FDCL Part III: Resistance and Diffusion on FDCL Graphs — Exact Flows and Harmonic Reduction

Abstract

We study resistance and diffusion on graph models associated with the six-digit FDCL construction. For the voxel graph, an explicit optimizing current gives the exact resistance between shorted horizontal faces. An orthogonal load decomposition and corrected planar currents prove that the prescribed uniform-face resistance is less than 1247/1152 at every level, while the corner-to-face resistance diverges at least linearly in the level. We derive exact harmonic reduction inequalities, a sharp anchored bound, and asymptotic equivalence of the harmonic and full first spectral gaps. For a distinct antidiagonal graph, a six-port recursion is exact for static boundary energy; nested cuts prove resistance divergence, reflection proves the spectral ordering, and an explicit mass derivative records the missing spectral parameter. Finite diffusion is treated with its speed measure and clock fixed. Exact cylinder averaging gives energy compactness at the 12n clock; the 6n clock remains conditional. A catalogue-free local harmonic contraction and exact residual enclosures separate local certificates from the remaining growing-cutoff conditions. The results preserve usable interfaces for companion papers without inferring an infinite-volume diffusion exponent from finite fits. Series and status. FDCL Part III of twelve, Version 1.0 (manuscript dated 7 October 2026, 39 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 and VIII as companion manuscripts. Files: the manuscript as PDF and a source archive (56 files) with the LaTeX source, exact finite resistance witnesses and 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).