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 spectral reliability potential Phi_k, derived from its estimation quality. Its exponential reliability weight exp(-Phi_k) governs both the fraction of the return signal and the fraction of the covariance deviation that survive regularisation. Reliable modes (low Phi_k) are preserved; unreliable modes (high Phi_k) relax to the reference. The logarithmic potential coordinate is characterised by a multiplicative composition law; 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.