The Geometry of AP Degradation: A Closed-Form, Zero-Parameter Theory of Localization Sensitivity

Average Precision (AP) is discontinuous in box coordinates because it thresholds a continuous quantity, IoU — but exactly where that discontinuity falls, and why it produces the empirical fragility practitioners observe (a single-pixel shift can drop mAP by double digits, hitting small objects hardest), has been missing. We derive closed-form IoU expressions under translation and corner-anchored scaling giving the exact shift at which a detection flips from correct to incorrect, $t^{\ast }(\tau ,W)=W\cdot \frac{1-\tau }{1+\tau }$, a zero-parameter formula in object size $W$ and IoU threshold $\tau$. This single formula unifies four otherwise-disconnected phenomena as corollaries of one mechanism: why small objects degrade fastest, why diagonal shifts hurt more than axis-aligned ones, why enlarging and shrinking have near-identical effect, and why averaged mAP degrades faster than ${\mathrm{AP}}_{50}$. We validate against official COCOeval on real COCO annotations (MAE $\le 0.032$, correlation $\ge 0.997$, transferring with no refitting to LVIS), and show the same closed form predicts real detectors' (Faster R-CNN, YOLOv11) relative degradation shape, once rescaled by their own baseline. The theory is prescriptive as well as explanatory: inverting the formula yields a size-adaptive threshold that cuts the size bias of AP degradation roughly twentyfold on the idealized protocol ($5.4\times$ on five real detectors), and a smoothed variant removes the discontinuity while leaving detector rankings unchanged.