Cohomology Bottleneck Regularizer

Insert the module immediately after a residual attention or MLP sublayer. Split a token representation z into three source tensors B1,B2,B3, each of width m, and construct two target tensors A1,A2. Implement four learned linear maps R11:B1→A1, R22:B2→A2, R31:B3→A1, and R32:B3→A2; concatenate them into a differential matrix D whose action is A1=R11(B1)+R31(B3) and A2=R22(B2)+R32(B3). Estimate the cohomology dimensions without an expensive SVD by using the soft nullity proxy q_tau(D)=sum_k sigmoid((tau-sigma_k(D))/eps), where sigma_k are singular values, tau is a threshold, and eps smooths the count. Add L_coh=lambda_e*relu(target_chi-(q_tau(D)-q_tau(D.T)))^2 or, more simply, penalize the squared Euler residual (||B||_2^2-||A||_2^2-delta)^2; use delta as a learned scalar per layer. A stronger version keeps the approximate cokernel basis from the smallest left singular vectors as an explicit residual feature, allowing the network to recover information that D cannot explain. The math is the kernel/cokernel decomposition and Euler difference; tau, eps, and delta are empirical hyperparameters. First test on a 4-layer MLP or ViT on CIFAR-10, comparing a matched-width baseline with and without the regularizer. Monitor gradient norm variance, rank collapse, calibration, and accuracy. Success is improved accuracy or calibration at equal parameters, together with fewer near-zero singular-value directions and smoother training loss.