Exact Robust Coordinate Reconstruction for a Fourth-Order Jordan Response Space over F₃: Finite Certificates, Optimal Repetition, and Reproducible Verification

This preprint establishes exact optimal measurement counts for robust reconstruction of the fourth-order response space associated with a six-dimensional unipotent Jordan block over the finite field F₃. Let J = I₆ + Σᵢ₌₁⁵ Eᵢ,ᵢ₊₁ and consider the formal expansion (J(I₆ + εX))⁹ − I₆ = Σₖ₌₁⁹ εᵏ Rₖ(X), where X is a trace-zero 6 × 6 matrix over F₃. The admissible perturbations satisfy R₁(X) = R₂(X) = 0, and W₄ denotes the linear span of their fourth-order responses. Assuming the independently certified result dim_F₃(W₄) = 24 from RT-JORDAN 2.7, we determine the exact minimum numbers of repeated individual matrix-entry measurements required for unique reconstruction: 24 measurements without errors, 42 measurements with one known erasure, and 66 measurements with one unknown substitution. The finite certificate identifies 15 essential coordinate functionals and 12 additional nonzero coordinates. An exhaustive exact-arithmetic audit examines all 4,096 optional coordinate supports, including 105 full-rank supports, and establishes the optimality of the proposed measurement multiplicities. The proof combines finite-field linear algebra, rank-based erasure and substitution criteria, exhaustive support enumeration, and independent verification algorithms. Two separately implemented Python programs reproduce the exact robustness optima. The accompanying reproducibility archive includes the mathematical manuscript, LaTeX source, verification programs, explicit optimal measurement multiplicities, proof audit, targeted literature audit, bibliography, author approval, AI-assistance disclosure, license files, and SHA-256 checksums. The study concerns a precisely defined finite algebraic observation model. It does not claim experimental physical validation or establish mathematical novelty relative to the complete existing literature. Publication status: Author: Ivo Vajc, Independent ResearcherVersion: 3.3Date: 11 November 2026Status: Author-approved, non-peer-reviewed mathematical preprint. Mathematical result: The exact optimal numbers of repeated coordinate measurements are N_clean = 24, N_erasure = 42, and N_substitution = 66. These results are conditional on the independently certified equality dim_F₃(W₄) = 24 established in the preceding RT-JORDAN research. Reproducibility: The accompanying package includes RT-JORDAN 2.6, 3.1, and 3.2 Python verification programs. The robustness calculations use exact arithmetic over F₃ and include exhaustive verification of 4,096 optional supports, of which 105 have full rank. Proof status: The mathematical conclusions are supported by finite computational certificates and separately implemented verification procedures. No external peer review is claimed. Licensing: The manuscript and accompanying scholarly documentation are released under Creative Commons Attribution 4.0 International (CC BY 4.0). The Python source code is released under the MIT License. Author authorization: The author reports reviewing the complete package, running the verification programs, checking the results, and explicitly granting approval for publication on Zenodo on 11 November 2026. AI assistance: OpenAI ChatGPT (GPT-6) assisted with mathematical analysis, algorithm development, Python implementation, verification workflows, literature-audit preparation, manuscript drafting, LaTeX formatting, and publication packaging. The author retains responsibility for the scientific content, verification, interpretation, and publication decisions. AI is not listed as a coauthor. Scope: The work concerns a stipulated finite algebraic model. No experimental validation or comprehensive claim of mathematical novelty is made.