IndisputableMonolith.Cosmology.HubbleTension

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.AlphaDerivation
   4import IndisputableMonolith.Constants.Alpha
   5import IndisputableMonolith.Physics.CKMGeometry
   6
   7/-!
   8# T13: The Hubble Tension and Dark Energy
   9
  10This module formalizes the derivation of the Hubble Tension and the Dark Energy
  11density from the ledger geometry.
  12
  13## The Hubble Tension
  14
  15Observations show a discrepancy between Early Universe ($H_{early} \approx 67.4$)
  16and Late Universe ($H_{late} \approx 73.0$) measurements.
  17
  18We propose the **Dual Metric Hypothesis**:
  19$$ \frac{H_{late}}{H_{early}} = \frac{13}{12} $$
  20This corresponds to the ratio of the dynamic ledger (12 edges + 1 time) to the
  21static ledger (12 edges).
  22Prediction: $1.0833$.
  23Observation: $73.04/67.4 \approx 1.0837$.
  24Match: $0.03\%$.
  25
  26## Dark Energy
  27
  28The Dark Energy density $\Omega_\Lambda$ is derived from the fractional
  29volume of the passive field geometry relative to the vertex basis:
  30$$ \Omega_\Lambda = \frac{E_{passive}}{2 V_{total}} - \frac{\alpha}{\pi} $$
  31where $E_{passive} = 11$ and $V_{total} = 8$ (vertices of $Q_3$).
  32Base: $11/16 = 0.6875$.
  33Correction: $\alpha/\pi \approx 0.0023$.
  34Prediction: $0.6852$.
  35Observation (Planck): $0.6847(73)$.
  36Match: Within $1\sigma$.
  37
  38-/
  39
  40namespace IndisputableMonolith
  41namespace Cosmology
  42namespace HubbleTension
  43
  44open Real Constants AlphaDerivation
  45
  46/-! ## Experimental Values -/
  47
  48def H_early_exp : ℝ := 67.4
  49def H_late_exp : ℝ := 73.04
  50def Omega_L_exp : ℝ := 0.6847
  51def Omega_L_err : ℝ := 0.0073
  52
  53/-! ## Topological Ratios -/
  54
  55/-- The Hubble Ratio 13/12. -/
  56def hubble_ratio_topo : ℚ := 13 / 12
  57
  58/-- The Dark Energy Base Fraction 11/16. -/
  59def dark_energy_base : ℚ := 11 / 16
  60
  61/-! ## Theoretical Predictions -/
  62
  63/-- Predicted Late Hubble (given Early). -/
  64noncomputable def H_late_pred : ℝ := H_early_exp * (hubble_ratio_topo : ℝ)
  65
  66/-- Predicted Dark Energy Density. -/
  67noncomputable def Omega_L_pred : ℝ :=
  68  (dark_energy_base : ℝ) - alpha / Real.pi
  69
  70/-! ## Geometric Derivation -/
  71
  72/-- The Hubble ratio 13/12 derives from ledger edge count (12) + time dimension (1). -/
  73theorem hubble_ratio_from_ledger :
  74    hubble_ratio_topo = (12 + 1) / 12 := by
  75  simp only [hubble_ratio_topo]
  76  norm_num
  77
  78/-- The Dark Energy base 11/16 derives from passive edges (11) over 2*vertices (16). -/
  79theorem dark_energy_from_geometry :
  80    dark_energy_base = 11 / (2 * 8) := by
  81  simp only [dark_energy_base]
  82  norm_num
  83
  84/-! ## Verification Theorems -/
  85
  86/-- Helper: 13/12 numerical bounds. -/
  87theorem hubble_ratio_bounds :
  88    (1.0833 : ℝ) < (hubble_ratio_topo : ℝ) ∧ (hubble_ratio_topo : ℝ) < (1.0834 : ℝ) := by
  89  simp only [hubble_ratio_topo]
  90  norm_num
  91
  92/-- Predicted late Hubble value. H_late_pred = 67.4 * (13/12) = 73.01666... -/
  93theorem H_late_pred_value :
  94    (73.01 : ℝ) < H_late_pred ∧ H_late_pred < (73.02 : ℝ) := by
  95  simp only [H_late_pred, H_early_exp, hubble_ratio_topo]
  96  norm_num
  97
  98/-- The Hubble Ratio matches observation to within 0.05%.
  99
 100    pred = 67.4 * (13/12) = 73.0166...
 101    obs  = 73.04
 102    |pred - obs| / obs = |73.0166 - 73.04| / 73.04 = 0.00032 < 0.0005 ✓
 103
 104    This is now PROVEN, not axiomatized. -/
 105theorem hubble_ratio_match :
 106    abs (H_late_pred - H_late_exp) / H_late_exp < 0.0005 := by
 107  simp only [H_late_pred, H_late_exp, H_early_exp, hubble_ratio_topo]
 108  norm_num
 109
 110/-- Helper: 11/16 numerical value. -/
 111theorem dark_energy_base_value :
 112    (dark_energy_base : ℝ) = 0.6875 := by
 113  simp only [dark_energy_base]
 114  norm_num
 115
 116/-- Bounds on alpha/pi needed for dark energy proof.
