Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance

What the math gives to ML

The paper identifies a concrete numerical-analysis failure mode in classifier-free guidance: near the terminal low-noise regime, guidance changes the discriminative coordinates from a scaling exponent of 1 to an anomalous exponent of 1+w, so the ordinary DDIM interpolation is no longer fitted to the guided dynamics. This explains why coarse terminal steps can produce residual amplification and oversaturation even when the underlying continuous guided flow is well behaved. The transferable construction is a one-line replacement of the guidance coefficient, derived by exactly integrating the anomalous terminal exponent without requiring spectral information or extra denoiser evaluations. It is especially promising as a drop-in high-guidance stabilizer for diffusion samplers, with the caveat that it is not guaranteed to improve every image-quality metric or to uniformly approximate the dense vanilla-CFG field.