IndisputableMonolith.Cosmology.EtaBPrefactorDerivation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Foundation.IntegrationGap
   4
   5/-!
   6# η_B Prefactor: Two-Sided 8-Tick Washout
   7
   8The order-one prefactor `c` in the baryogenesis formula
   9
  10  η_B = c × (algebraic content forced by integration-gap rung)
  11
  12is structurally derived as `c_RS = (1 − φ^(−8))^2`. The interpretive
  13argument is that the matter and antimatter sectors each carry one
  14integration-gap-worth of fermionic degrees of freedom and each is
  15washed out at rate `φ^(−8)` (one rung per fundamental tick at `D = 3`).
  16The squared form encodes two independent washout channels.
  17
  18We prove:
  19
  20* `φ^8 = 21φ + 13` via the Fibonacci identity `φ^(n+1) = F(n+1)φ + F(n)`.
  21* `φ^8 ∈ (46.81, 47.03)` via `φ ∈ (1.61, 1.62)`.
  22* `φ^(−8) ∈ (0.02126, 0.02137)`.
  23* `c_RS := (1 − φ^(−8))^2 ∈ (0.956, 0.959)` (numerical band).
  24* `φ^(−44) ∈ (6.37 × 10⁻¹⁰, 6.40 × 10⁻¹⁰)` via Fibonacci `F(44) = 701408733`.
  25* `c_RS · φ^(−44) ∈ (6.0 × 10⁻¹⁰, 6.2 × 10⁻¹⁰)`, the predicted η_B band.
  26
  27The Planck 2018 measurement `η_B = (6.10 ± 0.04) × 10⁻¹⁰` falls inside
  28the predicted band. Honest scope: the band-level theorem is
  29`THEOREM`-grade; the interpretive squared form is `HYPOTHESIS`-grade
  30pending a first-principles Boltzmann-equation derivation.
  31
  32## Status
  33
  34* `0 sorry`
  35* No theory-specific axioms; standard kernel only (plus `native_decide`
  36  for small Fibonacci arithmetic, which uses `Lean.ofReduceBool` and
  37  `Lean.trustCompiler`).
  38-/
  39
  40namespace IndisputableMonolith
  41namespace Cosmology
  42namespace EtaBPrefactorDerivation
  43
  44open Constants
  45
  46noncomputable section
  47
  48/-! ## φ^8 from the Fibonacci identity -/
  49
  50/-- `φ^8 = 21 φ + 13`, derived stepwise from `φ^2 = φ + 1`. -/
  51theorem phi_pow_8_fib : phi ^ (8 : ℕ) = 21 * phi + 13 := by
  52  have h2 : phi ^ 2 = phi + 1 := phi_sq_eq
  53  have h4 : phi ^ 4 = 3 * phi + 2 := by
  54    have hexp : phi ^ 4 = phi ^ 2 * phi ^ 2 := by ring
  55    rw [hexp, h2]
  56    ring_nf
  57    linarith [h2]
  58  have hexp : phi ^ 8 = phi ^ 4 * phi ^ 4 := by ring
  59  rw [hexp, h4]
  60  ring_nf
  61  linarith [h2]
  62
  63/-- `φ^8 > 46.81` (from `φ > 1.61`). -/
  64theorem phi_pow_8_lower : phi ^ (8 : ℕ) > 46.81 := by
  65  rw [phi_pow_8_fib]
  66  have hphi : phi > 1.61 := phi_gt_onePointSixOne
  67  linarith
  68
  69/-- `φ^8 < 47.03` (from `φ < 1.62`). -/
  70theorem phi_pow_8_upper : phi ^ (8 : ℕ) < 47.03 := by
  71  rw [phi_pow_8_fib]
  72  have hphi : phi < 1.62 := phi_lt_onePointSixTwo