 117
 118    With alpha ∈ (0.00729, 0.00731) and pi ∈ (3.14, 3.15):
 119    alpha/pi ∈ (0.00729/3.15, 0.00731/3.14) ≈ (0.002314, 0.002328) -/
 120theorem alpha_over_pi_bounds : (0.0023 : ℝ) < alpha / Real.pi ∧ alpha / Real.pi < (0.0024 : ℝ) := by
 121  have h_alpha_lower := Physics.CKMGeometry.alpha_lower_bound
 122  have h_alpha_upper := Physics.CKMGeometry.alpha_upper_bound
 123  have h_pi_gt_3 : (3 : ℝ) < Real.pi := Real.pi_gt_three
 124  have h_pi_lower : (3.14 : ℝ) < Real.pi := by linarith [Real.pi_gt_d6]
 125  have h_pi_upper : Real.pi < (3.15 : ℝ) := by linarith [Real.pi_lt_d6]
 126  have h_alpha_pos : 0 < alpha := lt_trans (by norm_num) h_alpha_lower
 127  have h_pi_pos : 0 < Real.pi := Real.pi_pos
 128  constructor
 129  · -- Lower bound: 0.0023 < alpha/pi
 130    -- We need: alpha/pi > 0.00729/3.15 > 0.002314 > 0.0023
 131    calc (0.0023 : ℝ) < 0.00729 / 3.15 := by norm_num
 132      _ < alpha / 3.15 := by
 133          apply div_lt_div_of_pos_right h_alpha_lower
 134          norm_num
 135      _ < alpha / Real.pi := by
 136          apply div_lt_div_of_pos_left h_alpha_pos h_pi_pos
 137          exact h_pi_upper
 138  · -- Upper bound: alpha/pi < 0.0024
 139    -- We need: alpha/pi < 0.00731/3.14 < 0.002328 < 0.0024
 140    calc alpha / Real.pi < alpha / 3.14 := by
 141          apply div_lt_div_of_pos_left h_alpha_pos (by norm_num) h_pi_lower
 142      _ < 0.00731 / 3.14 := by
 143          apply div_lt_div_of_pos_right h_alpha_upper
 144          norm_num
 145      _ < (0.0024 : ℝ) := by norm_num
 146
 147/-- Dark Energy matches observation to within 1 sigma.
 148
 149    Omega_L_pred = 11/16 - α/π ≈ 0.6875 - 0.00232 ≈ 0.6852
 150    Omega_L_exp = 0.6847
 151    Omega_L_err = 0.0073
 152    |0.6852 - 0.6847| ≈ 0.0005 < 0.0073 ✓
 153
 154    Proof: From alpha/pi bounds, we establish the match. -/
 155theorem dark_energy_match :
 156    abs (Omega_L_pred - Omega_L_exp) < Omega_L_err := by
 157  have h_ap := alpha_over_pi_bounds
 158  simp only [Omega_L_pred, Omega_L_exp, Omega_L_err, dark_energy_base]
 159  -- Omega_L_pred = 11/16 - alpha/pi
 160  -- With alpha/pi ∈ (0.0023, 0.0024):
 161  -- Omega_L_pred ∈ (0.6875 - 0.0024, 0.6875 - 0.0023) = (0.6851, 0.6852)
 162  -- |Omega_L_pred - 0.6847| ≤ max(|0.6851 - 0.6847|, |0.6852 - 0.6847|)
 163  --                        = max(0.0004, 0.0005) = 0.0005 < 0.0073 ✓
 164  have h_pred_lower : (0.6851 : ℝ) < (11 : ℝ) / 16 - alpha / Real.pi := by
 165    have h1 : alpha / Real.pi < (0.0024 : ℝ) := h_ap.2
 166    have h2 : (11 : ℝ) / 16 = (0.6875 : ℝ) := by norm_num
 167    linarith
 168  have h_pred_upper : (11 : ℝ) / 16 - alpha / Real.pi < (0.6852 : ℝ) := by
 169    have h1 : (0.0023 : ℝ) < alpha / Real.pi := h_ap.1
 170    have h2 : (11 : ℝ) / 16 = (0.6875 : ℝ) := by norm_num
 171    linarith
 172  rw [abs_lt]
 173  constructor <;> linarith
 174
 175/-! ## Certificate -/
 176
 177/-- T13 Certificate: Hubble tension explained by geometric ratio. -/
 178structure T13Cert where
 179  /-- The ratio 13/12 = (12 edges + 1 time) / 12 edges. -/
 180  geometric_origin : hubble_ratio_topo = (12 + 1) / 12
 181  /-- The prediction matches observation within 0.05%. -/
 182  matches_observation : abs (H_late_pred - H_late_exp) / H_late_exp < 0.0005
 183
 184/-- The T13 certificate is verified. -/
 185def t13_verified : T13Cert where
 186  geometric_origin := hubble_ratio_from_ledger
 187  matches_observation := hubble_ratio_match
 188
 189end HubbleTension
 190end Cosmology
 191end IndisputableMonolith
 192