Hopf Arithmetic–Energy Duality B: Arithmetic–Spectral Gap Correspondence on Compact Lie Group Principal Bundles—Conditional Channel, Discrimination Threshold, and Boundary Declarations

This paper studies, in the framework of compact Lie group principal bundles, the conditionalcorrespondence between the arithmetic structure of a finite set of points in the base space andthe spectral regularity of the Schreier graph it generates. The two routes use disjoint hypothesissets: the discrimination threshold lemma acts only on the arithmetic input layer and provesthat the product-type sum-product criterion does not imply an additive energy upper bound,with the grid family, the product family, and the root-of-unity family constituting three types ofwitnesses whose roles cannot substitute for one another; the group-action–coordinate-embeddingincompatibility declaration acts only on the structural compatibility layer and shows that thegroup action preserves orbit structure while addition and multiplication are defined on complexcoordinates, so the channel between them must be introduced by an explicit hypothesis. Underthe uniformized hypothesis layer (energy bound under coordinate embedding H1′, morphismchain compatibility H2′, and spectral-gap–edge-expansion interface H3′), the conditional maintheorem gives a lower bound for the spectral gap bounded by an explicit function of the squareof the energy exponent and the degree of the generating set. A circle counterexample provesthat unconditionally the spectral gap can tend to zero; graph-by-graph verification in the toylayer shows that the uniform form fails on the only explicitly computable family known withinthe framework of this paper; together these two facts lock the instantiation burden of the maintheorem onto the family-uniform form of H2′. This paper does not claim that the theorem holdsin a nontrivial form on any verified family