Causal Separation, Conditional Risk, and Projected Markowitz Portfolios

We formalize a single structural condition on a portfolio problem, causal separation: conditional on the realized path of a declared set of drivers through the investment horizon, asset returns are mutually independent. From this condition we derive the complete static portfolio theory it induces. Separation forces a diagonal-plus-low-rank conditional covariance through an exact tower decomposition, with the low-rank block identified as the response to driver innovations, and the constrained mean-variance problem admits a closed-form projected Markowitz solution in which the classical information matrix is replaced by its projection onto the constraint-compatible subspace. We prove uniqueness of the minimal sufficient separator as an information set; invariance of all derived objects under separator equivalence and reparametrization of the driver state; a conditional efficient-frontier theorem; a Hansen-Jagannathan bound whose gap to the unconstrained bound is an exact quadratic form in the shadow prices of the constraint geometry; a conditioning bound showing that separation regularizes estimation through the idiosyncratic variance floor; and exact first-order sensitivity bounds under approximate separation at a certified tolerance. The causal content is stated and proved rather than assumed: under an explicit structural margin the common causes form a separator, observational data identify their realized information and no more, and causal and correlational separators of equal fit are distinguished by interventions on non-parents. Seven reproducible experiments validate the theory, with structural identities holding at machine precision, and quantify its practical content against sample, shrinkage and principal-component covariances, including robustness, intervention invariance, and scaling to thousands of assets.