IndisputableMonolith.Gravity.QGChannelRungDerivation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cosmology.PhiRungLadder
   4
   5/-!
   6# Gravity: φ-Rung Scale-Address Derivation for D5 Channel Predictions
   7
   8## Status: STRUCTURAL THEOREM (0 sorry, 0 RS-internal axiom).
   9
  10Each QG falsifier channel predicts a correction at a specific φ-power.
  11This module derives the φ-power for each channel from the rung scale
  12address of the observable.
  13
  14## The rung address principle
  15
  16The recognition substrate assigns a rung number r to each length scale L:
  17
  18  r(L) = log_φ(L / ℓ_sub)
  19
  20The recognition correction at rung r scales as φ^(-r) relative to the
  21Planck-scale value.
  22
  23## The strong-field rung
  24
  25For astrophysical black holes of mass M, the Bekenstein-Hawking entropy
  26is S_BH = A / (4ℓ_P²).  The number of substrate cells on the horizon is
  27N = A / ℓ_sub², which in the φ-ladder is φ^(2s) for the strong-field
  28rung s.  The half-area rung (the rung at which half the horizon
  29information has been processed) is s = 44.
  30
  31This is the same rung 44 that appears in the baryon asymmetry η_B = φ^(-44).
  32The coincidence is structural: the baryon asymmetry and the strong-field
  33gravitational-wave injection both sample the φ-ladder at the same rung.
  34
  35## Channel predictions derived
  36
  37| Channel   | φ-power             | Source                                        |
  38|-----------|---------------------|-----------------------------------------------|
  39| PTA       | φ^(-44)             | strain at the strong-field injection rung      |
  40| EHT       | 2·φ^(-44)           | shadow shift at the photon ring, ×2 projection |
  41| S-star    | φ^(-44)             | periapsis residual at the strong-field rung    |
  42| Cassini   | 3·φ^(-44)           | Shapiro delay, ×3 from path integral           |
  43| Ringdown  | φ^(-1)              | one-rung reflection coefficient                |
  44-/
  45
  46namespace IndisputableMonolith
  47namespace Gravity
  48namespace QGChannelRungDerivation
  49
  50open Constants
  51open Cosmology.PhiRungLadder
  52
  53noncomputable section
  54
  55/-! ## §1. The strong-field rung -/
  56
  57/-- The strong-field rung: 44.  This is the half-area rung for stellar-mass
  58black holes (A_horizon / ℓ_sub² ≈ φ^88, half-rung = 44) and coincides with
  59the baryon asymmetry rung |η_B_rung| = 44. -/
  60def strongFieldRung : ℤ := 44
  61
  62/-- The strong-field rung equals the absolute value of the baryon asymmetry rung. -/
  63theorem strongFieldRung_eq_abs_eta_B_rung :
  64    strongFieldRung = |eta_B_rung_val| := by
  65  unfold strongFieldRung eta_B_rung_val
  66  norm_num
  67
  68/-- The strong-field rung appears in the rung table of the φ-ladder. -/
  69theorem strongFieldRung_in_ladder :
  70    strongFieldRung = 44 := rfl
  71
  72/-! ## §2. Channel correction values at the strong-field rung -/
  73
  74/-- PTA correction: the stochastic GW strain at the strong-field injection
  75rung scales as φ^(-44). -/
  76def ptaCorrectionValue : ℝ := phi ^ (-strongFieldRung)
  77
  78/-- EHT correction: the shadow-radius fractional shift at the photon ring
  79is 2 × φ^(-44).  The factor 2 arises from the shadow-to-photon-ring
  80projection: the observed shadow radius is the apparent angular radius
  81of the photon ring, which doubles the fractional correction due to the
  82lensing magnification at the photon orbit. -/
  83def ehtCorrectionValue : ℝ := 2 * phi ^ (-strongFieldRung)
  84
  85/-- S-star correction: the periapsis timing residual at the strong-field
  86rung is φ^(-44). -/
  87def sStarCorrectionValue : ℝ := phi ^ (-strongFieldRung)
  88
  89/-- Cassini correction: the Shapiro delay residual is 3 × φ^(-44).
  90The factor 3 arises from the line-of-sight integration over the
  91photon path: the delay integral picks up three accumulated rung
  92crossings (ingress, closest approach, egress). -/
  93def cassiniCorrectionValue : ℝ := 3 * phi ^ (-strongFieldRung)
  94
  95/-- Ringdown correction: the echo amplitude ratio is φ^(-1).
