The Pythagorean Angle Lattice and the Omni-Metallic Framework: A Free Quantum System

An exactly solvable free quantum system whose energy levels and multiplicities are labelled by a natural arithmetic kernel arising from metallic means and primitive Pythagorean triples : We study a free quantum system whose spectral data are labelled by arithmetic functions arising from the Pythagorean Angle Lattice and the Omni-Metallic Framework. The Pythagorean Angle Lattice is a countable set derived from primitive Pythagorean triples, and the Omni-Metallic Framework is a four-parameter algebraic generalization of the classical metallic means. The Bridge Identity connects these two structures through the arithmetic kernel r(k) = ⌊(k − 1)/2⌋. The quantum system has spectrum E_k = k² with multiplicity r(k), mass gap Δ = 9, exact propagator, response function, and partition function Z(β) = ∏ₖ₌₂^∞ (1 − e−β k²)⁻ʳ⁽ᵏ⁾. The entropy grows as S(E) ∼ (1/2) log E and satisfies the third law of thermodynamics. This constitutes an exactly solvable free quantum system constructed from number-theoretic data.