Education & Theory
Understanding exact fractional representations in physics
Compression Prior Hypothesis
Nature may prefer representations with minimal description length. Instead of arbitrary decimal expansions, fundamental constants might be encoded as simple rational fractions at specific renormalization schemes and energy scales.
Key Principles
- Constants are exact rational numbers p/q in lowest terms
- Complexity measured as L(x) = log₂(p) + log₂(q) bits
- Simpler fractions (fewer bits) are preferred by nature
Anti-Numerology Safeguards
Rigorous protocols prevent cherry-picking and ensure scientific validity:
Safeguard Rules
- R1: Search limited to primitive rationals with low complexity
- R2: Complexity measured by MDL framework
- R3: Scheme/scale tags prevent mixing different contexts
- R4: Freeze & score protocol - no post-hoc adjustments
- R5: Out-of-sample predictions for falsifiability
Practical Examples
Fine Structure Constant
α = 2639/361638
The electromagnetic coupling constant as an exact fraction eliminates all floating-point errors in QED calculations. This 31-bit representation provides 12+ significant figures of precision.
CKM CP Violation
sin 2β = 119/169
This exact rational emerges from the algebraic combination of CKM parameters ρ̄ = 3/20 and η̄ = 7/20, demonstrating how complex physics reduces to elementary arithmetic.
Neutrino Mixing
sin²θ₂₃ = 9/16
Maximal atmospheric neutrino mixing as an exact fraction. The 8-bit complexity suggests this is a fundamental ratio in the neutrino sector, corresponding to θ₂₃ = 45°.
Why This Approach Matters for Education
Conceptual Clarity
Students can follow every step of complex calculations using only basic arithmetic. No need for scientific calculators or numerical approximations - just fractions and elementary operations.
Error-Free Calculations
Eliminates the frustration of accumulated rounding errors in multi-step physics problems. Every intermediate result is exact, making it easy to check work and identify mistakes.
Deep Understanding
By seeing the exact algebraic relationships between physical parameters, students develop intuition for how different aspects of physics connect at a fundamental level.
Computational Confidence
Students learn that advanced physics is not about complex calculations but about understanding the underlying mathematical relationships - which can be surprisingly simple.
Learn More
Recommended Study Path
- Start with the Constants Reference to explore the complete catalog
- Try the CKM Calculator to see step-by-step calculations
- Experiment with the PMNS Calculator for neutrino physics
- Explore QED calculations with the fine structure constant
- Investigate cosmological parameters and their relationships
Each calculator includes theory sections explaining the physics behind the mathematics, making this tool suitable for learners from high school to graduate level.