The Cauchy Stress Tensor from the de Broglie Wave Equation: A Complete Derivation in Natural Coordinates

We derive the complete Cauchy stress tensor from the de Broglie relation λp = h using natural local coordinates δλi = h/δpi provided by wave geometry. The tensor has nine independent elements: three diagonal (normal pressures) determined bysingle-axis wavelengths δλi, and six off-diagonal (shear stresses) determined by pairwavelengths δλij ̸= δλji in general. The resulting tensor is not necessarily symmetric, departing from the classical Cauchy tensor. The asymmetric part generates vorticity, the anisotropy generates turbulence, and the full tensor eliminates the need for empirical viscosity, turbulence models, and loss coefficients. All quantities are derived from first principles using λp = h without empirical input.