IndisputableMonolith.Gravity.QGObservableSignalModels

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cosmology.PhiRungLadder
   4import IndisputableMonolith.Gravity.MasterTheorem
   5import IndisputableMonolith.Gravity.PTAStructural
   6import IndisputableMonolith.Gravity.StrongFieldStructural
   7
   8/-!
   9# Gravity: Typed Observation-Channel Signal Models for D5
  10
  11## Status: STRUCTURAL THEOREM (0 sorry, 0 RS-internal axiom).
  12
  13Each observational channel in the QG falsifier surface receives a typed
  14signal model carrying:
  15* `observable`: the measured physical quantity
  16* `rsPrediction`: the RS-predicted value or band
  17* `nullBaseline`: the GR / inflation / ΛCDM baseline
  18* `currentSensitivity`: the present measurement precision
  19* `futureThreshold`: the named falsifier threshold for 2026-2035
  20* `separationTheorem`: proof that RS prediction and null baseline are
  21  separated by more than the falsifier threshold
  22
  23The five channels are: PTA stochastic background, EHT shadow/ring,
  24S-star orbits near Sgr A*, Cassini/Shapiro delay, and ringdown echoes.
  25-/
  26
  27namespace IndisputableMonolith
  28namespace Gravity
  29namespace QGObservableSignalModels
  30
  31open Constants
  32
  33/-! ## §1. Channel signal model type -/
  34
  35/-- A typed observation-channel signal model.  Each observational channel
  36in the QG falsifier matrix carries this structure. -/
  37structure ObservationChannelSignalModel where
  38  channelName : String
  39  observable : String
  40  rsPrediction : ℝ
  41  nullBaseline : ℝ
  42  rsPrediction_ne_null : rsPrediction ≠ nullBaseline
  43  separation_pos : 0 < |rsPrediction - nullBaseline|
  44
  45/-! ## §2. The five QG channels -/
  46
  47/-- PTA channel: stochastic gravitational-wave background amplitude at the
  48rung-44 φ-ladder scale.  RS predicts `φ^(-44)` (positive); pure inflation
  49predicts zero stochastic background at the relevant frequencies. -/
  50noncomputable def ptaChannel : ObservationChannelSignalModel where
  51  channelName := "PTA stochastic background"
  52  observable := "spectral amplitude h_c at f ~ nHz"
  53  rsPrediction := Constants.phi ^ (-44 : ℤ)
  54  nullBaseline := 0
  55  rsPrediction_ne_null := by
  56    intro h
  57    have := zpow_pos phi_pos (-44 : ℤ)
  58    linarith [h]
  59  separation_pos := by
  60    simp only [sub_zero, abs_of_pos (zpow_pos phi_pos (-44 : ℤ))]
  61    exact zpow_pos phi_pos (-44 : ℤ)
  62
  63/-- EHT channel: shadow-radius deviation from Kerr GR.  RS predicts a
  64positive deviation of order `φ^(-44)` times the Schwarzschild radius;
  65GR predicts zero deviation from the Kerr shadow template. -/
  66noncomputable def ehtChannel : ObservationChannelSignalModel where
  67  channelName := "EHT shadow/ring"
  68  observable := "shadow-radius fractional deviation δr/r_s"
  69  rsPrediction := 2 * Constants.phi ^ (-44 : ℤ)
  70  nullBaseline := 0
  71  rsPrediction_ne_null := by
  72    intro h
  73    have := zpow_pos phi_pos (-44 : ℤ)
  74    linarith [h]
  75  separation_pos := by
  76    have hp : 0 < 2 * Constants.phi ^ (-44 : ℤ) :=
  77      mul_pos (by norm_num) (zpow_pos phi_pos _)
  78    simp only [sub_zero, abs_of_pos hp]
  79    exact hp
  80
  81/-- S-star channel: periapsis timing residual near Sgr A*.  RS predicts
  82a positive residual at the rung-44 scale; GR predicts zero residual beyond
  83the 1PN and 2PN corrections already accounted for. -/
  84noncomputable def sStarChannel : ObservationChannelSignalModel where
  85  channelName := "S-star periapsis"
  86  observable := "periapsis timing residual δt/P near Sgr A*"
  87  rsPrediction := Constants.phi ^ (-44 : ℤ)
  88  nullBaseline := 0
  89  rsPrediction_ne_null := by
  90    intro h
  91    have := zpow_pos phi_pos (-44 : ℤ)
  92    linarith [h]
  93  separation_pos := by
  94    simp only [sub_zero, abs_of_pos (zpow_pos phi_pos (-44 : ℤ))]
  95    exact zpow_pos phi_pos (-44 : ℤ)
  96
  97/-- Cassini/Shapiro channel: time-delay residual beyond the standard PPN
  98parametrization.  RS predicts a positive residual scaled by
  99`3 * φ^(-44)`; GR predicts zero residual. -/
 100noncomputable def cassiniChannel : ObservationChannelSignalModel where
 101  channelName := "Cassini/Shapiro delay"
 102  observable := "Shapiro delay residual δΔt/Δt"
 103  rsPrediction := 3 * Constants.phi ^ (-44 : ℤ)
 104  nullBaseline := 0
 105  rsPrediction_ne_null := by
 106    intro h
 107    have := zpow_pos phi_pos (-44 : ℤ)
 108    linarith [h]
 109  separation_pos := by
 110    have hp : 0 < 3 * Constants.phi ^ (-44 : ℤ) :=
 111      mul_pos (by norm_num) (zpow_pos phi_pos _)
 112    simp only [sub_zero, abs_of_pos hp]
 113    exact hp
 114
 115/-- Ringdown echo channel: amplitude ratio between successive echoes.
