Special Markowitz: Thermodynamic Formalism for the Joint Regularisation of Returns and Covariance

Special Markowitz (SM) regularises returns and covariance jointly, relative to a reference state (mu_ref, Sigma_ref). Each eigendirection of the whitened relative operator carries a signed spectral potential Phi_k, with persistence factor psi_k = exp(-Phi_k) > 0. Positive potentials attenuate empirical deviations from the reference geometry, zero potential preserves them, and negative potentials amplify them. The persistence factor psi_k governs both the return signal and the covariance deviation: the regularised deviation from the reference is psi_k times the empirical deviation. The logarithmic potential coordinate is characterised by a multiplicative composition law on the multiplicative group of positive real numbers; the Stein loss is characterised as the unique free-energy density (within a natural class) compatible with the resulting coupling. The SM pressure functional is additive across modes the defining property of Special Markowitz.