FDCL Part VIII: Hierarchical Networks and Feedback Control in FDCL Models — Exact Routing, Stability and Observation Conditions

Abstract

We develop explicit network, routing and feedback models motivated by FDCL geometry. Core–branch reductions preserve optimal congestion, while joint routing and capacity allocation on a general graph reduce to weighted shortest paths. A weighted graph field has an exact diffusion threshold including coupling to null modes; a quartic reaction stabilizes the linearly neutral regime with algebraic decay. Future-observable memory admits a sharp finite determination horizon. For the native FDCL word readouts classified in Part VII, exactly sixteen summary states preserve all future recurrent readouts, including the empty history. Routing certificates quantify supply deficits, the amount of randomization needed at an optimum, and abstraction error. A conditional quadratic certificate gives time-uniform probability bounds and a sharp reserve for changes in the selection law. Finally, a commuting dephasing model admits the exact range of rates compatible with prescribed diagonal noise, and bounded interventions have an explicit optimal allocation. These results use classical tools with their sources and assumptions stated; they certify the declared mathematical models, without asserting measured deployment performance. Series and status. FDCL Part VIII of twelve, Version 1.0 (manuscript dated 7 October 2026, 48 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, IV, V, VI, VII and IX as companion manuscripts. Files: the manuscript as PDF and a source archive (33 files) with the LaTeX source, six finite verifiers and their reports. 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).