FDCL Part IV: Recursive Dynamics and Observation on FDCL Graphs — Finite-State Laws and Four-Mode Reduction

We study recursive generation and finite observation of FDCL graphs. Exact incidence and birth systems distinguish counting quotients from letterwise observation. Geometric restriction and lifting give shift equivalences between origin-pattern automata at different radii. The seed cyclic module has four persistent modes 6n, n2n, 2n, 1, whereas the full transfer has a 33-dimensional invertible part. For radius R ≥ 2, the seed transient depth lies between ⌈log2 R⌉ and ⌈log2 R⌉ + 1. Exact weighted crop factorizations preserve the full nonzero Jordan structure. Two parallel leaf deletions give the Factory core; a radius-(r+3) occupancy window determines each rooted core r-ball. Consequently its finite-radius distributions converge to rational limits with error Or(n3−n). Supplied binary observation matrices retain a separate geometric-identification requirement. Complete output histories distinguish branching from extinction, while ambient grading, native quotients and integral certificates retain their descent conditions. The finite certificates and their verification rules accompany the proofs. Series and status. FDCL Part IV of twelve, Version 1.0 (manuscript dated 7 October 2026, 35 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, V, VII, VIII and IX as companion manuscripts. Files: the manuscript as PDF and a source archive (61 files) with the LaTeX source, the exact finite certificates required 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).