Hopf Arithmetic–Energy Duality A: An Arithmetic–Energy Fractal Uncertainty Principle on the Hopf Fibration

This paper studies the quantitative correspondence between the arithmetic structure of finitedirection sets on the three-dimensional direction sphere S3 and the analytic regularity of theirprojection measures. Through the composition of the Hopf map and stereographic projection, adirection set is encoded as a finite set in the complex plane and then connected to the frequencyside by a linear isometric embedding. We first prove a discrimination threshold lemma: aproduct-type sum-product criterion does not imply an additive energy bound, with the gridfamily as the main witness and the root-of-unity family and the direct-product family as controls;their exact integer arithmetic is recorded in the machine-check ledger. The hypothesis layermust adopt the energy form and must be explicitly stated as parallel set-type and measuretype clauses. Under this hypothesis layer—consisting of four clauses: regularity, additive energybound (two clauses), geometric position, and physical concentration—we prove that the operatornorm bound holds (the literature-transcription check of its final gain exponent is registered asdebt), and the gain exponent is determined by the energy exponent; moreover, the projectionmeasure satisfies scale-by-scale sticky mass bounds on the arithmetic frequency skeleton, andthe skeleton Riesz energy is finite. Full energy finiteness is proved as a conditional conclusion,holding only under an additional analytic Fourier decay assumption. All geometric constantsare explicit and proved sharp; bare l2 sticky decay is disproved by two counterexamples, and allsticky conclusions are stated only in scale-commensurate form; no general transfer from set-typeenergy bounds to measure-type energy bounds exists, and this is registered as a no-go asset; theparallel two-clause structure of the hypothesis layer is supported by this asset.