Abstract
<title>Abstract</title> <p>We ask whether representing the recent history of a financial instrument as a geometric field of normalized coeffcient vectors—organized by structural similarity rather than by chronology—improves directional forecasting. We formalize the Historical Coeffcient Field (HCF): the point cloud of open-relative session coeffcients together with a small set of geometric descriptors (local density, dispersion, trajectory stability, historical persistence, and a partition entropy) computed causally over a rolling window. We relate the construction to the established ideas it makes explicit—delay-coordinate reconstruction, analog forecasting, kernel density estimation, and the topological analysis of financial series—rather than presenting it as a new theory. We then test it under a strictly controlled, representation-only protocol: identical models (logistic regression, random forest, gradient boosting), identical walk-forward splits, four public instruments spanning very different price scales, with only the input representation varied. The result is negative and robust: no representation exceeds the majority-class base rate, and augmenting the coeffcient vectors with the geometric descriptors does not improve—and slightly degrades—directional accuracy relative to the coeffcients alone. We report this negative result as the paper’s contribution: on the assets and protocol tested, the geometric organization of market memory carries no directional information beyond that already present in simple baselines. On a deliberately more predictable target—volatility expansion—signal does appear, but simple rolling statistics outperform the geometric coeffcients, so the geometric organization is not merely redundant for direction but inferior where signal exists. We discuss why, and what a genuinely informative geometric representation would need to demonstrate.</p>