The Divergence Theorem for Regime Transition Detection: Multi-Scale Estimator Disagreement as a Cross-Domain Early Warning Signal

Abstract

Many fields anticipate regime transitions by comparing a fast, recent estimate of a system's variability to a slower baseline — the moving-average convergence/divergence indicator in finance, the Sahm Rule in macroeconomics, rising-variance indicators in ecology — without a shared theory. We study that comparison directly as the divergence operator: the difference between a fast-window and a slow-window trailing estimator of the same observable. We prove twenty-two results characterizing it, including its sensitivity, timing, and optimal windowing near a bifurcation; the Persistence-Sign Theorem, which fixes the sign of its forward prediction from the observable's autocorrelation alone; and a superset property establishing that critical slowing down is the special case of divergence restricted to bifurcation-type transitions, so the operator carries strictly more information. We test the theory across seven domains — equity volatility, equity correlation structure, hydrology, solar physics, epidemiology, weather, and macroeconomics — under a single pre-registered specification, and report a mixed verdict. The same-observable sign law holds in five of the seven domains and is contradicted in none, with solar physics its strongest confirmation; divergence's practical edge over critical slowing down is real but domain-dependent. The headline equity-volatility panel is the weakest case, with panel-mean correlation +0.0528 and failing its own pre-registered gate. Beyond the in-sample tests, the paper reports one clean out-of-sample check — a solar verification (Section 5.4) in which the operator's a priori negative sign, its specification fixed before 2009, persists on the held-out post-2008 data and is predicted to continue on data published after the paper; it is a verification of the locked operator, not a promoted live forecast. Verified rebuild under the Research-to-Publication Standard v1.8: every load-bearing number is registered in a machine-checked ledger (claims.lock) and regenerated on demand by verify.py on hash-pinned inputs, then independently reproduced from a clean checkout; a capped single-round adversarial review under a fix-or-rebut protocol 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. v1.0.1 is a metadata correction: the title page now carries the record's concept DOI (the v1.0 title page printed a DOI reserved by an earlier, never-published draft deposit). The paper's content is unchanged; the correction is logged in CORRECTIONS.md in the linked repository.