Act I — Geometry of high dimensions

Act 0 made execution inspectable: traffic has a boundary, streaming has a state, precision has a local grid, and a quadratic update has independent spectral clocks. What it cannot yet decide is which directions a large random system is likely to expose. Act I builds that arena’s instruments: a tail class that survives a union bound, the price of a continuum, two singular edges, a calibrated bulk, and a dimension that counts spectral mass instead of coordinates. The machine does not disappear. Its finite grids and finite samples remain the boundaries against which the geometric claims are audited. For general high-dimensional probability, Vershynin (2026) is the reference spine. This act’s narrower job is to turn that theory into calibrated nulls, finite-sample witnesses, and controls that can diagnose a training run.

Vershynin, Roman. 2026. High-Dimensional Probability: An Introduction with Applications in Data Science. 2nd ed. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html.