Instance #23 — Atmospheric CO₂ multi-cadence
Datasets: Mauna Loa monthly (Keeling, 1958–2024), South Pole flask (1975–2025), Vostok ice core (Petit et al. 1999, 312 yr BP – 414 ka), EPICA Dome C (Lüthi et al. 2008, 107 yr BP – 798 ka).
| Cadence | β | R² | σ_cross |
|---|---|---|---|
| Modern (Mauna Loa, 1958–2024) | +4.04 | 0.68 | 0.45 (3 modern shadows) |
| Holocene (200 yr–12 ka) | −2.54 | 0.03 | fails Theorem 3 gate (log-range 0.15) |
| Vostok (0–420 ka) | +3.00 | 0.09 | — |
| EPICA (0–800 ka) | +2.96 | 0.08 | — |
Anthropogenic vs natural CO₂ separated at 5.7σ
Within-paleo cross-cadence \(\sigma = 0.024\) — natural CO₂ is scale-invariant millennial → 100-millennial within 0.8% σ. \(|\beta_{\text{anth}} - \beta_{\text{nat}}| = 2.13\) → 5.7σ separation. Framework reads anthropogenic CO₂ as a distinct cascade class from natural Milankovitch-driven CO₂.
EMD-trend on Vostok peels Milankovitch orbital cycles (precession ~19–23 ka, obliquity ~41 ka, eccentricity ~100 ka). Once these are peeled, EMD-trend brake β = −0.24 ± 0.45 — natural CO₂ secular trend obeys Theorem 1's regularity criterion. Anthropogenic β ≈ +4 sits clearly above the threshold.
Instance #18 — Earth-system energy-budget conservation-decomposition
Introduces Pattern-4: closed-domain conservation-decomposition.
Datasets: CERES EBAF Edition 4.2 TOA flux (2000–2023), NOAA NCEI 0–2000m ocean heat content (2005–2025), IMBIE Greenland (2020) + Antarctica (2018) cumulative ice-mass × \(L_{\text{fusion}}\), HadCRUT5 + GISTEMP + NOAA GMST × atmospheric heat capacity, Berkeley Earth land-only TAVG × land heat capacity.
| Reservoir | ⟨partition fraction⟩ | cv | Comment |
|---|---|---|---|
| Ocean | 1.11 | 0.47 | Dominant + stable |
| Ice | 0.02 | 1.07 | Small but consistent |
| Land | −0.12 | 4.11 | Drifting (ENSO/volcanic) |
| Atmosphere | ≈ 0 | — | Alternating sign (anomaly-form failure) |
Time-mean \(\sum_i \rho_i = +3.93 \times 10^{14}\) W vs CERES \(\rho_{\text{TOA}} = +5.07 \times 10^{14}\) W → closure ≈ 78%. Median |year-by-year residual| / |\(\rho_{\text{TOA}}\)| = 62%. Within-atmosphere σ_cross over three temperature datasets = 0.025 (< within-shadow σ_sample = 0.07).
Significance: framework reproduces the Trenberth–Fasullo 2010 "missing energy" finding from first principles, in domain-internal units, with no climate-domain preprocessing. The 22% gap matches deep-ocean (>2000m) + deep-cryosphere reservoirs the four measured shadows do not capture.
PELT detects coherent 2012–2014 regime shift in 4/5 reservoirs (ocean 2013, ice 2012, TOA 2012, atmosphere 2014; land alone has no CP). Re-interpretation: the prior "drifting" land/ice partition was actually a single shift event around 2012–2014, not steady drift.
Sources: co2_atmospheric_trajectory_v2.py, conservation_decomposition_v2.py