Resonance Algebra: A Formal Framework for Vibrating Mathematical Objects
Abstract
We develop Resonance Algebra (Res), a formal algebraic structure for mathematical objects with vibrational dynamics. Each element r ∈ Res carries three real-valued observables: energy E(r) ≥ 0, damping rate γ(r) > 0, and bandwidth BW(r) > 0. The algebra is equipped with addition (superposition), multiplication (spectral convolution), and a symmetric bilinear resonance form ⟨r, s⟩.
We establish 13 theorems including: (1) Parseval identity — self-resonance equals energy; (2) Cauchy-Schwarz bound on resonance; (3) convolution preserves the unit energy ball; (4) bandwidth-energy tradeoff derived from the uncertainty principle. The framework is formalized in the Platonic proof language with 11 structural hypotheses and 5 primitive axioms, yielding 33 verified declarations.
Applications span turbulence cascades, quantum mechanical observables, and financial contagion networks — systems where spectral interaction governs dynamics.