correctness checking
   • Step-by-step verification
   • Error analysis
4. Score Calculation
   • Domain-specific metrics
   • Weighted aggregation
   • Confidence intervals

3. Technical Implementation

3.1 Core Components

• Python Evaluation Framework
  • Test case management
  • Response validation
  • Score calculation
• Solana Program Integration
  • Result verification
  • Score recording
  • Public accessibility

3.2 Validation Logic

Each response undergoes multi-stage validation:

1. Syntax Verification: Ensuring response format matches requirements
2. Semantic Analysis: Checking mathematical/logical correctness
3. Step Validation: Verifying solution methodology
4. Result Confirmation: Comparing final answers

4. Scoring System

4.1 Metrics

• Correctness (50%): Accuracy of final answer
• Methodology (30%): Proper solution steps
• Clarity (10%): Clear explanation
• Efficiency (10%): Optimal solution path

4.2 Score Aggregation

Final scores are calculated using weighted averages across all test cases, with adjustments for:

• Problem complexity
• Domain importance
• Response consistency
• Error margins

5. Future Development

5.1 Planned Improvements

• Expanded test case database
• Advanced validation algorithms
• Real-time evaluation capabilities
• Community contribution framework

5.2 Research Directions

• Automated test case generation
• Dynamic difficulty adjustment
• Cross-domain validation methods
• Uncertainty quantification

6. Conclusion

The Galileo Index provides a practical, implementable framework for measuring AI truthfulness. By focusing on verifiable test cases and leveraging blockchain technology for transparency, we enable objective comparison of AI models' capabilities in providing accurate information.