Covariance-complement uncertainty head

Integrate this into the uncertainty head of a graph neural network or graph VAE. Let the GNN output a mean feature tensor mu with shape n by d and dual log-variance parameters a with shape n by d, instead of a dense n by n covariance. Set d_s,i = softplus(b_i) + epsilon and parameterize the dual variance as v_tilde,i = d_s,i inverse times (1 - sigmoid(c_i)); this guarantees that v_tilde,i is positive and less than d_s,i inverse, and therefore gives positive primal variance v_i = d_s,i - d_s,i squared times v_tilde,i. Use v_i in the Gaussian negative log-likelihood, uncertainty-weighted reconstruction loss, or heteroscedastic node-regression head. If dual samples are available, estimate v_tilde,i online with an exponential moving average of (tilde_x_i - mean_tilde_x_i) squared; otherwise use the neural prediction. Add a consistency loss L_cc = (1 divided by n d) times the sum over i and c of (v_i + d_s,i squared times v_tilde,i - d_s,i) squared, with a small coefficient such as 1e-3. Never form a dense covariance matrix. On small graphs, compute exact diagonal entries of Q inverse by Cholesky solves and compare them with the predicted diagonal. Begin with synthetic cycle and grid graphs, node regression targets, and a two-layer GCN. Compare a diagonal Gaussian head, a dense exact Gaussian head, and this dual-complement head. Success means memory scaling of O(n d) rather than O(n squared d), lower negative log-likelihood than the diagonal baseline, and improved calibration measured by negative log-likelihood and expected calibration error.