IndisputableMonolith.Unification.CosmologicalPredictionsProved
The module assembles proved cosmological predictions from the Recognition Science phi-forcing chain, centering on the spectral index formula n_s = 1 - 2/φ³ together with explicit bounds 0.529 < n_s < 0.941. Cosmologists studying inflation observables would cite these structural results for their direct descent from the J-cost ledger. Each sibling theorem follows by algebraic reduction from the self-similarity fixed-point property and the RS time quantum.
claim$n_s = 1 - 2/φ^3$ where $4 < φ^3 < 4.25$, implying $0.529 < n_s < 0.941$; the Hubble parameter is positive and the powers φ², φ⁴, φ⁵ obey the stated bounds.
background
Recognition Science obtains all constants from a single functional equation whose self-similar solutions force the golden ratio φ as the unique fixed point of the J-cost map J(x) = (x + x^{-1})/2 - 1 on a discrete ledger. The upstream PhiForcing module proves this forcing step, while Constants supplies the native time quantum τ₀ = 1 tick. The present module converts those primitives into explicit cosmological expressions without additional dynamical assumptions.
proof idea
The module is a collection of short algebraic theorems. spectral_index_formula applies the J-uniqueness identity directly to the scale factor; the three bound theorems (lt_one, gt_half, bounds) follow by interval arithmetic on 4 < φ³ < 4.25; hubble_positive and the phi-power bounds are immediate sign checks and monotonicity arguments on the same ladder.
why it matters in Recognition Science
It supplies the first concrete bridge from the T6 phi-forcing step to observable cosmology, realizing the RS prediction for the scalar spectral index as a direct structural consequence. The module therefore feeds the unification program by furnishing numbers that can be compared with CMB data, while the accompanying note flags the need for higher-order corrections to reach the measured n_s ≈ 0.965.
scope and limits
- Does not derive the measured n_s ≈ 0.965 without further corrections.
- Does not model full inflationary dynamics or reheating.
- Does not include matter or dark-energy content.
- Does not address stability under perturbations.
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