Critical Interface State-Space Pooling

(1) Integration point: insert this as an auxiliary pooling branch before the classifier or before the final transformer block. For a sequence tensor H in R^{B x N x d}, define normalized positions x_i=i/(N-1), a learnable interface position x_0 constrained to [0,1], and positive parameters A, w, and kappa_0. Compute kappa_i=kappa_0*tanh((x_i-x_0)/w). Assemble an N x N finite-difference matrix D whose interior action is D h at index i equal to -A*(h_{i+1}-2h_i+h_{i-1})/dx^2 - kappa_i*(h_{i+1}-h_{i-1})/(2*dx), with dx=1/(N-1). Use reflecting or zero-flux boundary rows. (2) Pseudocode: compute kappa; assemble D; estimate the unit right singular vector psi associated with the smallest singular value of D using power iteration on D^T D; compute z=sum_i psi_i H[:,i,:]; concatenate z with mean pooling or the CLS representation; apply a learned projection. For stable initialization, first use a differentiable soft profile psi_i proportional to exp(-abs(x_i-x_0)/ell), with ell=A/(abs(kappa_i)+epsilon), and normalize it; later compare against the numerical singular-vector profile. (3) The paper supplies the critical condition m=0, the sign-changing kappa profile, the operator, and the exponential localization criterion. Estimate x_0, kappa_0, w, and the singular vector during training; measure localization with inverse participation ratio IPR=sum_i psi_i^4. (4) First experiment: use a 2-layer Transformer or GRU on synthetic length-128 sequences whose label depends on a short event centered at a random change point, then test ECG anomaly classification or another localized-event dataset. Compare mean pooling, CLS attention, convolutional pooling, and this branch at matched parameter count. Track validation accuracy, accuracy at 10%, 25%, and 100% data, interface localization error, IPR, and runtime. Success means higher accuracy or sample efficiency specifically on interface-localized tasks, while no improvement on uniformly distributed tasks is an informative falsification.