Geometric Early Warning and Local Stabilizing Screening in Damped Oscillatory Learning Dynamics

AI learning systems can remain apparently healthy even while their local update dynamics are already drifting toward oscillatory fragility, ineffective descent, or collapse. This creates a concrete control problem in nonlinear learning: before visible failure is fully realized, which admissible local parameter direction actually makes the local dynamics safer? This paper gives an exact local answer for a broad conditional class of collapse-prone learning dynamics that admit a dominant two-dimensional damped oscillatory block. On that block, let P = -Tr(J2), Q = det(J2), and R = Q/P^2. The main theorem proves that an admissible local parameter direction is geometrically stabilizing if and only if P dQ - 2Q dP < 0. This turns reduced spectral geometry into a control-oriented early-warning rule and a local ranking rule for admissible training-side interventions. The paper also gives a finite-window certification statement and a relinearized recertification scheme for repeated local updates. The scope is explicit and limited: the paper does not claim a full prognosis theory, intervention deadlines, or global guarantees. Its contribution is narrower and exact. Under stated local conditions, it determines which admissible local move decreases reduced geometric vulnerability before aggregate failure indicators become decisive. The results of this paper are limited to local theorem-level classification under the stated assumptions and do not by themselves constitute operational approval, safety assurance, legal advice, or any guarantee of realized performance.