FDCL Part VI: Zeta Functions and Trace Growth on FDCL Graphs — Exact Determinants and Infinite-Volume Bounds

We study metric spectral zeta functions, finite orbit determinants and nonbacktracking trace growth for explicitly distinguished FDCL graph models. For the unweighted Factory graph, edge-disjoint recursive copies make every anchor-normalized closed-walk count monotone and bounded. This proves existence of all fixed-length trace limits directly from the construction. The corresponding infinite-volume logarithmic determinant converges locally uniformly, and its power-series radius equals the reciprocal of the limiting finite-core Perron root. Radius contraction additionally gives an exact four-mode tail for every fixed walk length, with logarithmic onset. Finite integer certificates provide a narrow spectral window and explicit short traces. For the auxiliary metric tree, we derive the spectral zeta continuation, normalized determinant and an exact reciprocal spectral sum. Primitive orbit products, core pruning, weighted vertex reduction and subdivision retain their operator conventions. A uniform Bass-resolvent estimate and the lattice geometry identify maximum-walk growth with the same Perron limit, with an explicit polynomial prefactor. Every fixed-length maximum is attained at an explicit logarithmic level, and local positive weights approach the limit at a quantified rate. A certified diagonal metric gives uniform power and complex-resolvent constants, while a separate radial argument retains the full exterior Perron domain. Finally, carried-skeleton determinant budgets and exact finite valuations are separated from unproved per-prime density claims. Complete finite certificates accompany the numerical bounds. Series and status. FDCL Part VI of twelve, Version 1.0 (manuscript dated 7 October 2026, 39 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 and V as companion manuscripts. Files: the manuscript as PDF and the source material (53 files) with the LaTeX source, the complete finite inputs for each retained numerical certificate and their verification scripts, provided as four archives as described in the note below. The other parts of the series are archived separately. Note on record version 1.01. This version adds the source material, which could not be uploaded with record version 1.0. Because of its size (about 82 MB) it is provided as four archives that share one folder tree, FDCL_Part_VI_v1.0_Source. Archive part1 (ZIP) holds the LaTeX source, the check scripts, the smaller certificates, README.md and SHA256SUMS.txt. Archives part2 and part3 (ZIP) hold one large certificate each (cw_Q13_automaton.npz and cw_radius12_full.npz). Archive part4 holds the largest certificate (zr_lpf_cert_n8.npz); it is one ZIP file cut into five volumes (.zip.001 to .zip.005), which must be joined in order by plain concatenation before extraction (for example with cat on Linux or macOS, or copy /b on Windows; 7-Zip can also open the .001 volume directly). The joined part4 file has SHA-256 6c3a53936c55e6e13b6edd77aaad44079995fb2526d8a4c101f00b324686e280. Extracting all four archives into the same directory gives the complete tree of 53 files, which can be checked against SHA256SUMS.txt. The files are byte-identical to those in the author's source archive FDCL_Part_VI_v1.0_Source.zip; nothing was recompressed or altered. The manuscript PDF is unchanged: it is the same Version 1.0 file, dated 7 October 2026. 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).