We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincare inequality first reduces temporal risk volatility to derivative energy. A Jacobian-velocity theorem then supplies the corresponding pathwise control. Given explicit regularity and domination assumptions, the theorem identifies directional tangent energy along the deployment path as the governing quantity. 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 and beats isotropic smoothing there. It also gives validation-selected deployment gains on both real datasets, with the Air Quality subspace estimated from target-orthogonal sensor motion. Moderate drift-subspace misspecification is tolerable while orthogonal misspecification largely removes the benefit.

翻译：我们研究静态预测器在动态协变量漂移下的长期部署问题。首先，时域庞加莱不等式将时间风险波动性约化为导数能量；继而，雅可比-速度定理提供了相应的路径控制。在显式正则性与支配性假设下，该定理将沿部署路径的方向切线能量识别为控制量。对于低秩漂移，该量进一步约化为漂移子空间中的方向雅可比能量，由此引出漂移对齐切线正则化（DTR）及配套的监测代理方法。不同于各向同性网络平滑，DTR仅沿估测漂移方向惩罚敏感性。我们通过四项实验验证了从定理到方法的技术路线：时域不等式的合成基准测试、针对各向同性雅可比正则化的受控合成比较，以及基于UCI空气质量与得土安电力消耗数据集的两项静态部署研究。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。在两个真实数据集上，DTR均展现出经验证的部署增益，其中空气质量子空间通过目标正交传感器运动估测。适度的漂移子空间误设具有容忍性，而正交误设则基本消除收益。