Evidence Layer for Financial Early-Warning Systems

Market shocks are usually described by what they do to prices and volatility. Less is known about what they leave behind in the geometry of cross-asset dependence. We develop a theory of a market’s geometric history—the path of its correlation state on the elliptope after a shock—and ask whether that state returns, settles elsewhere, or settles at a point that depends on the history of the shock. We show that the evidence required for each answer is strictly ordered. A single event can establish relaxation or residual shift but never history dependence; within history dependence, memory of the peak, of the accumulated dose, and of the order of shocks are each identified only by designs that hold the other two fixed. Four structural results make these distinctions operational: an explicit sampling floor for distances between correlation estimates; invariance of relaxation times to the estimation window, with a resolution limit below which relaxation cannot be observed; a geometric decomposition of the relaxation curve when the residual is not aligned with the shock; and the blindness of every spectral functional to rearrangements of dependence. We derive tests for persistence, residual shift, settlement, window invariance, and peak, dose and order dependence, and calibrate them by simulation under heavy tails, volatility clustering and drift, documenting nine failure modes of naive procedures. All evidence reported here is simulated, and persistence in dependence does not, by itself, imply exploitable return predictability