How Far Can a Qutrit Beat a Qubit on Antipodal Tables? An Exact Qubit Test, Bounds, and a Conjectured Constant for Three Queries

Third note in a series on qubit–qutrit equivalence for centrally symmetric prepare-and-measure correlations (doi:10.5281/zenodo.24093433, doi:10.5281/zenodo.24121836). Setting: fixed binary measurements, no shared randomness, and tables of antipodal pairs of target expectation vectors. The question: by how much must a qutrit table be shrunk before a qubit can reproduce it? Proved: A qubit reproduces such a table exactly if and only if it factors through 3-dimensional real space with all rows and columns in the unit ball. With at most three queries this is the condition γ₂(T) ≤ 1 on the factorization norm, a single semidefinite program. Every qutrit table has γ₂(T) ≤ 2/√3, while the same argument gives exactly 1 for qubits. The worst-case factor κ(n) never decreases with the number of queries n. For three queries, γ₂(T)² ≥ I(T)/9, where I ≤ 9 is the qubit inequality of the previous note. Averaging a core over relabellings of the measurements can never raise the shrink factor. Certified in exact rational arithmetic (computer-assisted): κ(3) ≥ 1.002813276 and κ(4) ≥ 1.007479. For a fixed rational reference triple A*, close to the numerically best symmetric triple, the shrink factor lies in [1.002813276, 1.00281429]. The upper bound uses a new second-order covering of the direction sphere. Conjectured (not proved): κ(3) = √(I*)/3 = 1.0028142848…, and the worst triple is cyclically symmetric. No numerical search found a larger value. The note explains why the obvious proof strategies fail. Files: the preprint (PDF), and a reproducibility archive (ZIP) with five preregistered research stages, including failed and stopped runs, all certificates, and a SHA-256 manifest. After unpacking, two pure-Python exact checks reproduce the certificates in about 20 s, without any numerical solver. They check the certificates, not the surrounding proofs. Not peer reviewed. A large language model assisted with derivations, computations, certificate scripts and drafting; the author takes full responsibility for the content.