The Nature of Constants: Derivations from First Principles

This work is a critical expansion of the Pan framework, demonstrating that the fundamental constants of physics—across quantum, electromagnetic, gravitational, thermodynamic, cosmological, and even biological domains—can be derived from first principles using a unified mathematical structure. Serving as the foundational counterpart to “The Natural Emergence of Physics: Deriving Constants and Equations from the Pan Equation”, this paper delivers the full mathematical architecture behind those constants. It shows how values traditionally assumed to be empirical—such as the fine-structure constant, the Planck constant, the gravitational constant, and even Avogadro’s number—emerge naturally from structured resonance modulations governed by the Pan equation and weighted through the UMEC framework. This document also complements the Q-Epsilon series, particularly “Introduction to Q-Epsilon: Derivations from the Pan Equation”, by grounding the modulation structure (ε-dimension) in the direct derivation of constants and scaling relationships. In doing so, it confirms that constants across all known physical domains—including those previously treated as unrelated—arise from the same structured vacuum geometry and wave interaction logic. Readers seeking a deeper understanding of the Pan framework, resonance-based field derivations, or computational precision from first principles are encouraged to explore related works in the Pan series. In particular, UMEC: Non-Iterative Math is essential for understanding how this entire framework bypasses approximation and iteration, replacing them with entropy-weighted, instantly computable harmonic solutions. Questions, challenges, and collaborations are welcomed—readers are encouraged to reach out with inquiries.