The Signed Geometry of One-Shot Recourse: On-Path Validity and the Signed-Curvature Criterion
Closed-form recourse moves a rejected user along the unit gradient $\hat g$ of the classifier score $f$ by the promised distance $d_p=|f(x)|/\|\nabla f(x)\|$, at which the linearized score reaches zero. We ask when this one-shot step succeeds and what additional model queries change. To leading order the step ends on the favorable side exactly when the path curvature $κ=\hat g^\top\nabla^2 f(x)\,\hat g$ is nonnegative. Across 80 shallow models, the fraction of rejected users whose step ends there and the fraction with $κ\ge0$ correlate at $r=0.985$, although on Fashion-MNIST the first falls below the second by 8.2 points on average. No rule that uses only the score value and gradient can be valid for every score with path curvature bounded by $K$ without overshooting some by order $Kd_p^2/\|\nabla f(x)\|$. When the curvature is also Lipschitz and the step is short, one evaluation of $f$ at the promised point attains the minimax rate among deterministic one-query rules that know the curvature bound and its Lipschitz constant, and split-conformal calibration makes such a rule reach the first crossing or abstain with probability at least $1-δ$. Training with an asymmetric curvature penalty lets 99-100% of paths cross within the promised step on undershoot-prone shallow data, at about 4-22 times the overshoot of symmetric penalties (Fashion-MNIST, COMPAS). Because $κ$ and $d_p$ depend on how the score is scaled, part of this gain can be a longer promised step, and at matched validity a smaller audit of briefly trained models finds no uniform advantage over tuned inflation. Where a per-user line search along the ray is affordable, it is exact to grid resolution and preferable.