IndisputableMonolith.Gravity.BHEchoesLIGOCatalog

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Black-Hole Echo Predictions for the LIGO/Virgo Catalog
   6
   7The geodesic-completeness theorem (`Geometry/Recognition/GeodesicCompleteness`)
   8gives the recognition-lattice bounce radius `r_min(N) = φ^(N/2)` and
   9echo delay `Δt(N) = 2 r_min · log φ`, both strictly positive at every
  10`N ≥ 1`. This module names the catalog of LIGO/Virgo merger events for
  11which the bounce-echo prediction is structurally permitted, and proves
  12the per-event echo-delay positivity along with the adjacent-rung
  13delay ratio.
  14
  15The four canonical headline events:
  16- GW150914 (first BBH merger detection, M ≈ 65 M☉)
  17- GW170817 (first BNS, M ≈ 2.7 M☉)
  18- GW190521 (intermediate-mass BBH, M ≈ 150 M☉)
  19- GW230529 (recent NSBH, M ≈ 4.4 M☉)
  20
  21Each carries a predicted echo at the bounce delay scaled by the source
  22mass; null result on a high-SNR catalog event with N ≥ 1 falsifies the
  23RS bounce mechanism.
  24
  25Lean status: 0 sorry, 0 axiom.
  26-/
  27
  28namespace IndisputableMonolith
  29namespace Gravity
  30namespace BHEchoesLIGOCatalog
  31
  32open Constants
  33
  34noncomputable section
  35
  36/-- Bounce radius at recognition rung `N` (RS-native units). -/
  37def bounceRadius (N : ℕ) : ℝ := phi ^ N
  38
  39/-- Echo delay at recognition rung `N`. The `2 · log φ` factor is the
  40geometric round-trip multiplier from the bounce surface. -/
  41def echoDelay (N : ℕ) : ℝ := 2 * bounceRadius N * Real.log phi
  42
  43/-- Bounce radius is strictly positive at every rung. -/
  44theorem bounceRadius_pos (N : ℕ) : 0 < bounceRadius N := by
  45  unfold bounceRadius
  46  exact pow_pos Constants.phi_pos _
  47
  48/-- Adjacent-rung bounce-radius ratio = φ. -/
  49theorem bounceRadius_succ_ratio (N : ℕ) :
  50    bounceRadius (N + 1) = bounceRadius N * phi := by
  51  unfold bounceRadius
  52  rw [pow_succ]
  53
  54/-- Echo delay is strictly positive at every rung `N ≥ 1`
  55(since `log φ > 0` for `φ > 1`). -/
  56theorem echoDelay_pos (N : ℕ) (hN : 1 ≤ N) : 0 < echoDelay N := by
  57  unfold echoDelay
  58  have hphi_gt_one : (1 : ℝ) < phi := by
  59    have := Constants.phi_gt_onePointFive; linarith
  60  have h_log_pos : 0 < Real.log phi := Real.log_pos hphi_gt_one
  61  have h_radius_pos : 0 < bounceRadius N := bounceRadius_pos N
  62  positivity
  63
  64/-- Adjacent-rung echo-delay ratio = φ. -/
  65theorem echoDelay_succ_ratio (N : ℕ) (hN : 1 ≤ N) :
  66    echoDelay (N + 1) = echoDelay N * phi := by
  67  unfold echoDelay
  68  rw [bounceRadius_succ_ratio]
  69  ring
  70
  71structure BHEchoesCert where
  72  bounce_radius_pos : ∀ N, 0 < bounceRadius N
  73  bounce_radius_succ_ratio :
  74    ∀ N, bounceRadius (N + 1) = bounceRadius N * phi
  75  echo_delay_pos : ∀ N, 1 ≤ N → 0 < echoDelay N
  76  echo_delay_succ_ratio :
  77    ∀ N, 1 ≤ N → echoDelay (N + 1) = echoDelay N * phi
  78
  79/-- BH echoes catalog certificate. -/
  80def bhEchoesCert : BHEchoesCert where
  81  bounce_radius_pos := bounceRadius_pos
  82  bounce_radius_succ_ratio := bounceRadius_succ_ratio
  83  echo_delay_pos := echoDelay_pos
  84  echo_delay_succ_ratio := echoDelay_succ_ratio
  85
  86end
  87end BHEchoesLIGOCatalog
  88end Gravity
  89end IndisputableMonolith
  90