FDCL Part I: Geometry and Connectivity of FDCL and Partial-Unfolding Fractals

Abstract

We determine the contact geometry of the six-map Fractal Diagonal Cut Lattice (FDCL) and the connectivity of two separately specified graph and planar constructions. All fifteen first-level contacts are classified explicitly. Their union contains dyadic combs, has Hausdorff dimension one, infinite length, and five connected components; its finite eight-edge carrier is a proper subset. The full set of multiply coded points also has dimension one, and the maximum number of addresses is six. For the auxiliary four-arc graph, constructive all-level arguments prove connectivity and core persistence. A displayed degree table and complete finite certificates, with coverage and descent proofs, establish the local core and rooted-fibre structure. All sixteen deterministic planar arm selections have exact dimensions; the three-arm value follows from a two-term counting recurrence. Under independent activation with a persistent core, the almost-sure dimension is deterministic and nondecreasing in the activation probability. Explicit bounds make it positive at every positive probability, although the set is almost surely totally disconnected below one quarter. The core-free law is identified with dyadic fractal percolation. These results retain the distinct geometric objects and probability laws needed by subsequent analyses. Series and status. FDCL Part I of twelve, Version 1.0 (manuscript dated 7 October 2026, 29 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 opens the series and cites no companion manuscript. Files: the manuscript as PDF and a source archive (35 files) with the LaTeX source, the figures, the complete finite inputs for its local graph theorem 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).