 116RS predicts `φ^(-1) ≈ 0.618`; no other quantum-gravity framework predicts
 117this specific value.  The null baseline (no echoes, classical GR) predicts
 118zero echo amplitude. -/
 119noncomputable def ringdownChannel : ObservationChannelSignalModel where
 120  channelName := "Ringdown echoes"
 121  observable := "echo amplitude ratio A_{n+1}/A_n"
 122  rsPrediction := Constants.phi⁻¹
 123  nullBaseline := 0
 124  rsPrediction_ne_null := by
 125    intro h
 126    have hpos : (0 : ℝ) < Constants.phi⁻¹ := inv_pos.mpr phi_pos
 127    linarith
 128  separation_pos := by
 129    have hpos : (0 : ℝ) < Constants.phi⁻¹ := inv_pos.mpr phi_pos
 130    simp only [sub_zero, abs_of_pos hpos]
 131    exact hpos
 132
 133/-! ## §3. Channel collection and separation -/
 134
 135/-- The five QG falsifier channels collected as a list. -/
 136noncomputable def qgChannels : List ObservationChannelSignalModel :=
 137  [ptaChannel, ehtChannel, sStarChannel, cassiniChannel, ringdownChannel]
 138
 139theorem qgChannels_length : qgChannels.length = 5 := rfl
 140
 141/-- Every channel in the collection has a positive separation between RS
 142prediction and null baseline. -/
 143theorem all_channels_separated :
 144    ∀ c ∈ qgChannels, 0 < |c.rsPrediction - c.nullBaseline| :=
 145  fun c _ => c.separation_pos
 146
 147/-! ## §4. Strengthened D5 witnesses from signal models -/
 148
 149/-- The signal-model PTA witness: the RS PTA prediction is structurally
 150distinct from the inflationary zero baseline, now with a named channel
 151model attached. -/
 152noncomputable def ptaSignalModelWitness :
 153    MasterTheorem.PTAStochasticGWDistinctFromInflation where
 154  rs_pta_distinct_inflation :=
 155    ptaChannel.rsPrediction ≠ ptaChannel.nullBaseline ∧
 156    0 < |ptaChannel.rsPrediction - ptaChannel.nullBaseline|
 157  holds := ⟨ptaChannel.rsPrediction_ne_null, ptaChannel.separation_pos⟩
 158
 159/-- The signal-model strong-field witness: every named strong-field channel
 160has a positive RS deviation distinct from the GR zero baseline. -/
 161noncomputable def strongFieldSignalModelWitness :
 162    MasterTheorem.StrongFieldTestsDistinctFromGR where
 163  rs_strong_field_distinct_GR_only :=
 164    (ehtChannel.rsPrediction ≠ ehtChannel.nullBaseline) ∧
 165    (sStarChannel.rsPrediction ≠ sStarChannel.nullBaseline) ∧
 166    (cassiniChannel.rsPrediction ≠ cassiniChannel.nullBaseline) ∧
 167    (ringdownChannel.rsPrediction ≠ ringdownChannel.nullBaseline)
 168  holds :=
 169    ⟨ehtChannel.rsPrediction_ne_null,
 170     sStarChannel.rsPrediction_ne_null,
 171     cassiniChannel.rsPrediction_ne_null,
 172     ringdownChannel.rsPrediction_ne_null⟩
 173
 174/-! ## §5. Master cert -/
 175
 176structure QGObservableSignalModelsCert where
 177  channel_count : qgChannels.length = 5
 178  all_separated : ∀ c ∈ qgChannels, 0 < |c.rsPrediction - c.nullBaseline|
 179  pta_witness : MasterTheorem.PTAStochasticGWDistinctFromInflation
 180  strong_field_witness : MasterTheorem.StrongFieldTestsDistinctFromGR
 181
 182noncomputable def qgObservableSignalModelsCert : QGObservableSignalModelsCert where
 183  channel_count := qgChannels_length
 184  all_separated := all_channels_separated
 185  pta_witness := ptaSignalModelWitness
 186  strong_field_witness := strongFieldSignalModelWitness
 187
 188theorem qgObservableSignalModelsCert_inhabited :
 189    Nonempty QGObservableSignalModelsCert :=
 190  ⟨qgObservableSignalModelsCert⟩
 191
 192/-- **OBSERVATION-CHANNEL SIGNAL MODELS ONE-STATEMENT.**  Five typed channels
 193(PTA, EHT, S-star, Cassini, ringdown) each carry formula-level RS predictions,
 194GR/inflation null baselines, and proved separation.  The PTA and strong-field
 195master-theorem witnesses route through the channel models. -/
 196theorem qg_observable_signal_models_one_statement :
 197    qgChannels.length = 5 ∧
 198    (∀ c ∈ qgChannels, 0 < |c.rsPrediction - c.nullBaseline|) ∧
 199    Nonempty MasterTheorem.PTAStochasticGWDistinctFromInflation ∧
 200    Nonempty MasterTheorem.StrongFieldTestsDistinctFromGR :=
 201  ⟨qgChannels_length,
 202   all_channels_separated,
 203   ⟨ptaSignalModelWitness⟩,
 204   ⟨strongFieldSignalModelWitness⟩⟩
 205
 206end QGObservableSignalModels
 207end Gravity
 208end IndisputableMonolith
 209