FDCL Part VII: Arithmetic Readouts and Prime-Set Structure in FDCL Models — Exact Obstructions and Finite-Field Reconstructions

Abstract

We investigate what arithmetic information is retained by readouts of FDCL graphs, address automata and explicitly declared finite-field reconstruction models. A fixed additive length group cannot supply arbitrarily many independent prime logarithms, but a one-dimensional determinant sequence already has infinite multiplicative rank. We give factorization-free rank certificates and sharp count-vector bounds. Every nonempty native address-word transformation is eventually idempotent, and its fixed-point count depends only on its letter support, at every finite radius. The resulting languages are star-free; transient images can still distinguish word order. Natural endomorphism rings, covering predicates and reduction of power series are treated with their exact categories and coefficient rings. An explicit reachable orbit disproves a component-only Frobenius test; disjoint strata, point bounds and coverage give a sufficient criterion. Ternary digit models yield exact minimal observable state counts and rationality and algebraicity prime sets; all 162 characteristic-three moment types have determined degrees, genera and curve-local zeta functions. Elliptic supersingular primes provide an infinite density-zero control with unbounded minimal CM witness size. Classical analytic controls and exact obstructions distinguish completion symmetry and self-adjointness from arithmetic identification. Series and status. FDCL Part VII of twelve, Version 1.0 (manuscript dated 7 October 2026, 40 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, VIII and IX as companion manuscripts. Files: the manuscript as PDF and a source archive (111 files) with the LaTeX source and finite exact verifiers with their inputs and 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).