Optimal Conjugate Probes and an Exact Exceptional Kernel for a Jordan System over F₃

This mathematical preprint studies exact identification of traceless 3 × 3 matrices over the finite field F₃ using selectable conjugates of the regular unipotent Jordan matrix q = I + E₁₂ + E₂₃. The unknown perturbation X remains fixed while the known probe orientation changes. Observations are the full formal coefficient matrices R₁, R₂, and R₃ of [q_g(I + εX)]³ − I.Two explicit orientations identify all 6,561 possible perturbations when coefficients through degree three are available. A single conjugate orientation is insufficient, so two fixed orientations are optimal for this measurement model. For the stated pair, coefficients through degree two determine X uniquely exactly outside a four-dimensional joint first-response kernel K containing 81 states. Inside K, three explicit quadratic forms give eight fibers of sizes 3, 6, 6, 6, 12, 12, 12, and 24. Scalar ambiguity prevents complete identification through degree two even when arbitrary numbers of conjugate orientations are available.The proof package supplies 80 affine linear collision systems and 80 unit identities in finite-field coordinate rings with relations tᵢ³ = tᵢ. A separate, self-contained Python checker verifies the stored proof objects without importing the certificate generator. Exhaustive numerical censuses and an adaptive equation decoder provide additional verification on all 6,561 states. The decoder requests third-order responses only for the 81 exceptional states.The deposit includes the manuscript PDF, LaTeX sources, certificate data, standard-library Python programs, verification outputs, licenses, and a SHA-256 manifest. The results constitute a finite computer-assisted algebraic classification for the specified model over F₃. No extension to other fields or dimensions, noise robustness, or optimality of individual scalar measurements is claimed. This version has not undergone external peer review. The manuscript, certificate data, and accompanying non-code documentation are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Python source code is licensed under the MIT License. These licenses apply to their respective components; they do not offer alternative licenses for every file. See LICENSE_CONTENT.txt and LICENSE_CODE.txt in the source archive.