Elliptope-aware spectral correlation head

Integrate this into any module that predicts an n-by-n similarity or correlation matrix: a graph-NN edge-affinity head, a covariance head in a VAE, or an attention variant that learns token-to-token correlations. Let the network emit an unconstrained matrix X in R^{n-by-r}; form G=X X^T+epsilon I, then normalize it as C=diag(G)^(-1/2) G diag(G)^(-1/2). This guarantees C is PSD and has unit diagonal without an eigendecomposition. During training add the log-determinant barrier L_barrier=-logdet(C+epsilon I)-n log(1+epsilon), computed using Cholesky. Use epsilon=1e-3 initially and sweep alpha over {1e-4,1e-3,1e-2}; if full matrices are too large, use a low-rank Woodbury log-determinant computation. Pseudocode: X=network(h); G=X@X.T+eps*I; s=sqrt(diag(G)); C=G/(s[:,None]*s[None,:]); barrier=-cholesky_logdet(C+eps*I)-n*log1p(eps); loss=task_loss+alpha*barrier; backpropagate. The math supplies the PSD/unit-diagonal construction and the spectral barrier; estimate empirically the actual minimum eigenvalue, condition number, and gradient norm. First test on a four-layer transformer with sequence length 64 and a synthetic correlated-denoising task, comparing against unconstrained symmetric logits followed by eigenvalue clipping. Log minimum eigenvalue, Cholesky failures, validation loss, and wall-clock overhead. Success is fewer numerical failures and lower validation loss at equal parameter count, with the minimum eigenvalue staying above the chosen floor rather than collapsing toward the random-elliptope scale O(n^-2).