Cascade primitives mapped to climate
| Symbol | Meaning in framework | Mapping in climate |
|---|---|---|
| \(\Phi\) | cascading value at a moment | tipping observable: Arctic sea-ice extent, AMOC streamfunction, Amazon NPP, atmospheric CO₂, GMSL, etc. |
| \(\tau\) | gap to the next moment | cadence: annual for CMIP6 monthly aggregates; ka for paleoclimate; daily for NSIDC |
| \(\rho\) | local rate of change | finite-differenced d\(\Phi\)/dt, smoothed with 5-yr centred kernel |
| \(\beta\) | brake exponent | slope of \(\log|\rho|\) against \(\log|\Phi|\) per model trajectory |
| \(\sigma\) | ensemble dispersion | cross-realisation std of \(\rho\) within a single CMIP6 model |
| \(\mu\) | shadow consensus | cross-model median \(\rho\) (Theorem 2 concentration) |
| \(\sigma_{\text{cross}}\) | cross-shadow brake dispersion | std of \(\beta\) across CMIP6 models per SSP |
Named objects
- \(\mathcal{C}\) — the cascade: a stream of \((\Phi, \tau, \rho)\) over an ordered moment-set. In climate, the trajectory of an observable under a given forcing.
- \(\mathcal{S}_j(\mathcal{C})\) — a shadow: the same trajectory under climate model \(j\). The CMIP6 ensemble provides 19–25 model shadows per observable.
- \(\{\mathcal{C}_i\}_{i=1..N}\) — an ensemble: \(N\) realisations of the same cascade (model seeds for the same climate model under the same SSP).
Operators applied to climate cascades
- \(\mathcal{B}(\mathcal{S}_j)\) — brake exponent on a single CMIP6 model's trajectory; computed by polyfit OLS, GLS-AR(1), or Bayesian-AR(1).
- \(\mathcal{S}(\{\mathcal{B}(\mathcal{S}_j)\})\) — cross-model dispersion of brake exponents = \(\sigma_{\text{cross}}\). The framework's universality call.
- \(\mathcal{M}(\{\mathcal{S}_j\})\) — median trajectory across models. Concentrates exponentially in model count by Theorem 2.
- \(\mathcal{P}(\mathcal{C}, c)\) — spectral primitive: decomposes a cadence-stamped cascade into trend, resonance, residual sub-cascades. Used in instance #22 (Mid-Pleistocene Transition) to separate the 41-kyr → 100-kyr resonance shift from the brake-rate evolution.
- \(\mathfrak{A}(\{\mathcal{S}_\alpha\}; \mathcal{C})\) — anti-shadow detector. Empirically anchored by instance #25 (stratospheric ozone): hole-minimum and zonal-mean each pass T6 individually, but \(\mathfrak{A} = 22.4 \gg \tau_{T3}\) — joint admissibility fails.
- \(\mathscr{A}\) — scope-reporter (Theorem 12): every climate finding publishes its own \(\mathscr{A}\) tuple (tested cascade/shadows/cadence, excluded conditions, \(\tau_{T3}\) precision floor, gauge group).