"On a Hidden Harmonic–Geometric Structure Linking Planck's Constant and π"

This work presents a unified harmonic framework in which structured wave superposition leads to energy localization and emergent mass formation. A dimensionless scaling parameter K = 0.413334 governs spectral compression, harmonic coupling, and attenuation. The framework demonstrates that harmonic energy distributions follow a convergent structure, enabling stable mass-like configurations through phase coherence. A geometric interpretation of K is developed, linking it to curvature-induced attenuation under near-horizon conditions. Closed-form expressions for total energy and mass are derived, establishing direct dependence on the scaling parameter. A modified energy–frequency relation E = K h ν is introduced as an effective scaling form under constrained conditions. The model provides a conceptual bridge between harmonic wave systems, geometry, and emergent mass phenomena.Harmonic structures are fundamental to physical systems, appearing in wave mechanics, quantum theory, and field dynamics. This work proposes a unified framework in which energy and mass arise from structured harmonic superposition governed by a scaling parameter. Rather than treating mass as a fundamental property, it is interpreted as an emergent result of localized energy distributions.