We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincaré inequality reduces temporal risk volatility to derivative energy, and a Jacobian-velocity theorem identifies directional tangent energy along the deployment path as the governing quantity under explicit along-path regularity and domination assumptions. Under low-rank drift, that quantity reduces to directional Jacobian energy in the drift subspace, motivating drift-aligned tangent regularization (DTR) and a matched monitoring proxy. Rather than smoothing the network isotropically, DTR penalizes sensitivity only along estimated drift directions. We validate the theorem-to-method pipeline in four experiments: a synthetic benchmark for the time-domain inequality, a controlled synthetic comparison against isotropic Jacobian regularization, and two frozen-deployment studies on the UCI Air Quality and Tetouan power-consumption datasets. DTR reduces risk volatility and directional gain in the controlled low-rank regime, beats isotropic smoothing there, and gives validation-selected deployment gains on both real datasets when the Air Quality drift subspace is estimated from target-orthogonal sensor motion. Moderate drift-subspace misspecification is tolerable while orthogonal misspecification largely removes the benefit.

翻译：我们研究冻结预测器在动态协变量偏移下的长期部署问题。时域庞加莱不等式将时间风险波动性简化为导数能量，而雅可比-速度定理识别出沿部署路径的方向切向能量作为主导量，该量建立在明确的沿路径正则性和控制假设之上。在低秩漂移下，该量简化为漂移子空间中的方向雅可比能量，从而激发漂移对齐切线正则化(DTR)及其匹配的监控代理。不同于各向同性平滑网络，DTR仅沿估计的漂移方向惩罚敏感性。我们通过四个实验验证了从理论到方法的完整流程：针对时域不等式的合成基准测试、与各向同性雅可比正则化的受控合成对比，以及基于UCI空气质量与得土安电力消耗数据集的两项冻结部署研究。在受控低秩场景下，DTR降低了风险波动性与方向增益，优于各向同性平滑方法；当从目标正交传感器运动估计空气质量漂移子空间时，DTR在两个真实数据集上均带来了经过验证选择的部署收益。适度的漂移子空间误规范是可容忍的，而正交误规范则基本消除了收益。