  73  linarith
  74
  75/-! ## φ^(−8) bounds -/
  76
  77private lemma phi_zpow_neg8_eq_inv : phi ^ (-8 : ℤ) = (phi ^ (8 : ℕ))⁻¹ := by
  78  rw [show ((-8 : ℤ)) = -((8 : ℕ) : ℤ) from by norm_num, zpow_neg, zpow_natCast]
  79
  80theorem phi_zpow_neg8_lower : phi ^ (-8 : ℤ) > 0.02126 := by
  81  rw [phi_zpow_neg8_eq_inv]
  82  have hupper : phi ^ (8 : ℕ) < 47.03 := phi_pow_8_upper
  83  have hpos : (0 : ℝ) < phi ^ (8 : ℕ) := pow_pos phi_pos 8
  84  have h1 : (phi ^ (8 : ℕ))⁻¹ > (47.03 : ℝ)⁻¹ := by
  85    rw [gt_iff_lt, inv_lt_inv₀ (by norm_num : (0:ℝ) < 47.03) hpos]
  86    exact hupper
  87  have h2 : (47.03 : ℝ)⁻¹ ≥ 0.02126 := by norm_num
  88  linarith
  89
  90theorem phi_zpow_neg8_upper : phi ^ (-8 : ℤ) < 0.02137 := by
  91  rw [phi_zpow_neg8_eq_inv]
  92  have hlower : phi ^ (8 : ℕ) > 46.81 := phi_pow_8_lower
  93  have hpos : (0 : ℝ) < phi ^ (8 : ℕ) := pow_pos phi_pos 8
  94  have h1 : (phi ^ (8 : ℕ))⁻¹ < (46.81 : ℝ)⁻¹ := by
  95    rw [inv_lt_inv₀ hpos (by norm_num : (0:ℝ) < 46.81)]
  96    exact hlower
  97  have h2 : (46.81 : ℝ)⁻¹ ≤ 0.02137 := by norm_num
  98  linarith
  99
 100/-! ## The two-sided washout prefactor -/
 101
 102/-- The single-tick correction factor `1 − φ^(−8)`. -/
 103def correctionFactor : ℝ := 1 - phi ^ (-8 : ℤ)
 104
 105/-- The two-sided 8-tick washout prefactor: `c_RS = (1 − φ^(−8))^2`. -/
 106def c_RS : ℝ := correctionFactor ^ 2
 107
 108theorem c_RS_expanded : c_RS = (1 - phi ^ (-8 : ℤ)) ^ 2 := by
 109  unfold c_RS correctionFactor
 110
 111theorem one_minus_phi_neg8_lower : (1 - phi ^ (-8 : ℤ)) > 0.978 := by
 112  have h := phi_zpow_neg8_upper
 113  linarith
 114
 115theorem one_minus_phi_neg8_upper : (1 - phi ^ (-8 : ℤ)) < 0.979 := by
 116  have h := phi_zpow_neg8_lower
 117  linarith
 118
 119theorem c_RS_pos : 0 < c_RS := by
 120  unfold c_RS correctionFactor
 121  have h := one_minus_phi_neg8_lower
 122  positivity
 123
 124theorem c_RS_lt_one : c_RS < 1 := by
 125  rw [c_RS_expanded]
 126  have hl : (1 - phi ^ (-8 : ℤ)) > 0.978 := one_minus_phi_neg8_lower
 127  have hu : (1 - phi ^ (-8 : ℤ)) < 0.979 := one_minus_phi_neg8_upper
 128  nlinarith [hl, hu]
 129
 130/-- `c_RS > 0.956`. -/
 131theorem c_RS_lower : c_RS > 0.956 := by
 132  rw [c_RS_expanded]
 133  have hl : (1 - phi ^ (-8 : ℤ)) > 0.978 := one_minus_phi_neg8_lower
 134  have hu : (1 - phi ^ (-8 : ℤ)) < 0.979 := one_minus_phi_neg8_upper
 135  nlinarith [hl, hu]
 136
 137/-- `c_RS < 0.959`. -/
 138theorem c_RS_upper : c_RS < 0.959 := by
 139  rw [c_RS_expanded]
 140  have hl : (1 - phi ^ (-8 : ℤ)) > 0.978 := one_minus_phi_neg8_lower
 141  have hu : (1 - phi ^ (-8 : ℤ)) < 0.979 := one_minus_phi_neg8_upper
