Hopf Arithmetic–Energy Duality E: Cross-Categorical Interfaces in the Kakeya Context

This paper performs a structural audit, in the context of the Kakeya problem, of threegroups of cross-categorical interfaces: combinatorial projection and additive energy, tube-bundleintersection complexes and Riesz energy, and filtered-chain algebras and viscous decay. Theconclusion of the audit takes the form of discrimination and registration, not theorem assembly.At the object level, the Hopf-type encoding of direction sets on the two-dimensional sphere mustpass through a non-canonical spinor lift; the coordinate image carries arithmetic structure, andthe position condition guarantees uniform control of metric distortion, and this condition cannotbe removed. At the discrimination level, passage of the product-type sum-product criterion doesnot imply that additive energy drops from the B3 scale; exact closed-form computations forthe grid family and the root-of-unity family close this inference chain; all arithmetic inputsmust therefore take energy form. At the assumption level, the skeleton mass bound conditionaltheorem holds under four explicit assumptions—regularity, additive energy bound, geometricposition, and operator bound—and registration (R) (finiteness of the scaled density norm),in which the operator bound is an explicit assumption rather than a conclusion derived fromother assumptions; the general higher-dimensional form is an open problem. At the relocationlevel, the homotopy type of the intersection complex does not determine finiteness of the Rieszenergy of measures on it; this counterexample is an exact computation and closes the claimthat “stabilization is equivalent to energy finiteness”; the linear formula for Krull dimensionwas withdrawn in type review because no derivation chain exists between the objects at itstwo ends, and its legitimate form is registered as an open problem with a dimension-alignmentcondition. The main asset of this paper is no-go results: four closed failures are entered inno-go form, and their adjudication value equals that of positive theorems. The paper producesno new Kakeya dimension or volume bounds, claims no cross-categorical equivalence, and thelegitimate form of cross-categorical correspondence is only conjectures or open problems withgap declarations.