Rest Energy from Density-Weighted Loop Curvature: A Covariant Locking PrincipleRichard J. Reyes – November 11, 2025 GitHub Repository: github.com/rickyjreyes/geometry_of_resonance This paper develops a covariant formulation of rest energy as an invariant of curvature-regulated wave confinement. Starting from the Wave Confinement Theory (WCT) Lagrangian, it derives the density-weighted loop identity connecting the effective wavenumber keff to curvature and phase locking along a closed path. The resulting relation Erest = ħc⟨σ⟩w , m = (ħ/c)⟨σ⟩w defines inertial mass as the density-weighted mean curvature of a confined field. Established in the paperCovariant Locking Principle:∂sφ = σ + α / w(s)with the closure condition ∮Γσ ds = 2πn, yielding finite, stable, and unique mass–energy eigenvalues. Key results• The rest-energy law Erest = ħc⟨σ⟩w recovers E = mc² as a special case.• The variational functional ensures strict stability and eliminates ultraviolet divergence as L → 0.• The formalism extends naturally to gauge and spin-covariant derivatives, linking curvature locking to electromagnetism and gravity.• Quantitative bounds connect imperfect lock error, mass variance, and holonomy. ContributionProvides a first-principles derivation of rest energy and mass as emergent invariants of wave confinement. The work unifies curvature, phase, and energy through a covariant locking mechanism, demonstrating how finite mass and inertia arise without singularities. It advances the WCT program toward a geometric–field correspondence that bridges microscopic confinement and macroscopic gravitation. Keywordswave confinement; rest energy; curvature locking; covariant invariant; density-weighted curvature; holonomy; inertial mass; geometric origin of energy; variational stability; Wave Confinement Theory. For correspondence regarding this work, please contact Richard J. Reyes at reyes.ricky30@gmail.com.ORCID iD: 0009-0005-5975-8718.