 142  nlinarith [hl, hu]
 143
 144/-! ## φ^(−44) bounds via the Fibonacci identity -/
 145
 146/-- `φ^(n+1) = F(n+1) · φ + F(n)`. (Standard Fibonacci-φ identity.) -/
 147private theorem phi_pow_fib (n : ℕ) :
 148    phi ^ (n + 1) = (Nat.fib (n + 1) : ℝ) * phi + (Nat.fib n : ℝ) := by
 149  induction n with
 150  | zero =>
 151    simp only [Nat.fib_zero, Nat.cast_zero, add_zero]
 152    rw [show Nat.fib 1 = 1 from rfl]; simp
 153  | succ n ih =>
 154    have hfib : Nat.fib (n + 2) = Nat.fib n + Nat.fib (n + 1) := Nat.fib_add_two
 155    calc phi ^ (n + 1 + 1) = phi ^ (n + 1) * phi := by ring
 156      _ = ((Nat.fib (n + 1) : ℝ) * phi + (Nat.fib n : ℝ)) * phi := by rw [ih]
 157      _ = (Nat.fib (n + 1) : ℝ) * phi ^ 2 + (Nat.fib n : ℝ) * phi := by ring
 158      _ = (Nat.fib (n + 1) : ℝ) * (phi + 1) + (Nat.fib n : ℝ) * phi := by rw [phi_sq_eq]
 159      _ = ((Nat.fib (n + 1) : ℝ) + (Nat.fib n : ℝ)) * phi + (Nat.fib (n + 1) : ℝ) := by ring
 160      _ = (↑(Nat.fib n + Nat.fib (n + 1)) : ℝ) * phi + (Nat.fib (n + 1) : ℝ) := by
 161            simp only [Nat.cast_add]; ring
 162      _ = (Nat.fib (n + 2) : ℝ) * phi + (Nat.fib (n + 1) : ℝ) := by rw [hfib]
 163
 164theorem phi_pow_44_fib :
 165    phi ^ (44 : ℕ) = (701408733 : ℝ) * phi + 433494437 := by
 166  have hfib := phi_pow_fib 43
 167  have hf44 : Nat.fib 44 = 701408733 := by native_decide
 168  have hf43 : Nat.fib 43 = 433494437 := by native_decide
 169  simp only [hf44, hf43] at hfib
 170  exact hfib
 171
 172private lemma phi_zpow_neg44_eq_inv : phi ^ (-44 : ℤ) = (phi ^ (44 : ℕ))⁻¹ := by
 173  rw [show ((-44 : ℤ)) = -((44 : ℕ) : ℤ) from by norm_num, zpow_neg, zpow_natCast]
 174
 175theorem phi_zpow_neg44_lower : phi ^ (-44 : ℤ) > 6.37e-10 := by
 176  rw [phi_zpow_neg44_eq_inv]
 177  have hupper : phi ^ (44 : ℕ) < 1.5698e9 := by
 178    rw [phi_pow_44_fib]
 179    have hphi_lt : phi < 1.62 := phi_lt_onePointSixTwo
 180    nlinarith
 181  have hpos : (0 : ℝ) < phi ^ (44 : ℕ) := pow_pos phi_pos 44
 182  have h1 : (phi ^ (44 : ℕ))⁻¹ > (1.5698e9 : ℝ)⁻¹ := by
 183    rw [gt_iff_lt, inv_lt_inv₀ (by norm_num : (0:ℝ) < 1.5698e9) hpos]
 184    exact hupper
 185  have h2 : (1.5698e9 : ℝ)⁻¹ ≥ 6.37e-10 := by norm_num
 186  linarith
 187
 188theorem phi_zpow_neg44_upper : phi ^ (-44 : ℤ) < 6.40e-10 := by
 189  rw [phi_zpow_neg44_eq_inv]
 190  have hlower : phi ^ (44 : ℕ) > 1.5627e9 := by
 191    rw [phi_pow_44_fib]
 192    have hphi_gt : phi > 1.61 := phi_gt_onePointSixOne
 193    nlinarith
 194  have hpos : (0 : ℝ) < phi ^ (44 : ℕ) := pow_pos phi_pos 44
 195  have h1 : (phi ^ (44 : ℕ))⁻¹ < (1.5627e9 : ℝ)⁻¹ := by
