The Adaptation Rate Theorem: Predictable Lead Times in Multi-Scale Trailing Measures

We prove that for any quantity measured by trailing averages at two different timescales, the faster measure detects regime changes before the slower measure, with a lead time at the midpoint detection threshold of (w_slow − w_fast)/2 periods and, more generally, α·(w_slow − w_fast) at threshold fraction α. The bound is exact for step transitions under the simple moving average, extends to general weighted averages through an effective-lag formula, and is a property of weighted-sum arithmetic rather than of the statistical structure of the data. We test this on 24 pre-registered tests over 23 series across three domains under sign-test decision rules: the directional prediction holds on 24 of 24 tests (combined two-sided sign-test probability 1.2 × 10⁻⁷; 9.5 × 10⁻⁷ after collapsing near-duplicate instruments to 21 effective tests). On the headline external-event test, across 332 forward-volatility events the divergence's median advance notice is 90.0 calendar days versus 57.0 for the slow measure's own threshold crossing, and the divergence beats the slow measure on 13 of 13 instruments. On randomly permuted data the divergence fires before the onset at a pooled rate of 99.98% — the mechanism is arithmetic, and the controls say so directly. We state the conditions under which the directional prediction fails (the Aliasing Condition), show within the average-lag framework of Zakamulin (2017) that the commonly quoted Hull moving-average lag heuristic overstates the lag-additive value by a factor of two to three depending on window convention, and document a two-thirds law for the exponential moving average's realized lead. This paper was produced under the Lagging Truth 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 from SHA-256-hashed inputs — no script, no number. The manuscript passed a two-round capped adversarial review before certification; the full review transcript, the claims ledger, the verification suite, and a public corrections log are committed in the linked repository. Code in the linked repository is MIT-licensed; the plain-English companion is CC BY-NC 4.0. See the repository LICENSE file.JEL Classification: C58; C22; G17; E37