Hyperbolic Ring-Closure Regularizer

(1) Integrate this into a CNN, vision transformer patch grid, or graph neural network immediately after a scalar feature head produces u in R^{H times W}. Use one scalar channel or a small learned projection of the feature tensor; do not apply the loss independently to every high-dimensional channel initially. Convert each feature to R_i=softplus(a u_i+b)+R_min, with R_min=10^{-3}. Choose q=sigmoid(q_0)(1-10^{-4}), and compute r_i=arcosh(q cosh(R_i)); clamp the arcosh argument to at least 1+10^{-6}. (2) Pseudocode: R=softplus(a*u+b)+R_min; q=sigmoid(q_0)*(1-eps); r=acosh(q*cosh(R)); for every directed grid edge i to j, compute phi[i,j]=atan(tanh(R[j])/sinh(r[i])), psi[i,j]=atan(tanh(r[j])/sinh(R[i])), and theta[i,j]=phi[i,j]+psi[i,j]; accumulate closure[i]=sum over j adjacent to i of theta[i,j]; compute Lring=mean over edges of (cosh(r[i])/cosh(R[i])-cosh(r[j])/cosh(R[j])) squared plus mean over vertices of (closure[i]-2*pi) squared. Add lambda_ring times Lring to the task loss. (3) The geometric formulas are computed exactly; q, a, and b are learned or fixed, while the closure target 2*pi and grid adjacency are fixed. Log the minimum r_i, maximum R_i, closure error, and gradient norm to detect numerical saturation. (4) First experiment: train a small CIFAR-10 CNN with a scalar feature head, comparing cross-entropy alone, a standard total-variation penalty, a standard Laplacian penalty, and this ring loss at equal parameter count and training steps. Test lambda_ring in {10^{-4},10^{-3},10^{-2},10^{-1}}. Success means lower validation loss or improved corruption robustness at the same training budget, with closure error decreasing during training and no increase in gradient explosions.