Convergent Pearson-Correlation Recurrent Layer

1. Integration point: for each training example, take three learned feature vectors z_1,z_2,z_3 from an MLP, CNN, transformer token group, or graph-neighborhood sampler. Stack them as rows of Z and define P_0=C(Z); then apply K recurrent updates P_{t+1}=C(P_t), with an epsilon-stabilized Pearson denominator. Feed P_K, its leading eigenvector, or the sign pattern of its off-diagonal entries to a downstream classifier or contrastive loss. Keep P symmetric by replacing it with (P+P^T)/2 and set the diagonal to one after each update. 2. Pseudocode: P = pearson_rows(Z); for t in range(K): P = pearson_rows(P); P = symmetrize_and_unit_diagonal(P); loss = task_loss(head(P), y). Estimate the kernel vector q as the eigenvector of P with smallest eigenvalue, compute a projective ratio such as r_t=max_{i,j}|q_i/q_j| in a fixed sign chart, and stop when both ||P_{t+1}-P_t||_F and the change in r_t are below tolerance. 3. Computed from the paper: the rank bound, elliptope invariance, kernel update, and the prediction that only patterned fixed points are attracting. Estimated empirically: finite-precision Jacobian eigenvalues, convergence rate, and which basin a learned Z enters. Use a straight-through estimator or unroll K steps; detach q and stopping decisions if gradients become unstable. 4. First cheap experiment: train a three-token MLP or one-layer transformer on synthetic data where each example contains one of three planted sign-pattern clusters, comparing a standard learned 3-by-3 affinity matrix against this recurrent correlation layer and a single Pearson layer. Sweep random initializations and K. The falsifiable signature is a sharp collapse of the smallest eigenvalue toward zero after the first update, followed by convergence of P_t to one of seven numerically identifiable fixed points; for generic initial states, the fraction not converged by K should decay rapidly, while points initialized near the four unstable fixed points should form a vanishingly small exceptional set. Verify the predicted rank bound to numerical tolerance and compare measured fixed-point attraction against finite-difference Jacobian eigenvalues.