We consider radar sensing with affine frequency division multiplexing (AFDM), a chirp-based waveform recently proposed for high-mobility integrated sensing and communication. While numerical Cramér-Rao bounds for AFDM radar are available in the literature, no closed-form Fisher information analysis has so far revealed how the waveform's chirp structure shapes delay-Doppler estimation accuracy.In this paper, we provide such an analysis. We identify a cancellation in the AFDM likelihood: the frequency drift introduced by the chirp modulation is exactly compensated by a discrete phase correction built into the chirp-periodic prefix, leaving only a small residual. Exploiting this cancellation, we derive an exact closed-form Fisher information matrix that depends on the AFDM chirp structure through a single scalar, and from it we obtain closed-form Cramér-Rao bounds for joint delay and Doppler estimation.Three consequences follow. AFDM is provably less delay-Doppler-coupled than OFDM for any nonzero chirp rate. The delay Cramér-Rao bound improves quadratically with the chirp rate, while the Doppler bound is unaffected by it. Finally, our framework reduces continuously to the classical OFDM result as the chirp vanishes, certifying it as a strict generalization of OFDM radar sensing theory.Overall, our work shows that the chirp-periodic prefix -- until now studied only as a channel-equalization device -- is the structural element that decouples delay and Doppler in AFDM sensing, and that AFDM's superior sensing performance can be characterized analytically rather than through numerical bounds alone. Numerical experiments at realistic vehicular and low-Earth-orbit parameters validate all closed-form expressions.