Volume-Threshold Contracting State Layer

Integrate this as an experimental recurrent or state-space block between an embedding layer and the output head. Represent A as Q diag(tanh(a_1),...,tanh(a_d)) Q^{-1}, or use a normal or orthogonal parameterization initially so that its spectral radius and determinant are cheap to compute; initialize eigenvalue magnitudes below one. Add a bounded driver x_t=sigmoid(w_x^T u_t+b_x), optionally followed by x_{t+1}=ell x_t modulo one, and compute h_{t+1}=A h_t+phi_theta(x_t,u_t). Pseudocode: estimate rho(A) by power iteration; compute g=log(ell)+log(abs(det(A))); if rho(A)>=1, project eigenvalue magnitudes below 1-epsilon; run the state update; record hidden-state singular values, pairwise distances, and covariance rank. The theorem-level ingredients taken from the paper are the fiber-contraction requirement rho(A)<1 and the volume exponent g; neural training must estimate effective dimension empirically. The first cheap experiment should use a width-8 recurrent layer on sequential MNIST or permuted sequential MNIST, comparing a standard RNN against layers with ell times abs(det(A)) equal to 0.5, 0.9, 1.1, and 2.0, while keeping rho(A)=0.95. The quantitative prediction is a change near log(ell)+log(abs(det(A)))=0: below the boundary, covariance participation ratio and estimated correlation dimension should remain submaximal and often collapse with sequence length; above it, state-space occupancy and covariance rank should increase, while gradients remain bounded because rho(A)<1. Test whether the empirical transition lies within 20 percent of ell times abs(det(A))=1.