FDCL Part XI: Dynamical SU(2) Gauge Fields on FDCL Graphs — Hierarchical Curvature Completion and Local Spectral Bounds

Abstract

We specify a dynamical SU(2) gauge model on finite FDCL voxel graphs, with Gauss invariance at every vertex. Unit plaquettes miss a central flux whose shortest detecting loop has length 2n + 4. An explicit hierarchical catalogue completes nonabelian flatness with the minimum number of added relations and the minimum possible longest loop. Its cell topology and the sharp interaction-moment constants follow from direct geometric arguments. General depth weights separate flatness detection from quantitative decay. The standard compact-link and spin-network construction gives a self-adjoint physical Hamiltonian with a unique positive ground state on each fixed graph. We identify both the exact weighted electric gap and its first eigenspace. For two squares sharing an edge, we match the lowest-spin compression to the established two-plaquette model and retain the full spin space at asymmetric couplings. Energy-resolved Schur comparisons use the actual finite coupling shell to control every omitted representation. A center-parity symmetry improves the explicit second-order gap expansion to a fourth-order remainder and determines its sign away from the critical directions. Exact rational certificates distinguish operator ordering, independent-square replacement and small-cutoff gaps. Classical sign-repair bounds, vacuum-annihilation criteria and sharp state-cutoff estimates distinguish geometric, form and quantum errors. A positive transfer operator supplies reflection-positive finite time cylinders and a fixed-graph continuous-time limit. A uniform interacting gap across graph levels and a continuum Yang–Mills construction remain unproved. Series and status. FDCL Part XI of twelve, Version 1.0 (manuscript dated 7 October 2026, 49 pages); unsubmitted working paper. The FDCL series studies the Fractal Diagonal Cut Lattice, the three-dimensional self-similar set generated by six dyadic corner maps, and the graph, operator and gauge models associated with it. This part cites Parts I, VIII, IX and XII as companion manuscripts. Files: the manuscript as PDF and a source archive (56 files) with the LaTeX source, the exact finite records supporting its numerical claims and their verification scripts. The other parts are archived separately. AI use disclosure. Generative AI (GPT-6.0, OpenAI; Claude Opus 5.5, Anthropic) was used substantively in preparing this work, including literature comparison, the development and checking of proofs and counterexamples, exact computations and the writing and running of verification code, and drafting and editing. The research questions, framework and final claims were directed and reviewed by the author, who takes full responsibility for the content, including the accuracy of all references and reported numbers. Repository metadata were prepared with assistance from Claude (Anthropic).