The strategic claim
The framework's strongest pattern — cross-shadow universality — maps onto a major open question in climate science: which apparent climate tipping signals in CMIP6 model ensembles are robust across structurally different models, and which are model-artefacts? This question is formalised by Theorem 10 (joint-admissibility \(\mathfrak{A}\)) and Theorem 11 (cross-shadow σ_cross convergence rate).
Instance #16 — CMIP6 SSP5-8.5 tipping cross-shadow consensus
Datasets: CMIP6 SSP5-8.5 (2015–2100) via Pangeo cloud catalogue. Three tipping observables:
| Element | Models | β median | σ_cross | Direction agreement | Verdict |
|---|---|---|---|---|---|
| Arctic sea-ice (siconc) | 21/21 | −0.341 | 0.694 | 1.000 | robust |
| AMOC (msftmz) | 19/19 | +0.309 | 1.545 | 1.000 | partially-robust |
| Amazon NPP | 25/25 | +1.717 | 1.874 | 0.800 | inconclusive |
Methodological finding: at annual cadence Layer-A R² ≈ 0.02–0.03 across all three elements. Annual climate ρ is dominated by interannual variability. The framework reports this honestly under Law V; the Layer-B direction-and-rate-sign cross-shadow consensus carries the signal at this cadence.
Source: cmip6_tipping_cross_shadow.py
Instance #17 — Arctic sea-ice observational (NSIDC, 1979–2025)
47-year record. β = −1.05 ± 0.6 across all four estimators with R² ≈ 0.05. Cross-shadow σ = 0.557.
Theorem 3 precision-floor empirical confirmation
\(\Phi_{\max}/\Phi_{\min} \approx 2.1\) → \(\log^2\) precision floor ≈ 0.55. Var(\(\hat\beta\)) is intrinsically large at this log-range; the framework correctly produces an estimate that respects its own theorem and refuses to over-claim. The framework's self-knowledge is the validated property here.
PELT change-point detection (BIC penalty, no imported threshold) rediscovers regime breaks at 1990 and 2007 in September extent — independent corroboration of the literature "Arctic Dipole Shift" hypothesis. Source script.
Instance #19 — Wunderling cascading-tipping test
8-model intersection (sea-ice + AMOC + Amazon NPP). Pairwise correlations of β across models. SVD PC1 = 76.4% variance, axis (-0.10, +0.60, -0.80) — dominantly AMOC↔Amazon antiphase mode. Outcome B with C-leaning: at n = 8 the framework cannot distinguish moderate cross-element coupling from sampling noise. Qualitative pattern matches Wunderling et al. 2024 but quantitatively inconclusive.
Material correction under Bayesian-AR(1): AMOC↔Amazon r collapses −0.55 → −0.18 (the prior antiphase-mode signal was largely autocorrelation artefact). Only sea-ice↔Amazon r = +0.61 survives leave-one-out single-model removal.
Instance #29 — RAPID AMOC observational
RAPID array overturning streamfunction at 26.5°N, 2004–2024 monthly (n = 240). Layer-A doesn't apply at 20-yr record length (Theorem 3 binds, log²(\(\Phi_{\max}/\Phi_{\min}\)) = 1.755).
PELT detects ZERO change-points in the 20-yr record — the Smeed et al. 2008–2012 AMOC weakening claim does NOT survive framework-native BIC test. EMD residue collapses cleanly — no detectable secular monotonic trend; AMOC fluctuates around its 17 Sv mean rather than declining.
Layer-B clean: RAPID β at quantile 0.21 within CMIP6 central 68% band — framework verdict MODELS CAPTURE OBSERVED AMOC DYNAMICS.