Exact Third-Order Reconstruction and Optimal Linear Detection for a Six-Dimensional Unipotent Jordan System over F₃

We investigate third-order reconstruction responses associated with a six-dimensional unipotent Jordan block over the finite field F₃. Let J₆ be the standard unipotent Jordan block and let X range over the trace-zero matrices in sl₆(F₃). We define the reconstruction coefficients Rₖ(X) through the formal expansion [J₆(I + εX)]⁹ − I = Σₖ₌₁⁹ εᵏRₖ(X). On the admissible perturbation locus defined by R₁(X) = R₂(X) = 0, we prove that the linear span of the third-order responses R₃(X) has dimension exactly four over F₃. The upper bound is established through exact cubic coefficient extraction and degree-three reduction modulo the quadratic constraints defining the admissible locus. The lower bound is certified by four explicit admissible perturbations with linearly independent third-order responses. We further establish that every nonzero projective direction in the four-dimensional response space is attained by an admissible perturbation. Explicit certificates cover all 40 projective directions and are checked using exact finite-field arithmetic. Consequently, four independent linear scalar probes are both necessary and sufficient for global zero detection and complete linear recovery of third-order responses. A concrete optimal set is provided by the matrix entries (1,1), (1,4), (1,5), and (1,6). The results are specific to the six-dimensional system and the stated linear observation model. Reproducible Python code and computational certificates accompany the preprint. No general theorem for arbitrary Jordan block dimensions is claimed. Research context: This work forms part of the Reconstruction Theory research programme, specifically the RT-JORDAN series (versions 1.0–1.9). Mathematical scope: The main theorem concerns a six-dimensional unipotent Jordan block over F₃, third-order polynomial reconstruction responses, and linear scalar observation maps. It does not establish an arbitrary-dimensional classification. Computational reproducibility: The accompanying standalone Python verifier uses exact arithmetic over F₃ and requires no external Python packages. It is designed to run offline, including under Pydroid 3 on Android. Verification status: Algebraic upper bounds, explicit response witnesses, and all 40 projective-direction certificates have been checked computationally. The preprint has not undergone external peer review. AI assistance disclosure: OpenAI ChatGPT was used substantially in mathematical exploration, programming, certificate design, and manuscript drafting. The named author is responsible for reviewing the mathematics, checking the released files, verifying citations and originality claims, and deciding whether to submit the manuscript.