Hopf Arithmetic–Energy Duality D: Encoding of Adaptive Mesh Refinement Direction Sets, Viscous Control, and Skeleton Energy

This paper studies the conditional channel between the arithmetic structure of cell directionsets in adaptive mesh refinement and the analytic regularity of projection measures. At thegeometric layer, three-dimensional directions lie on the two-dimensional sphere, and the use ofthe Hopf fibration requires an explicit spinor lift; under this lift, the intermediate compositionis the identity map on the coordinate plane, and the fourfold composition degenerates to thestereographic projection itself, so that the arithmetic structure lives on the coordinate image. Atthe arithmetic discrimination layer, the grid family and the root-of-unity family witness, from thehigh-energy and low-energy ends respectively, that the product-type sum-product criterion doesnot imply an additive energy bound; the arithmetic progression family and the uniform Cartesiangrid direction set exclude, from the two directions of “non-decay of energy” and “non-expansionof the direction set,” the inference that refinement rules automatically produce an energy bound.Under the four external assumptions of regularity (H1), additive energy bound (H2), geometricposition (H3), and operator bound (H4), the scaled viscous mass satisfies slicewise decay, andthe skeleton Riesz energy is finite in the two-dimensional exponent range 0 < s < 2; finiteness ofthe full energy holds conditionally only under an additional analytic Fourier decay assumption.All inferences toward finite element stability are registered as open problems at the morphismlayer and do not enter the list of conclusions.