Moving Averages Follow Price: A Mathematical Proof and Empirical Validation of the Adaptation Property in Trailing Averages

We argue that when a time series deviates from any causal trailing average, the subsequent gap closure is, on the data we examine, predominantly consistent with the mechanical adaptation of the average rather than reversion of the series toward the average. The result is algebraic: for any weighted moving average satisfying standard non-negativity conditions and any permanent displacement, the average converges to the new level with 100% of the gap closure attributable to filter movement and 0% to series movement (Lemmas 1–3). We extend this deterministic result to stochastic processes, proving that the aggregate filter share equals 1 under a random walk (Theorem 2) and providing convergence bounds in the adaptation regime for general stationary ergodic processes (Theorem 3). We validate the theory on 44 instrument-filter combinations spanning 11 financial instruments, six asset classes, four continents, and up to 155 years of daily data. The aggregate moving average share of gap closure exceeds 50% on all 44 combinations tested and exceeds 100% (indicating the series moved further away while the average converged) on 33 of 44 (75%). The next-bar direction test shows no systematic tendency for price to move toward its moving average: under Hull-50, toward rates cluster near 50% on 8 of 11 instruments with mild attraction (~53%) on three (uncorrected for multiplicity); under SMA-200, three instruments show mild repulsion (~47%), of which only one retains significance after a full-sample drift adjustment. Synthetic random walk and GARCH-calibrated controls confirm the decomposition methodology is approximately unbiased under zero attraction, so the elevated shares observed on real instruments are not metric artifacts. A Schelling Point test across 137 instrument-window-timeframe combinations finds no detectable self-fulfilling prophecy effect at popular moving average windows (𝛿 = −0.005 pp, two-sided 𝑝 = 0.78). Two findings carry forward-looking predictive content: large displacement from the moving average predicts elevated forward realized volatility (Spearman 𝜌 = +0.54 average across 11 instruments, positive on all 11; 33 of 38 tests — including every full-sample and out-of-sample test — remain significant under serial-dependence-corrected inference), and the S&P 500 shows genuine quarterly mean-reversion (𝑟 = −0.28 at 126 days), which operates at a different timescale and through a different mechanism than the mechanical adaptation that dominates at shorter horizons. The qualifications to “predominantly” are reported in full: the aggregate share falls below 50% in 5 of 18 rolling decade-windows on SPX (one negative, 1991–2001), and the modern era (1990–2026) shows mild but statistically significant attraction (toward rate 53.1%, 𝑝 = 0.0089, not adjusted for multiplicity). The adaptation result itself is an attribution of past gap closure, not a forward predictor. Verified rebuild under the Research-to-Publication Standard v1.3: every load-bearing number is registered in a machine-checked ledger (claims.lock) and regenerated on demand by verify.py on SHA-256-hashed inputs; the two-round adversarial review record and a public corrections log are committed in the repository. The paper is licensed CC BY-NC-ND 4.0; the accompanying reconstruction and verification code is MIT-licensed.