FDCL Part V: Gauge Hamiltonians and Scattering on FDCL Graphs — Exact Reductions and Observable Quotients

Abstract

We develop exact operator reductions for several graph models associated with FDCL, keeping their underlying graphs, mass metrics and boundary conventions explicit. A harmonic extension yields a two-sided spectral-gap bound with its induced mass; a coordinate trial function rules out a proposed level-independent static comparison. For a voxel SU(2) connection, we prove holonomy and pairing identities, the sharp uniform base-cell ground-energy floor, its optimal bound sin4θ/6, and an exact ground quartet. For an ordered bowtie hierarchy, a sum of squares reduces endpoint multiplicity to signed constraints and gives a complete local-mode basis from level four onward, with positive limiting density. Local screening maps admit rational response and global logarithmic contraction proofs, while continuously encoded full framed responses obey a dimension obstruction and exact seed transport identities. Weighted scattering is conjugated to rational coordinates, yielding arithmetic, phase-selection and reciprocal determinant laws without radical calculations. Finally, observable quotients separate representation, form and grading descent, and prove the consequences of supplied exact matrix certificates for two finite rational transport models. Direct proofs replace numerical extrapolation where possible; passive complex responses, Ward identities and dual load certificates retain their masses and sources. Finite certificates and remaining all-level questions are identified separately. Series and status. FDCL Part V of twelve, Version 1.0 (manuscript dated 7 October 2026, 38 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, II, III and IV as companion manuscripts. Files: the manuscript as PDF and a source archive (150 files) with the LaTeX source, the selected finite certificates used by the paper 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).