FDCL Part IX: Proof Methods and Exact Certification for FDCL Models — From Finite Evidence to Uniform Theorems

We develop explicit proof and certification interfaces for the FDCL program using classical geometric, linear-algebraic and finite-state methods. Reducible pressure bounds and sharp multiplicity examples separate symbolic growth from geometric dimension. Mass-orthogonal constraint repair, residual minimization and a norm-preserving symmetric completion sharpen finite error certificates; inertia supplies the index information absent from residual proximity. General mass Schur pencils have a strictly negative derivative wherever the eliminated block is invertible. Modular rank upper bounds require an explicit minor-height budget, while rational acceptance retains all-row identity checks. As a concrete FDCL consequence of the revised Part VII support theorem, we derive the full weighted transfer determinant at every radius and an exact count of eventually resetting words. These conclusions concern the full native state space, separately from a seed-observable quotient. Coupled tail budgets and normalized energy-profile bounds quantify two further evidence transfers. The general implications are proved here; the FDCL specialization explicitly identifies its companion premise. Exact finite witnesses illustrate the proofs without substituting for their uniform hypotheses. Series and status. FDCL Part IX of twelve, Version 1.0 (manuscript dated 7 October 2026, 43 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 VIII as companion manuscripts. Files: the manuscript as PDF and a source archive (34 files) with the LaTeX source, six exact finite checkers 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).