IndisputableMonolith.Unification.CosmologicalPredictionsProved
This module derives explicit cosmological predictions in Recognition Science from the golden ratio φ forced by self-similarity. A physicist working on RS unification would cite it for the structural formula of the scalar spectral index and its interval bounds. The argument consists of direct algebraic reductions from the imported PhiForcing lemmas together with interval estimates on φ³.
claimThe spectral index satisfies $n_s = 1 - 2/φ^3$. When $4 < φ^3 < 4.25$ the value lies in the open interval $(0.529, 0.941)$.
background
The module imports Constants, which fixes the RS time quantum τ₀ = 1 tick, and PhiForcing, whose doc-comment states that it proves φ is forced by self-similarity in a discrete ledger with J-cost. The local setting is the application of the Recognition Composition Law and the T5–T6 forcing chain to cosmology, yielding structural formulas for CMB observables. The supplied doc-comment notes that the formula is calculated directly from φ while the observed n_s ≈ 0.965 requires further corrections from the full RS derivation.
proof idea
The module is organized as a collection of lemmas (spectral_index_formula, spectral_index_bounds, etc.) that perform algebraic substitution of the phi-forcing identity into the expression for n_s, followed by elementary interval arithmetic on the known bounds 4 < φ³ < 4.25. No complex tactics are required; each step is a direct rewriting from the upstream PhiForcing results.
why it matters in Recognition Science
The module supplies the first concrete cosmological output of the Recognition Science framework in the Unification domain. It applies the phi-forcing result (T6) to produce the spectral-index prediction cited in the module doc-comment. Because used_by is empty, the immediate parent is the larger unification effort that will incorporate these calculated predictions into the full RS cosmology.
scope and limits
- Does not derive the observed n_s ≈ 0.965 without additional corrections.
- Does not address other cosmological parameters such as the Hubble constant.
- Does not reprove the forcing of φ itself, which is imported from PhiForcing.
- Does not claim numerical agreement with Planck data beyond the stated structural interval.
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