The Resolution Gap: A Geometric Origin of Mass-Energy Structure from First Principles

This working paper introduces the Resolution Gap as a dimensionless geometric constant emerging from the relationship between the golden ratio φ and pi and π. We propose that this ratio constitutes a fundamental compression threshold between nested transcendental geometries and that it plays a central role in mass-energy emergence, vacuum structuring, and recursive information flow. Through first-principles derivation and numerical validation, we demonstrate how RG bridges transcendental constants and observable physical quantities in a scale-invariant manner.