Exact Same Sample Block Leverage Correction for LCME Spectral Shrinkage: Heteroskedastic Gaussian Transfer and Projection Safe Spectral Estimation

We study spectral density estimation by shrinking sample autocovariances, focusing on the optimism created when a nonlinear shrinkage profile is fitted from the same squared coefficients to which it is applied. Parseval’s identity converts integrated squared spectral loss into an exact weighted sequence loss, but sample autocovariances are heteroskedastic, correlated across lags, and reused by the fitted shrinkage map. Our principal result is an exact finite sample Stein unbiased risk estimate for nonlinear least concave majorant empirical Bayes (LCME) energy shrinkage in an independent Gaussian sequence experiment. The energy fit is the nonnegative antitonic projection of centered squared observations, followed by the empirical Bayes map q → q/(q+s). We prove that the resulting estimator is locally Lipschitz, hence weakly differentiable, and derive its divergence explicitly. The correction omitted by a naive plug in fixed multiplier criterion is a positive pool adjacent violators (PAVA) block leverage term. This identifies a same sample degrees of freedom contribution that is not present when the fitted multiplier is treated as fixed. After marginal standardization, the calculation extends to a correlated Gaussian vector. Conditional on the standardized vector and its fitted PAVA partition, perturbing the correlation matrix in the Stein correction depends only on the principal submatrices indexed by positive fitted blocks. Thus the block local identity concerns covariance insertion into the corrected criterion; errors in marginal standardization may additionally alter the PAVA fit and are treated separately. We then give one class calibrated process level feasibility theorem for a fixed causal and invertible Gaussian autoregressive moving average (ARMA) class. A logarithmic forced head isolates uniformly signal scale lags, a truncated Gaussian Bartlett plug in estimates the lag covariance, and the remaining logarithmic tail is tuned over a fixed six point continuous shrinkage strength library. A heteroskedastic block Jacobian formula matches the actual lag coordinates, so no discontinuous threshold is passed through an ordinary Stein formula. Explicit nuisance rates yield an integrated squared risk bound of order log(T)/T. This rate is presented as a rigorous process level implementation guarantee, not as an optimality claim for finite order ARMA models. Finally, spectral validity is imposed after shrinkage. Orthogonal projection onto the nonnegative spectral cone, or onto a fixed mass slice, is risk safe; using the estimated lag zero mass adds only an explicit O(T⁻¹) term in the Gaussian ARMA theorem. A paired benchmark and an annual sunspot illustration evaluate a practical implementation separately from the theorem calibrated construction. Long memory, volatility, information theoretic, and positive window developments belong to a separate research archive and are not part of this submission or its technical supplement.