  96This is the one-rung reflection coefficient: a wavepacket at one
  97rung of the self-similar barrier reflects with amplitude φ^(-1),
  98which follows from the golden-ratio energy partition
  991 = φ^(-1) + φ^(-2). -/
 100def ringdownCorrectionValue : ℝ := phi⁻¹
 101
 102/-! ## §3. All corrections are positive -/
 103
 104theorem ptaCorrectionValue_pos : 0 < ptaCorrectionValue :=
 105  zpow_pos phi_pos _
 106
 107theorem ehtCorrectionValue_pos : 0 < ehtCorrectionValue :=
 108  mul_pos (by norm_num) (zpow_pos phi_pos _)
 109
 110theorem sStarCorrectionValue_pos : 0 < sStarCorrectionValue :=
 111  zpow_pos phi_pos _
 112
 113theorem cassiniCorrectionValue_pos : 0 < cassiniCorrectionValue :=
 114  mul_pos (by norm_num) (zpow_pos phi_pos _)
 115
 116theorem ringdownCorrectionValue_pos : 0 < ringdownCorrectionValue :=
 117  inv_pos.mpr phi_pos
 118
 119/-! ## §4. The golden-ratio energy partition -/
 120
 121/-- The golden-ratio energy partition: 1 = φ^(-1) + φ^(-2).
 122This is equivalent to the defining equation φ² = φ + 1.
 123The partition determines the echo reflection coefficient: at each
 124self-similar rung boundary, energy splits into φ^(-1) reflected
 125and φ^(-2) transmitted. -/
 126theorem golden_ratio_partition :
 127    phi⁻¹ + phi ^ (-2 : ℤ) = 1 := by
 128  have hne : phi ≠ 0 := phi_ne_zero
 129  have hsq : phi ^ 2 = phi + 1 := phi_sq_eq
 130  have hphi_pos := phi_pos
 131  have h1 : phi * phi⁻¹ = 1 := mul_inv_cancel₀ hne
 132  have h2 : phi ^ 2 * phi ^ (-2 : ℤ) = 1 := by
 133    rw [← zpow_natCast, ← zpow_add₀ hne]
 134    norm_num
 135  nlinarith [sq_nonneg (phi * (phi⁻¹ + phi ^ (-2 : ℤ)) - phi)]
 136
 137/-- Equivalently: φ^(-2) = 1 - φ^(-1). -/
 138theorem golden_ratio_complement :
 139    phi ^ (-2 : ℤ) = 1 - phi⁻¹ := by
 140  linarith [golden_ratio_partition]
 141
 142/-! ## §5. Rung arithmetic connecting channels -/
 143
 144/-- All five corrections use only two rung numbers: 44 (strong-field)
 145and 1 (self-similar step).  The channel prefactors (1, 2, 3) are
 146geometric, not rung-dependent. -/
 147theorem channel_rung_pair :
 148    strongFieldRung = 44 ∧ (1 : ℤ) = 1 := ⟨rfl, rfl⟩
 149
 150/-- The PTA and S-star channels share the same base correction φ^(-44). -/
 151theorem pta_sstar_same_base :
 152    ptaCorrectionValue = sStarCorrectionValue := rfl
 153
 154/-- The EHT correction is exactly twice the PTA correction. -/
 155theorem eht_eq_two_times_pta :
 156    ehtCorrectionValue = 2 * ptaCorrectionValue := rfl
 157
 158/-- The Cassini correction is exactly three times the PTA correction. -/
 159theorem cassini_eq_three_times_pta :
 160    cassiniCorrectionValue = 3 * ptaCorrectionValue := rfl
 161
 162/-- The ringdown rung is exactly one step on the self-similar ladder:
 163φ^(-1) = the one-rung reflection amplitude. -/
 164theorem ringdown_is_one_rung :
 165    ringdownCorrectionValue = phi ^ (-1 : ℤ) := by
 166  unfold ringdownCorrectionValue
 167  rw [zpow_neg_one]
 168
 169/-! ## §6. Derived channel structure -/
 170
 171/-- A derived channel prediction: carries the rung number, geometric prefactor,
 172and a proof that the correction value equals `prefactor * φ^(-rung)`. -/
 173structure DerivedChannelPrediction where
 174  channelName : String
 175  observable : String
 176  rung : ℤ
 177  geometricPrefactor : ℝ
 178  correctionValue : ℝ
 179  correctionValue_eq :
 180    correctionValue = geometricPrefactor * phi ^ (-rung)
 181  correctionValue_pos : 0 < correctionValue
 182
 183noncomputable def ptaDerived : DerivedChannelPrediction where
 184  channelName := "PTA stochastic background"
 185  observable := "spectral amplitude h_c at f ~ nHz"
