approachCount
plain-language theorem explainer
The theorem establishes that the inductive type of geoengineering approaches contains exactly five constructors. Climate researchers applying J-cost bounds to intervention risks would cite this cardinality to fix the dimension of the approach space at five. The proof is a one-line decision procedure that enumerates the constructors and confirms the finite cardinality.
Claim. The finite type consisting of the five geoengineering approaches SAI, MCB, OIF, DAC and enhanced weathering has cardinality five: $|A| = 5$ where $A = $ {SAI, MCB, OIF, DAC, enhancedWeathering}.
background
In the Recognition Science climate module, geoengineering interventions perturb the recognition ratio r = (post-intervention flux)/(baseline flux) and thereby add a J-cost term. The module lists five canonical approaches: stratospheric aerosol injection (SAI), marine cloud brightening (MCB), ocean iron fertilisation (OIF), direct air capture (DAC) and enhanced weathering (EW). These are encoded as the constructors of the inductive type GeoengineeringApproach, which derives Fintype so that cardinality statements become decidable. The upstream inductive definition in GeoengineeringApproachesFromConfigDim supplies a parallel enumeration of five interventions under different constructor names.
proof idea
The proof is a one-line wrapper that applies the decide tactic directly to the Fintype.card expression. The tactic inspects the five constructors of the inductive type and returns the equality with 5.
why it matters
The result supplies the five_approaches field inside the downstream geoengineeringRiskCert definition that certifies risk thresholds. It implements the configDim D = 5 slot required by the E4 Climate Engineering section, where the five approaches must keep induced J-costs inside the canonical phi band. The declaration touches the open question of whether further approaches can be added without violating J-uniqueness from the forcing chain.
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