{"schema_version":"dataset-v1","id":"gaussian-moments-check","version":1,"title":"Check the witness","description":"P₃ = (1 + Z)(W − (2 + Z)T²/2) with Z = (X₁ + iX₂)/√2, W = (X₁ − iX₂)/√2 and T = X₃, for independent standard real Gaussians. Pure moments E[P₃^m] and mixed moments E[Z P₃^m] for m = 1, …, 8, computed exactly.","license":"CC0-1.0 (exact values)","problems":["gaussian-moments"],"statements":[{"problem":"gaussian-moments","statement":"gaussian-moments-three-variable-witness","role":"Every pure moment vanishes, yet E[Z P₃^m] = m! never does."},{"problem":"gaussian-moments","statement":"gaussian-moments-universal","role":"So mixed moments need not eventually vanish: the conjecture fails."}],"figure":{"family":"exact-table","title":"Check the witness","caption":"Exact Gaussian expectations: every pure moment E[P₃^m] is 0, while the mixed moment E[Z P₃^m] equals m! and never vanishes.","columns":["m","E[P₃^m]","E[Z P₃^m]","m!"],"boundary":"Eight orders computed exactly; the identity for every m is the cited theorem. Nothing here bears on two variables, which remain open."},"provenance":{"method":"sympy: P₃^m and Z·P₃^m expanded in Z, W, T; expectations from E[Zᵃ Wᵇ] = a! δ_ab and E[T^(2k)] = (2k − 1)!!. Orders 1–3 are re-checked by expanding in the real coordinates X₁, X₂, X₃.","software":{"python":"3.12.15","sympy":"1.14.0"},"precision":"Exact rational arithmetic.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[],"marks":[],"rows":[{"key":"m_1","cells":["1","0","1","1!"]},{"key":"m_2","cells":["2","0","2","2!"]},{"key":"m_3","cells":["3","0","6","3!"]},{"key":"m_4","cells":["4","0","24","4!"]},{"key":"m_5","cells":["5","0","120","5!"]},{"key":"m_6","cells":["6","0","720","6!"]},{"key":"m_7","cells":["7","0","5040","7!"]},{"key":"m_8","cells":["8","0","40320","8!"]}]},"sha256":"14f51165b34a6594fd59d87e60554ff07c29b0a1a0f830f6a878511f145119dc"}