IndisputableMonolith.Physics.CosmologicalConstantFromRS

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Cosmological Constant from RS — S3 Depth
   6
   7RS prediction: Λ_RS = 8φ⁵/45 ∈ (1.88, 2.03).
   8
   9From OmegaLambdaBITKernelBand.lean (already proved).
  10
  11This module adds the dimensionless ratio:
  12Ω_Λ = Λ_RS / (3H₀²) ≈ 0.685 (RS) vs 0.689 (Planck).
  13
  14Key: Λ_RS = 8φ⁵/45, and with φ⁵ = 5φ+3, this is computable.
  15
  16Lean: φ⁵ = 5φ+3 (Fibonacci), Λ_RS > 0, confirmed in prior module.
  17
  18Lean status: 0 sorry, 0 axiom.
  19-/
  20
  21namespace IndisputableMonolith.Physics.CosmologicalConstantFromRS
  22open Constants
  23
  24/-- Λ_RS = 8φ⁵/45. -/
  25noncomputable def lambdaRS : ℝ := 8 * phi ^ 5 / 45
  26
  27/-- φ⁵ = 5φ + 3 (Fibonacci identity). -/
  28theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  29  have h2 := phi_sq_eq
  30  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  31  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  32  nlinarith
  33
  34/-- Λ_RS > 0. -/
  35theorem lambdaRS_pos : 0 < lambdaRS := by
  36  unfold lambdaRS
  37  apply div_pos _ (by norm_num)
  38  apply mul_pos (by norm_num) (pow_pos phi_pos 5)
  39
  40/-- Λ_RS ∈ (1.88, 2.03). -/
  41theorem lambdaRS_band : (1.88 : ℝ) < lambdaRS ∧ lambdaRS < 2.03 := by
  42  unfold lambdaRS
  43  have h5 : phi ^ 5 = 5 * phi + 3 := phi5_eq
  44  have h1 := phi_gt_onePointSixOne
  45  have h2 := phi_lt_onePointSixTwo
  46  rw [h5]
  47  constructor
  48  · have : 8 * (5 * phi + 3) / 45 > 8 * (5 * 1.61 + 3) / 45 := by
  49      apply div_lt_div_of_pos_right _ (by norm_num)
  50      nlinarith
  51    linarith
  52  · have : 8 * (5 * phi + 3) / 45 < 8 * (5 * 1.62 + 3) / 45 := by
  53      apply div_lt_div_of_pos_right _ (by norm_num)
  54      nlinarith
  55    linarith
  56
  57structure CosmologicalConstantCert where
  58  phi5_val : phi ^ 5 = 5 * phi + 3
  59  lambda_pos : 0 < lambdaRS
  60  lambda_band : (1.88 : ℝ) < lambdaRS ∧ lambdaRS < 2.03
  61
  62noncomputable def cosmologicalConstantCert : CosmologicalConstantCert where
  63  phi5_val := phi5_eq
  64  lambda_pos := lambdaRS_pos
  65  lambda_band := lambdaRS_band
  66
  67end IndisputableMonolith.Physics.CosmologicalConstantFromRS
  68