Gravity and the Arch: Implementation Tests and Findings

The Gravity paper claims a recovered growth rate — the arch — for an investor who holds a baseline allocation and exercises stored safety after damage. This paper tests that claim in simulation and then explains what the simulation found. The arch is an exact long-run law: under the bang-bang shortfall rule studied here, for baselines below the market the gap to the arch line is positive recurrent with a closed-form stationary law, the fraction of time spent at Kelly converges to p* = y₀/(2−y₀), and mean log-growth converges to the arch; at the market seat there is no stationary law and the convergence is degenerate. At the 20- and 50-year horizons tested, the arch is kept by the median and by the dollar mean but not yet by the mean of log-wealth, and the shortfall is a transient whose clock is computable. Against the natural skeptic — a static portfolio at twice the baseline, or at the best static seat for the same goal — the moving portfolio loses on growth and on dispersion and wins on the only thing it was built for: the probability of keeping the line — with less average exposure, and for a computable number of years: 67 at 30/70, 78 at 50/50, 91 at 70/30, 122 at the market. The boundary T*(y₀) is exactly characterized — the static side in closed form, the moving side by an explicit Laplace transform — and evaluated numerically; it is a curve, drawn in years. What remains outside the theorem — which paths fall deep, and how deep, before the law asserts itself — is the price, and the price is itself a finding: exact, at a price, over time, within limits.