Hopf Arithmetic–Energy Duality C: Viscous Spectrum and Viscous Dimension—Scaled Definitions, Discrimination Thresholds, and Conditional Inequality Chains

This paper establishes a theory of viscous dimension within the energy-method framework ofthe fractal uncertainty principle. Two baseline facts are fixed first: the first is the discriminationthreshold lemma; the grid family, the root-of-unity family, and the product family prove thatthe product-type sum-product criterion does not imply an additive energy bound. The second isthe bare-viscosity double counterexample; the Dirac measure and the cube measure prove thatl2 viscous decay without a scale parameter does not hold. On these two baselines, the viscousdimension is defined in scaled form: it depends on a scale parameter h, an arithmetic frequencyskeleton, and an operator norm bound, and its decay exponent is given by the energy-boundexponent through an explicit channel; without an energy bound, the viscous dimension existsonly as a formal symbol, and no well-posedness is claimed. This paper proves that the inequalitychain among the viscous dimension, the Fourier dimension, and the Hausdorff dimension holdsin conditional form: the left end is obtained under an analytic Fourier decay assumption, whilethe right end distinguishes skeleton energy from full energy; finiteness of the full energy holdsonly under an additional high-frequency decay assumption. The examples of random directionsets and arithmetic-structure direction sets are given as conditional numerical evidence in theform of a machine-check ledger, and no general theorem is claimed.