Stock Price Movements as Geodesics on a Curved Manifold: A Causal Schrödinger-Reference and Conditional-Tail Forecast of Two-Sided Price Events

We examine a causally specified geometric reference for forecasting a two-sided path event: whether the absolute cumulative log return from the forecast origin reaches 0.05 at any of the next 20 trading observations. The Geometric Curve Manifold (GCM) formulation studied here combines a harmonic Schrödinger expectation, a stipulated price–illiquidity–volatility geodesic that conditions residual scale, and a peaks-over-threshold model for fully matured path-error amplitudes. Raw GCM (POT) probabilities are calibrated using only outcomes available at each forecast origin. On 3874 exactly matched forecasts for 95 equities and five exchange-traded funds, the calibrated GCM (POT) equal-instrument Brier score is 0.15563, compared with 0.15912 for raw HS (historical simulation) and 0.16026 for raw GARCH-filtered HS. An exploratory raw Gaussian GARCH(1,1) sensitivity scores 0.16198 on the same keys. Paired 20-date-block bootstrap intervals on a smaller balanced calendar include zero for all three comparisons. Symmetric recalibration removes a unique lowest-score ranking, and calibrated GCM (POT) downside differences from the calibrated GARCH benchmarks are not significant under the reported bootstrap and finite-sample HAC tests. GCM offers a distinct, testable alternative for event-probability forecasting, without demonstrated superiority.