Multi-Sensor Convergence as a Regime Detection Signal: The Sensor Convergence Theorem

Abstract

When several independent instruments observing one system begin to move together, that agreement carries information about the system’s near future. We formalize this as the Sensor Convergence Theorem: under a latent factor model with three scope conditions — sensor independence, persistent regime structure, and positive loading — the average pairwise correlation of sensor divergences is a monotonically increasing function of latent regime magnitude (numeric stress test: true; symbolic verification of the companion closed forms: true). We test the theorem across four natural domains where it should hold — financial markets, river hydrology, solar physics, and economics — and a fifth, public‑health surveillance, registered to fail, with a pre‑registered design, fingerprinted input data, and a machine‑verified claims ledger. The thesis concerns independent sensors; the evidence that bears on it most directly comes from the multi‑instrument tests, and we are careful throughout to separate that evidence from single‑instrument constructions, which we find are largely a repackaging of trailing volatility. Solar physics offers the cleanest case: three physically distinct instruments yield Spearman ρ = 0.0972 against forward solar variability (conservative block‑shuffle z = 2.171), while a matched set of four transforms of a single instrument — an engineered independence violation — yields ρ = ‑0.2319, uninformative as the theorem predicts. Three weekly economic indicators, placed on an equal footing, yield only ρ = 0.09 (block‑shuffle z = 0.969, not significant); an earlier construction in which one large‑scale indicator dominated the target overstated this, so economics sits at the theorem’s boundary, where the internal‑ facet and cross‑indicator tests both fail; two previously claimed single‑indicator exceptions did not survive reconstruction and are withdrawn. A market‑wide count of simultaneously converging instruments anticipates risk‑asset volatility but not currencies or rates. By con‑ trast, single‑instrument convergence — eight futures markets (mean ρ = 0.0677, 7 of 8 posi‑ tive; S&P 500 e‑mini ρ = 0.2571) and two rivers (Connecticut ρ = 0.1885, block z = 3.957; Ohio ρ = 0.1089, block z = 2.88) — correlates with forward variability, but controlling for trailing volatility collapses it (S&P partial ρ = 0.0307): at the single‑instrument level the signal is largely redundant with recent turbulence, and we do not claim incremental skill there. Even the apparent rise in equity‑sensor convergence during crises is largely mechanical: a Forbes–Rigobon heteroskedasticity correction turns the genuine equity component sharply negative (adjusted dispersion measure ‑1.4277), while the same correction leaves a robust genuine component in crude oil (1.5905). The signal’s non‑redundant content is structural — the cross‑system count, the scope contrasts above, and the Detection Frontier, whose closed‑form difficulty ranking is negatively rank‑correlated with empirical detection strength across 96 configurations (partial Spearman = ‑0.4743; the ranking survives removal of window‑width, controlled Spearman = ‑0.5066). Derivative transformations of the signal carry almost nothing (mean ρ = 0.0162 and 0.0261): the information lives in the level. A weather battery registered to fall short instead weakly clears the honest block‑shuffle null (ρ = 0.0997, z = 1.826), so the predicted scope‑ failure is not cleanly confirmed; the boundary of the regime‑structure condition appears soft at seasonal timescales. A fifth domain, influenza surveillance, is registered to fail and does: its four near‑redundant sensors all track influenza prevalence and so violate the independence condition, and convergence is anti‑predictive of forward influenza activity (ρ = ‑0.248; principled failure confirmed true). A falsifiable forward prediction for 2026–2031 is registered; at the level of non‑overlapping elevated‑convergence episodes (33 historical episodes, each separated by at least one forward‑window length), the in‑sample hit rate is 0.6364 against a base rate of 0.5 — suggestive but not significant once the windows are required not to overlap (one‑ sided binomial p = 0.0814). The Connecticut result survives deseasonalization (ρ = 0.1846). This work is not the first to link rising co‑movement to impending regime change — it shares that core with the critical‑slowing‑down and systemic‑risk literatures (Section 5.1); our contribution is the explicit, separately tested independence condition, the closed‑form detection frontier, and a fully reproducible verification contract. Every number in this paper regenerates from committed scripts on hash‑pinned data; the verification gate (verify.py) is green.Verified rebuild under the Research-to-Publication Standard v1.4: every load-bearing number is registered in a machine-checked ledger (claims.lock) and regenerated on demand by verify.py on hash-pinned inputs; the two-round adversarial review (plus a targeted remediation re-review) 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.