cmb_span_configDim
The declaration establishes that the five CMB observables form a set of cardinality five, aligning them with the recognition parameter space dimension. Cosmologists applying Recognition Science to CMB data would cite this to equate observable count with configDim. The proof is a direct one-line reduction to the decidable cardinality of the finite inductive type.
claimThe set of CMB observables has cardinality five: $|T, n_s, r, Omega_b, Omega_Lambda| = 5$.
background
The module treats CMB temperature via Recognition Science, identifying five canonical observables (temperature, spectral index, tensor-scalar ratio, baryon density, dark energy) that span the five-dimensional recognition parameter space. The inductive definition CMBObservable enumerates exactly these five cases and derives Fintype, DecidableEq, and related instances. The upstream theorem cmbObservableCount computes the cardinality directly via the decide tactic on this finite type.
proof idea
One-line wrapper that applies the upstream cardinality theorem cmbObservableCount.
why it matters in Recognition Science
The result confirms that the five observables match the configuration dimension stated in the module, supporting the structural link between T_CMB and universe age through phi powers. It sits within the CMB temperature derivation from RS and aligns with the broader forcing chain where configDim appears as five for this observable set. No open questions are flagged in the supplied material.
scope and limits
- Does not compute numerical values for any observable.
- Does not incorporate measurement error or observational bounds.
- Does not derive the five-dimensional count from the spatial dimension D=3.
- Does not address relations to the J-cost or phi-ladder mass formulas.
formal statement (Lean)
37theorem cmb_span_configDim : Fintype.card CMBObservable = 5 := cmbObservableCount
proof body
38