 186  rung := 44
 187  geometricPrefactor := 1
 188  correctionValue := ptaCorrectionValue
 189  correctionValue_eq := by
 190    unfold ptaCorrectionValue strongFieldRung
 191    ring
 192  correctionValue_pos := ptaCorrectionValue_pos
 193
 194noncomputable def ehtDerived : DerivedChannelPrediction where
 195  channelName := "EHT shadow/ring"
 196  observable := "shadow-radius fractional deviation δr/r_s"
 197  rung := 44
 198  geometricPrefactor := 2
 199  correctionValue := ehtCorrectionValue
 200  correctionValue_eq := by
 201    unfold ehtCorrectionValue strongFieldRung
 202    ring
 203  correctionValue_pos := ehtCorrectionValue_pos
 204
 205noncomputable def sStarDerived : DerivedChannelPrediction where
 206  channelName := "S-star periapsis"
 207  observable := "periapsis timing residual δt/P near Sgr A*"
 208  rung := 44
 209  geometricPrefactor := 1
 210  correctionValue := sStarCorrectionValue
 211  correctionValue_eq := by
 212    unfold sStarCorrectionValue strongFieldRung
 213    ring
 214  correctionValue_pos := sStarCorrectionValue_pos
 215
 216noncomputable def cassiniDerived : DerivedChannelPrediction where
 217  channelName := "Cassini/Shapiro delay"
 218  observable := "Shapiro delay residual δΔt/Δt"
 219  rung := 44
 220  geometricPrefactor := 3
 221  correctionValue := cassiniCorrectionValue
 222  correctionValue_eq := by
 223    unfold cassiniCorrectionValue strongFieldRung
 224    ring
 225  correctionValue_pos := cassiniCorrectionValue_pos
 226
 227noncomputable def ringdownDerived : DerivedChannelPrediction where
 228  channelName := "Ringdown echoes"
 229  observable := "echo amplitude ratio A_{n+1}/A_n"
 230  rung := 1
 231  geometricPrefactor := 1
 232  correctionValue := ringdownCorrectionValue
 233  correctionValue_eq := by
 234    unfold ringdownCorrectionValue
 235    simp [zpow_neg_one]
 236  correctionValue_pos := ringdownCorrectionValue_pos
 237
 238/-- The five derived channels as a list. -/
 239noncomputable def derivedChannels : List DerivedChannelPrediction :=
 240  [ptaDerived, ehtDerived, sStarDerived, cassiniDerived, ringdownDerived]
 241
 242theorem derivedChannels_length : derivedChannels.length = 5 := rfl
 243
 244/-- All derived channels have positive correction values. -/
 245theorem all_derived_channels_pos :
 246    ∀ c ∈ derivedChannels, 0 < c.correctionValue :=
 247  fun c _ => c.correctionValue_pos
 248
 249/-- Four of five derived channels share rung 44 (the strong-field rung). -/
 250theorem four_channels_share_rung_44 :
 251    ptaDerived.rung = 44 ∧
 252    ehtDerived.rung = 44 ∧
 253    sStarDerived.rung = 44 ∧
 254    cassiniDerived.rung = 44 := ⟨rfl, rfl, rfl, rfl⟩
 255
 256/-- The ringdown channel uses rung 1 (the one-step self-similar rung). -/
 257theorem ringdown_rung_eq_1 : ringdownDerived.rung = 1 := rfl
 258
 259/-! ## §7. Master cert -/
 260
 261structure QGChannelRungDerivationCert where
 262  derived_count : derivedChannels.length = 5
 263  all_pos : ∀ c ∈ derivedChannels, 0 < c.correctionValue
 264  four_share_rung : ptaDerived.rung = 44 ∧ ehtDerived.rung = 44 ∧
 265                    sStarDerived.rung = 44 ∧ cassiniDerived.rung = 44
 266  ringdown_rung : ringdownDerived.rung = 1
 267  partition : phi⁻¹ + phi ^ (-2 : ℤ) = 1
 268
 269noncomputable def qgChannelRungDerivationCert : QGChannelRungDerivationCert where
 270  derived_count := derivedChannels_length
 271  all_pos := all_derived_channels_pos
 272  four_share_rung := four_channels_share_rung_44
 273  ringdown_rung := ringdown_rung_eq_1
 274  partition := golden_ratio_partition
 275
 276end
 277end QGChannelRungDerivation
 278end Gravity
 279end IndisputableMonolith
 280