Synchronized spherical dimension bridge

Integrate the bridge directly after a contrastive embedding head, retrieval encoder, or transformer final hidden state. Given h in R^D, first normalize it to h divided by its Euclidean norm, then split it into K blocks h_k. Normalize each block to u_k. For every block, learn a dimension-changing matrix A_k from the original block dimension d_k to target dimension e_k, and compute v_k = A_k u_k divided by the Euclidean norm of A_k u_k. Choose nonnegative join weights w_k satisfying the sum of w_k squared equals one, either fixed from block widths or learned through a softmax followed by square-root normalization. Produce the final unit representation z by concatenating w_k v_k. In pseudocode: u = split(normalize(h)); v_k = normalize(A_k @ u_k); w = normalize_positive_weights(); z = concat(w_k * v_k). For each training pair, compute original angular distance d_ij = arccos(clamp(h_i dot h_j, -1 + epsilon, 1 - epsilon)) and adapted angular distance t_ij = arccos(clamp(z_i dot z_j, -1 + epsilon, 1 - epsilon)). Add lambda times the mean absolute defect |d_ij - t_ij| to the task or contrastive loss. Log both mean and maximum batch defect. Compute the spherical formulas exactly; estimate global distortion empirically using random pairs and hard negatives. Begin with a two-layer MLP on CIFAR-10 or a small sentence-retrieval encoder, comparing against a single dense projection with the same output dimension. Success is lower memory at equal recall, stable maximum angular defect, and error that does not grow linearly with the number of blocks.