Spline Reorganization Without Grokking: Stress-Testing Knot Entropy as a Progress Measure in KANs

Internal training statistics are often interpreted as progress measures for latent or future model behavior, but responsiveness to training alone does not validate that inference. We introduce Knot Entropy, the normalized Shannon entropy of curvature mass across spline knot intervals in Kolmogorov--Arnold Networks (KANs), and establish boundedness, scale invariance, affine invariance, and a curvature-concentration bound. We stress-test its interpretation as a grokking progress measure on modular addition and multiplication modulo 97. Across five seeds and 100,000 optimization steps, a scalar-input fixed-grid KAN memorizes addition (train accuracy $1.000$) without comparable generalization (test accuracy $0.285 \pm 0.033$), while an MLP pipeline positive control reaches $0.991 \pm 0.004$ test accuracy. On multiplication, the KAN nearly memorizes ($0.993 \pm 0.003$ train accuracy) but remains near chance ($0.024 \pm 0.002$ test accuracy); the MLP also fails to generalize. Knot Entropy decreases by $0.049 \pm 0.006$ on addition and $0.042 \pm 0.001$ on multiplication before partially rebounding, although none of the ten tested KAN runs satisfies the grokking criterion. These results separate metric sensitivity from construct validity: Knot Entropy reliably detects spline reorganization, but its decrease alone does not justify an inference that the model is progressing toward grokking.