Fitted CFG: Stabilization Bands and Contraction Matching

Classifier-free guidance (CFG) changes the flow that a diffusion sampler integrates, but the DDIM update was designed for the unguided flow. In an ordered, commuting Gaussian model, guidance strengthens contraction along discriminative directions, and a coarse DDIM step can reverse such a coordinate. On this model, a band of guidance coefficients prevents the reversal and controls discriminative residual amplification without prescribing the exact contraction; it contains the sign cap and a parameter range of deterministic CFG++. Terminal matching fixes the factor left free and selects fitted CFG, a one-coefficient change at no extra denoiser cost. Fitted CFG and the caps share first-order absolute accuracy; matching additionally bounds the accumulated log-relative contraction error independently of the noise horizon, for fixed guidance and maximum log step. On CIFAR-10, fitted CFG lowers FID in all cells of two nine-cell grids and in a 50k comparison (27.87 to 20.88); the sign cap matches this stabilization in paired controls. A calibrated finite-scale extension improves FID, KID and saturation on 30k SDXL samples at some cost in text alignment. The framework separates three design criteria: stability, consistency with the prescribed field, and contraction accuracy.