IndisputableMonolith.Cosmology.SIConversion

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# SI Calibration Seam for Cosmological Observables
   6
   7RS derives all physics in native units (c = ℓ₀ = τ₀ = 1). To compare
   8predictions with observations in SI units (meters, seconds, km/s/Mpc),
   9we need a single calibration anchor.
  10
  11## The Reporting Seam
  12
  13The Planck scale provides the bridge:
  14  ℓ_P = √(ℏG/c³) ≈ 1.616255 × 10⁻³⁵ m
  15  t_P = ℓ_P/c ≈ 5.391247 × 10⁻⁴⁴ s
  16
  17In RS-native units, ℓ_P = 1/√π (proved in UniverseSize.lean).
  18The SI value of ℓ_P is an EXTERNAL FACT — it depends on the human-
  19defined meter and second, not on RS theory.
  20
  21## What This Module Provides
  22
  231. SI anchor constants (ℓ_P, t_P, Mpc in SI)
  242. Conversion formulas for cosmological observables
  253. The RS-to-SI bridge ratio
  26
  27## Epistemic Status
  28
  29The SI numerical values (ℓ_P_SI, t_P_SI, etc.) are CODATA-sourced
  30experimental numbers. They are NOT RS predictions — they are the
  31calibration seam through which RS predictions are expressed in human
  32units. The theoretical content is in the RATIOS (R_obs/ℓ_P, t_age/t_P),
  33not in the SI values themselves.
  34
  35## Status: 0 sorry, 0 axiom
  36-/
  37
  38namespace IndisputableMonolith.Cosmology.SIConversion
  39
  40noncomputable section
  41
  42/-! ## Part 1: Planck Scale in SI -/
  43
  44/-- Planck length in meters (CODATA 2018).
  45    ℓ_P = √(ℏG/c³) = 1.616255 × 10⁻³⁵ m.
  46    Uncertainty: ±0.000018 × 10⁻³⁵ m (relative: 1.1 × 10⁻⁵). -/
  47def planck_length_SI : ℝ := 1.616255e-35
  48
  49/-- Planck time in seconds (CODATA 2018).
  50    t_P = ℓ_P/c = 5.391247 × 10⁻⁴⁴ s. -/
  51def planck_time_SI : ℝ := 5.391247e-44
  52
  53/-- Speed of light in m/s (exact since 2019 SI redefinition). -/
  54def c_SI : ℝ := 299792458
  55
  56/-- 1 Megaparsec in meters (IAU 2012 definition).
  57    1 pc = 648000/π AU, 1 AU = 149597870700 m (exact). -/
  58def Mpc_SI : ℝ := 3.0857e22
  59
  60/-- 1 light-year in meters. -/
  61def ly_SI : ℝ := 9.461e15
  62
  63/-- 1 Gyr (billion years) in seconds. -/
  64def Gyr_SI : ℝ := 3.1557e16
  65
  66theorem planck_length_SI_pos : 0 < planck_length_SI := by
  67  unfold planck_length_SI; norm_num
  68
  69theorem planck_time_SI_pos : 0 < planck_time_SI := by
  70  unfold planck_time_SI; norm_num
  71
  72theorem c_SI_pos : 0 < c_SI := by
  73  unfold c_SI; norm_num
  74
  75theorem Mpc_SI_pos : 0 < Mpc_SI := by
  76  unfold Mpc_SI; norm_num
  77
  78/-! ## Part 2: Conversion Functions -/
  79
  80/-- Convert a distance from Planck lengths to meters. -/
  81def planck_to_meters (r_planck : ℝ) : ℝ := r_planck * planck_length_SI
  82
  83/-- Convert a time from Planck times to seconds. -/
  84def planck_to_seconds (t_planck : ℝ) : ℝ := t_planck * planck_time_SI
  85
  86/-- Convert a Hubble parameter from 1/t_P to km/s/Mpc.
  87    H₀ [1/t_P] × (t_P [s] / Mpc [m]) × (1 km) = H₀ [km/s/Mpc]. -/
  88def hubble_to_kms_mpc (h_planck : ℝ) : ℝ :=
  89  h_planck * planck_time_SI⁻¹ * (1000 / Mpc_SI)
  90
  91/-- Convert seconds to Gyr. -/
  92def seconds_to_Gyr (t_s : ℝ) : ℝ := t_s / Gyr_SI
  93
  94/-- Convert meters to billion light-years (Gly). -/
  95def meters_to_Gly (r_m : ℝ) : ℝ := r_m / (ly_SI * 1e9)
  96
  97/-! ## Part 3: Observed Values for Comparison -/
  98
  99/-- Observed comoving radius of the observable universe (meters).
 100    Source: Planck 2018 ΛCDM fit.
 101    46.5 Gly ≈ 4.40 × 10²⁶ m. -/
 102def obs_radius_m : ℝ := 4.40e26
 103
 104/-- Observed age of the universe (seconds).
 105    13.787 ± 0.020 Gyr (Planck 2018). -/
 106def obs_age_s : ℝ := 4.354e17
 107
 108/-- Observed age in Gyr. -/
 109def obs_age_Gyr : ℝ := 13.787
 110
 111/-- Early-universe Hubble measurement (km/s/Mpc).
 112    Planck 2018: 67.36 ± 0.54. -/
 113def obs_H0_early : ℝ := 67.36
 114
 115/-- Late-universe Hubble measurement (km/s/Mpc).
 116    SH0ES 2022 (Riess et al.): 73.04 ± 1.04. -/
 117def obs_H0_late : ℝ := 73.04
 118
 119/-! ## Part 4: SI Certificate -/
 120
 121structure SICalibrationCert where
 122  planck_length_positive : 0 < planck_length_SI
 123  planck_time_positive : 0 < planck_time_SI
 124  consistency : planck_length_SI / c_SI < planck_time_SI * 1.01
 125  consistency2 : planck_time_SI * 0.99 < planck_length_SI / c_SI
 126
 127theorem si_calibration_cert : SICalibrationCert where
 128  planck_length_positive := planck_length_SI_pos
 129  planck_time_positive := planck_time_SI_pos
 130  consistency := by unfold planck_length_SI c_SI planck_time_SI; norm_num
 131  consistency2 := by unfold planck_length_SI c_SI planck_time_SI; norm_num
 132
 133end
 134
 135end IndisputableMonolith.Cosmology.SIConversion
 136