An Unbounded Gap Between Identification and Identifiable Elicitation: Central Moments

For $n\ge 2$, two expectation equations identify the mean together with the $n$th central moment. How many coordinates must a report have if a one-observation loss uniquely elicits it and the report determines that moment? We prove that $n$ coordinates are necessary and sufficient among all identifiable intermediaries. The result holds on every open convex set of positive distributions on any $n+1$ distinct real outcomes. It requires no regularity of the loss, report map, identification function, or link. Thus ${\mathrm{elic}}_{\mathcal{I}}({\mu }_n)=n$ while $\mathrm{iden}({\mu }_n)\le 2$, giving an unbounded gap between identifiable elicitation and indirect identification. This resolves the central-moment conjecture of Frongillo and Kash on the stated domains. The proof constructs a $C^1$ manifold of supporting coefficients from the affine fibers of a convex negative Bayes risk. Translations along that manifold force an $n$-dimensional tangent space. Standardized moments of order at least three have the same exact complexity. The bound also extends to convex distribution classes with a full-dimensional raw-moment image.