The Dyadic–Nome Bridge: A Structural Classification Theorem for Admissible Scale Connections

A persistent difficulty in foundational physics is the comparison of quantities defined at different scales. Ultraviolet–infrared relations, vacuum hierarchies, and canonically completed spectral data become well-defined only after the normalization structure of the underlying source has been specified. In hierarchy and naturalness problems, this appears as the problem of distinguishing scale-invariant structural data from quantities whose values depend on a chosen normalization, subtraction, or parametrization scheme. The prior question is therefore not whether quantities at different scales have the same numerical value, but what structure makes their comparison invariant rather than convention-dependent. This paper answers that question by proving a structural classification theorem for admissible scale connections: cross-scale comparison laws satisfying zero-scale normalization, semigroup composition, continuity, and infinitesimal factorization through a phase weight and source response invariant. The main result shows that every admissible scale connection follows a canonical exponential transport law in a real nome parameter. This identifies the real nome as the intrinsic multiplicative coordinate for lawful scale comparison, rather than a branch-specific convention. When the dyadic phase lock and completion-level nome are jointly inherited from the same completion-locked source, this classification specializes to the dyadic–nome bridge. The dyadic side comes from the circle/Hopf phase structure, where the diagonal Hopf-fiber geometry determines the dyadic half-angle lock. The nome side comes from the heat and Fourier structure of the circle. The fixed-point-normalized Berezinskii–Kosterlitz–Thouless (BKT) critical law selects the response-side locked nome, while the primitive Archimedean determinant calculation identifies the matching completion-side normalization. The canonical-lock-inheritance theorem then shows that the response branch and Archimedean completed determinant channel inherit the same completion lock. Thus the dyadic–nome bridge identifies when scale comparison is canonical: the compared quantities must descend from one completion-locked source through jointly inherited phase and completion normalizations. Cross-scale comparison becomes a canonical operation on source-level data rather than a downstream adjustment of conventions. The appendix records representative instantiations of the same dyadic–nome transport law in fine-structure constant normalization, cosmological-constant vacuum transport, and Hilbert–Pólya spectral matching. Across these derivations, the bridge functions as an upstream lock, so that scale comparison is governed by source-level phase and completion data rather than by downstream normalization, fitting, or spectral matching conventions. License note: Distributed under CC BY-NC-ND 4.0.