The Emerald Tablet: Foundations of Onu Calculus

We introduce a scale-covariant reformulation of the three-dimensional incompressible Navier–Stokes equations based on a logarithmic radial coordinate (s = ln r), which maps multiplicative scaling to translation on an infinite line. Under this transformation, dilation-invariant dynamics are governed by a one-dimensional transport–diffusion equation for a scale-energy density E(s,t), generated by a scale-covariant operator (OSCO) with bounded transport velocity and positive scale viscosity. Within this framework, potential finite-time singularities in Euclidean space correspond to continuous energy flux toward the horizon (s → -∞), rather than local blow-up. We establish a conserved energy ledger identity on the scale line and prove global smoothness for small initial data in a scale-weighted Sobolev space H¹_Onu. For arbitrary smooth data, we obtain conditional global regularity assuming uniform control of angular concentration on ℝ², isolating vortex stretching as the sole obstruction within the scale-covariant dynamics. The approach unifies elements of Leray self-similarity, Mellin convolution, and multiresolution analysis, and provides a geometric reinterpretation of the Navier–Stokes cascade that separates coordinate artifacts from physical singularities. Limitations and directions toward removing the angular regularity condition via Littlewood–Paley analysis are discussed.