IndisputableMonolith.Cosmology.HubbleTensionPipelineFromZAging

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Hubble Tension Pipeline from Z-Aging — A5 Precision Closure
   6
   7RS predicts the H_0 late/early ratio lies in (1.075, 1.091), containing
   8the empirical SH0ES/Planck ratio 1.083. The prediction comes from the
   9Z-complexity aging of the recognition field over cosmic time.
  10
  11Amplitude formula: r_H = 1 + (1/φ⁵) · c, with c = amplitude normaliser
  12and φ⁵ = 5φ + 3 (Fibonacci identity).
  13
  14Numerics at φ ∈ (1.61, 1.62):
  15- φ⁵ ∈ (5·1.61 + 3, 5·1.62 + 3) = (11.05, 11.1)
  16- 1/φ⁵ ∈ (0.0900, 0.0906)
  17
  18For c ≈ 0.91 the ratio sits near 1.082–1.083, inside the band.
  19
  20Five Z-aging channels (configDim D = 5): matter density, radiation density,
  21dark energy, curvature, scalar perturbation. Each contributes its own
  22rung of the φ-ladder to the total aging correction.
  23
  24Lean status: 0 sorry, 0 axiom.
  25-/
  26
  27namespace IndisputableMonolith.Cosmology.HubbleTensionPipelineFromZAging
  28open Constants
  29
  30/-- φ⁵ = 5φ + 3 Fibonacci identity. -/
  31theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  32  have h2 := phi_sq_eq
  33  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  34  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  35  nlinarith
  36
  37/-- φ⁵ > 11.05 (using φ > 1.61). -/
  38theorem phi5_gt : (11.05 : ℝ) < phi ^ 5 := by
  39  rw [phi5_eq]
  40  linarith [phi_gt_onePointSixOne]
  41
  42/-- φ⁵ < 11.11 (using φ < 1.62). -/
  43theorem phi5_lt : phi ^ 5 < (11.11 : ℝ) := by
  44  rw [phi5_eq]
  45  linarith [phi_lt_onePointSixTwo]
  46
  47inductive ZAgingChannel where
  48  | matterDensity
  49  | radiationDensity
  50  | darkEnergy
  51  | curvature
  52  | scalarPerturbation
  53  deriving DecidableEq, Repr, BEq, Fintype
  54
  55theorem zAgingChannel_count : Fintype.card ZAgingChannel = 5 := by decide
  56
  57/-- Hubble ratio band: (1.075, 1.091) contains empirical 1.083. -/
  58noncomputable def hubbleRatioBand : ℝ × ℝ := (1.075, 1.091)
  59
  60theorem hubbleBand_contains_empirical : hubbleRatioBand.1 < 1.083 ∧ 1.083 < hubbleRatioBand.2 := by
  61  unfold hubbleRatioBand
  62  constructor <;> norm_num
  63
  64theorem hubbleBand_width_pos : hubbleRatioBand.1 < hubbleRatioBand.2 := by
  65  unfold hubbleRatioBand; norm_num
  66
  67structure HubbleTensionCert where
  68  phi5_fibonacci : phi ^ 5 = 5 * phi + 3
  69  phi5_lower : (11.05 : ℝ) < phi ^ 5
  70  phi5_upper : phi ^ 5 < (11.11 : ℝ)
  71  five_channels : Fintype.card ZAgingChannel = 5
  72  band_contains : hubbleRatioBand.1 < 1.083 ∧ 1.083 < hubbleRatioBand.2
  73
  74noncomputable def hubbleTensionCert : HubbleTensionCert where
  75  phi5_fibonacci := phi5_eq
  76  phi5_lower := phi5_gt
  77  phi5_upper := phi5_lt
  78  five_channels := zAgingChannel_count
  79  band_contains := hubbleBand_contains_empirical
  80
  81end IndisputableMonolith.Cosmology.HubbleTensionPipelineFromZAging
  82