Structural Limits of Ellipticity: Obstructions to Mean–Variance Sufficiency in Financial Markets

Elliptical distributions provide one of the most powerful justifications for mean– variance portfolio analysis. Under joint ellipticity, every portfolio return shares a common standardized shape, so a broad class of law-invariant risk measures reducesto a function of location and scale. This paper studies the structural content of that assumption and identifies the settings in which it cannot be maintained. The analysis separates generator-free implications from restrictions that require additionalassumptions on support, temporal aggregation, radial tails, or regime structure.The paper develops three principal results. First, limited liability combined with symmetry imposes a mirror bound that is difficult to reconcile with observed asset returns. Second, mixtures with a common location and generator remain elliptical only when their dispersion matrices are proportional; for two Gaussian regimes, the paper gives a complete classification that also excludes nontrivial location switching. Third, a baseline-free first variation of expected shortfall isolates the relevant tail functional before any Gaussian Edgeworth approximation is introduced. This distinction clarifies when co-skewness corrections are theoretically justified and when an elliptical reference distribution must first absorb symmetric higher-order shape.Additional results connect ellipticity to convex payoffs, capital structure, multiplicative compounding, horizon invariance, common tail indices, tail-dependence rigidity, and radial–angular independence. Exact tests are derived for central symmetry and radial–angular independence when nuisance parameters are known. For dependent returns, a block sign-flip calibration re-estimates filtering, centering, and thresholds within each resample. Under its maintained dynamic sign-invariance condition, theprocedure has appropriate simulated size across persistent-volatility, GARCH, and Student elliptical nulls. Applied to a frozen sample of United States equity portfolios from 1949 through 2017, it rejects central symmetry across nine pre-specified paneland threshold combinations after Holm correction. Throughout, a claims register separates proved, classical, conditional, numerical, and empirical statements, making the scope of each conclusion explicit. The results support treating ellipticity as aconditional, horizon-specific approximation whose practical adequacy depends on whether structural misspecification is economically material relative to the estimation error required to model it.