IndisputableMonolith.Unification.CosmologicalPredictionsProved
This module assembles theorems deriving the spectral index and related cosmological quantities directly from the golden ratio φ in Recognition Science. Cosmologists modeling the early universe would cite it for the explicit RS prediction n_s = 1 - 2/φ³ together with its numerical bounds. The structure consists of algebraic reductions that apply the self-similarity fixed point and J-cost ledger from the imported PhiForcing module to produce the listed formulas and inequalities.
claim$n_s = 1 - 2/\phi^3$ with $4 < \phi^3 < 4.25$ yielding $0.529 < n_s < 0.941$; additional theorems establish positivity of the Hubble parameter and bounds on powers of φ.
background
The module imports Constants, which fixes the RS time quantum τ₀ = 1 tick, and PhiForcing, whose doc-comment states that φ is forced by self-similarity in a discrete ledger with J-cost. These supply the J-uniqueness relation J(x) = (x + x^{-1})/2 - 1 and the self-similar fixed point that generate the phi-ladder used throughout.
The local setting is the unification domain, where the forcing chain (T5 J-uniqueness through T6 phi) is applied to cosmological observables. The module doc-comment supplies the target formula n_s = 1 - 2/φ³ and notes that the observed value requires further corrections from the full RS derivation.
proof idea
The module is a collection of sibling theorems. spectral_index_formula states the structural prediction n_s = 1 - 2/φ³. The three bound theorems (spectral_index_lt_one, spectral_index_gt_half, spectral_index_bounds) follow by direct substitution of the interval 4 < φ³ < 4.25. hubble_positive and hubble_formula_structure apply the same phi-ladder arithmetic to the Hubble parameter. All steps are algebraic reductions that invoke the phi-forcing lemmas imported from PhiForcing.
why it matters in Recognition Science
The module supplies the explicit cosmological predictions required by the unification program. It fills the step that converts the T5-T6 forcing chain into an observable spectral index, as stated in the module doc-comment. No downstream theorems are listed, but the collection directly supports the broader claim that RS derives n_s and Hubble structure from the same J-cost ledger that forces φ and D = 3.
scope and limits
- Does not derive the observed n_s ≈ 0.965 without additional corrections.
- Does not address tensor-to-scalar ratio or other CMB parameters.
- Does not incorporate the full RS mass formula or Berry threshold.
- Does not prove the numerical value of φ itself; that remains in PhiForcing.
depends on (2)
declarations in this module (12)
-
theorem
spectral_index_formula -
theorem
spectral_index_lt_one -
theorem
spectral_index_gt_half -
theorem
spectral_index_bounds -
theorem
hubble_positive -
theorem
hubble_formula_structure -
theorem
phi_squared_bounds -
theorem
phi_fourth_bounds -
theorem
phi_fifth_bounds -
structure
CosmologicalPredictionsCert -
theorem
cosmological_predictions_cert_exists -
theorem
cosmological_calculated_proofs_summary