CMBTempCert
CMBTempCert encodes the assertion that the cosmic microwave background is characterized by exactly five observables in the Recognition Science model. Cosmologists working inside the RS framework cite this structure to confirm the match to configuration dimension D=5. The definition is a direct structural wrapper around the Fintype cardinality of the inductive type whose five constructors list temperature, spectral index, tensor-scalar ratio, baryon density, and dark energy.
claimLet CMBObservable be the inductive type with constructors temperature, spectralIndex, tensorScalar, baryonDensity, darkEnergy. Then CMBTempCert is the structure whose field asserts that the cardinality of this finite type equals 5.
background
Recognition Science derives CMB temperature from a structural relation T_CMB ≈ 1/(φ^45 × τ₀ × constant) tied to the age of the universe via Wien's law. The module states that five canonical observables (temperature, spectral index, tensor/scalar ratio, baryon density, dark energy) equal configDim D=5, with zero-sorry Lean status. CMBObservable is the inductive enumeration of these five quantities, carrying DecidableEq, Repr, BEq, and Fintype instances derived from its constructors.
proof idea
The declaration is a structure definition. Its single field states the equality Fintype.card CMBObservable = 5, which holds by the Fintype instance automatically derived for the inductive type CMBObservable.
why it matters in Recognition Science
CMBTempCert supplies the certified observable count that the downstream cmbTempCert construction applies directly. It realizes the RS claim that five CMB observables align with configuration dimension D=5, supporting the structural derivation of T_CMB from the phi-ladder. The definition closes the enumeration step in the CMB temperature certification chain inside the Physics.CosmicMicrowaveBackgroundTemperatureFromRS module.
scope and limits
- Does not compute the numerical value of T_CMB.
- Does not derive any relation between temperature and universe age.
- Does not invoke the phi-ladder mass formula or RS constants.
- Does not address spectral properties or tensor modes beyond the count.
formal statement (Lean)
39structure CMBTempCert where
40 five_observables : Fintype.card CMBObservable = 5
41