Certified Finite Robust Reconstruction of Fibonacci-Modulo-9 Translation Classes: Exact Channel Minima in Two Prescribed 24-Channel Observation Families

Certified Finite Robust Reconstruction of Fibonacci-Modulo-9 Translation Classes: Exact Channel Minima in Two Prescribed 24-Channel Observation Families We investigate exact and robust reconstruction of translation-equivalence classes of a three-dimensional periodic Fibonacci-residue product field modulo nine. The underlying field has spatial period 24 in each coordinate, and its translation action yields 3,456 distinct equivalence classes. Two explicitly prescribed observation families are studied: toroidal meridian sampling curves and independently constructed spatial wave-loop sampling curves. Each family consists of 24 observation channels, with 12 sampled lattice sites per channel. The resulting observations define finite codes whose distinguishability and robustness are characterized by minimum Hamming distance. For both observation families, we establish the exact minimum numbers of retained channels required to achieve minimum Hamming distances 1, 2, 3, and 5: m(1) = 6, m(2) = 7, m(3) = 9, and m(5) = 13. These thresholds correspond respectively to error-free identification, tolerance of one known erasure, correction of one arbitrary substitution, and correction of two arbitrary substitutions. The optimality results are supported by explicit channel constructions, exhaustive pairwise separation analysis, exact combinatorial exclusion of smaller channel subsets, and an independent verification implementation. The accompanying reproducibility package includes Python source code, fixed integer sampling offsets, computational audit materials, and documentation. The results are exact within the two specified finite observation families. They do not establish global optimality over arbitrary observation geometries, physical realizability, or superiority of toroidal sampling. The underlying coding-theoretic and multicover methods have established precedents; the contribution is the certified finite analysis of the explicitly defined Fibonacci-modulo-nine observation systems. Keywords: Fibonacci modulo 9; Pisano period; finite reconstruction; robust identification; Hamming distance; test cover; set multicover; computational proof; optimal channel selection. Author-approved release — 8 November 2026. This preprint is released with the explicit approval of the named author, Ivo Vajc, following author review of the mathematical manuscript and execution of the accompanying computational programs. The release includes the final manuscript, LaTeX source, Python implementations, an independent finite verification procedure, fixed sampling coordinates, a literature audit, and reproducibility documentation. The manuscript is licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). The accompanying Python source code is licensed under the MIT License. The mathematical results concern explicitly stipulated finite observation models and have not undergone external peer review. No empirical physical validation or unrestricted claim of mathematical novelty is asserted. Declaration of AI Assistance OpenAI ChatGPT was used substantially in mathematical exploration, algorithm development, Python programming, computational certificate design, literature review assistance, and manuscript drafting. The named author, Ivo Vajc, reviewed the mathematical content, tested the accompanying programs, examined the final release materials, and explicitly approved the manuscript for publication. The author retains responsibility for the accuracy of the mathematical statements, the released source files, bibliographic references, originality claims, and the decision to publish. AI assistance does not constitute independent human peer review or experimental validation.