A Complex Summand in a Real Algebra Generated by Conditional Expectations

This preprint classifies the unital real *-algebra generated by weighted conditional-expectation projections for a fixed family of partitions of a nine-point set. For strictly positive weights, the algebra is M₂(ℂ) ⊕ ℝ³ when the three pair masses are equal and the three singleton masses are equal. Otherwise it is M₅(ℝ) ⊕ ℝ². The construction reduces to six atoms. For finite families of partition expectations with a common strictly positive measure, we prove that six atoms and three conditional expectations are sharp, jointly attainable lower bounds for the existence of a nonzero skew-adjoint central element. In the nine-point family, three expectations suffice and are necessary for each fixed measure. A single generating subfamily that works for every positive measure requires four expectations. The accompanying reproducibility archive contains the LaTeX source, an exact-arithmetic verifier, its reference output, and SHA-256 checksums.