 196    rw [inv_lt_inv₀ hpos (by norm_num : (0:ℝ) < 1.5627e9)]
 197    exact hlower
 198  have h2 : (1.5627e9 : ℝ)⁻¹ < 6.40e-10 := by norm_num
 199  linarith
 200
 201/-! ## The corrected η_B prediction -/
 202
 203/-- The corrected RS prediction for η_B: `c_RS · φ^(−44)`. -/
 204def eta_B_corrected : ℝ := c_RS * phi ^ (-44 : ℤ)
 205
 206theorem eta_B_corrected_pos : 0 < eta_B_corrected := by
 207  unfold eta_B_corrected
 208  exact mul_pos c_RS_pos (zpow_pos phi_pos (-44))
 209
 210/-- The corrected η_B prediction is greater than `6.0 × 10⁻¹⁰`. -/
 211theorem eta_B_corrected_lower : eta_B_corrected > 6.0e-10 := by
 212  unfold eta_B_corrected
 213  have hc : c_RS > 0.956 := c_RS_lower
 214  have hphi : phi ^ (-44 : ℤ) > 6.37e-10 := phi_zpow_neg44_lower
 215  have hcpos : (0 : ℝ) < c_RS := c_RS_pos
 216  have hphi_pos : (0 : ℝ) < phi ^ (-44 : ℤ) := zpow_pos phi_pos (-44)
 217  nlinarith [hc, hphi, hcpos, hphi_pos]
 218
 219/-- The corrected η_B prediction is less than `6.2 × 10⁻¹⁰`. -/
 220theorem eta_B_corrected_upper : eta_B_corrected < 6.2e-10 := by
 221  unfold eta_B_corrected
 222  have hc : c_RS < 0.959 := c_RS_upper
 223  have hphi : phi ^ (-44 : ℤ) < 6.40e-10 := phi_zpow_neg44_upper
 224  have hcpos : (0 : ℝ) < c_RS := c_RS_pos
 225  have hphi_pos : (0 : ℝ) < phi ^ (-44 : ℤ) := zpow_pos phi_pos (-44)
 226  nlinarith [hc, hphi, hcpos, hphi_pos]
 227
 228/-- The η_B band: `c_RS · φ^(−44) ∈ (6.0 × 10⁻¹⁰, 6.2 × 10⁻¹⁰)`. -/
 229theorem eta_B_corrected_in_observed_band :
 230    eta_B_corrected > 6.0e-10 ∧ eta_B_corrected < 6.2e-10 :=
 231  ⟨eta_B_corrected_lower, eta_B_corrected_upper⟩
 232
 233/-- The Planck 2018 central value `6.10 × 10⁻¹⁰` lies inside the predicted band. -/
 234theorem observed_in_predicted_band :
 235    (6.0e-10 : ℝ) < 6.10e-10 ∧ (6.10e-10 : ℝ) < 6.2e-10 := by
 236  constructor <;> norm_num
 237
 238end -- noncomputable section
 239
 240/-! ## Master certificate -/
 241
 242structure EtaBPrefactorCert where
 243  prefactor_in_unit : 0 < c_RS ∧ c_RS < 1
 244  prefactor_band : c_RS > 0.956 ∧ c_RS < 0.959
 245  expanded : c_RS = (1 - phi ^ (-8 : ℤ)) ^ 2
 246  prediction_band :
 247    eta_B_corrected > 6.0e-10 ∧ eta_B_corrected < 6.2e-10
 248  observed_inside : (6.0e-10 : ℝ) < 6.10e-10 ∧ (6.10e-10 : ℝ) < 6.2e-10
 249
 250theorem etaBPrefactorCert : EtaBPrefactorCert where
 251  prefactor_in_unit := ⟨c_RS_pos, c_RS_lt_one⟩
 252  prefactor_band := ⟨c_RS_lower, c_RS_upper⟩
 253  expanded := c_RS_expanded
 254  prediction_band := eta_B_corrected_in_observed_band
 255  observed_inside := observed_in_predicted_band
 256
 257end EtaBPrefactorDerivation
 258end Cosmology
 259end IndisputableMonolith
 260