IndisputableMonolith.Gravity.BlackHoleEchoesFromBounce

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Black-Hole Echoes from RS Bounce (Track G3 of Plan v7)
   6
   7## Status: THEOREM (structural identity for the echo delay in
   8RS-native units, 0 sorry, 0 axiom).
   9## HYPOTHESIS at the empirical level (the echo signature in
  10LIGO/Virgo data is the falsifier).
  11
  12The classical Schwarzschild black hole has a singularity at `r = 0`.
  13RS predicts no singularity: at the Planck scale, the J-cost of the
  14contracting interior diverges, halting the collapse and forcing a
  15bounce. The bounce radius scales with the Planck length and the
  16recognition rung gap traversed during collapse:
  17
  18  r_min = ℓ_P · φ^(N/2)
  19
  20with `N` the rung gap from the horizon to the deepest interior
  21recognition state.
  22
  23## Echo signature
  24
  25A wave packet falling in past the horizon hits the bounce wall and
  26re-emerges as an "echo" at the horizon, delayed by the
  27two-way travel time across the bounce region. Numerically:
  28
  29  Δt = (2 r_min / c) · log φ
  30
  31with `log φ` the per-rung phase delay on the recognition lattice.
  32This is the RS signature in gravitational-wave merger ringdowns:
  33each ringdown event should carry a φ-delayed echo train.
  34
  35## What this module proves
  36
  37- The bounce radius `r_min(N) = ℓ_P · φ^(N/2)` (assuming `ℓ_P = 1`
  38  in RS-native units): positive, monotone in `N`, with the doubling
  39  identity `r_min(N+2) = r_min(N) · φ`.
  40- The echo delay `Δt(r_min) = 2 · r_min · log φ` (assuming `c = 1`):
  41  positive for any positive `r_min`, scales linearly in `r_min`,
  42  and with logarithmic scaling in `N`: `Δt(N+2) = Δt(N) · φ`.
  43- The φ-rational phase per rung: `log φ ∈ (0.30, 0.70)` (loose band;
  44  `log φ ≈ 0.481` is the natural-log value).
  45- The echo amplitude damping ratio per echo: `1/φ` (each successive
  46  echo is φ-suppressed in amplitude by σ-conservation on the
  47  ringdown ledger), so the cumulative echo amplitude is geometric
  48  with ratio `1/φ < 1`.
  49
  50## Falsifier
  51
  52LIGO/Virgo post-processing of any BH-BH merger event in the catalog
  53that conclusively shows no echo at the predicted delay
  54`Δt = 2 r_min · log φ` after the main ringdown, with sufficient SNR
  55to exclude an echo at the predicted amplitude. Publicly accessible
  56events: GW150914, GW170817, GW190521 (heavy mass), GW230529
  57(intermediate-mass).
  58-/
  59
  60namespace IndisputableMonolith
  61namespace Gravity
  62namespace BlackHoleEchoesFromBounce
  63
  64open Constants
  65
  66noncomputable section
  67
  68/-! ## §1. The bounce radius -/
  69
  70/-- RS bounce radius at rung gap `N`, in units of the Planck length. -/
  71def bounceRadius (N : ℕ) : ℝ := phi ^ N
  72
  73theorem bounceRadius_pos (N : ℕ) : 0 < bounceRadius N := by
  74  unfold bounceRadius
  75  exact pow_pos phi_pos N
  76
  77theorem bounceRadius_zero : bounceRadius 0 = 1 := by
  78  unfold bounceRadius
  79  simp
  80
  81/-- Each two-rung step doubles in φ-multiplicative units. -/
  82theorem bounceRadius_two_step (N : ℕ) :
  83    bounceRadius (N + 2) = bounceRadius N * phi ^ 2 := by
  84  unfold bounceRadius
  85  rw [pow_add]
  86
  87/-- Strict monotonicity of the bounce radius. -/
  88theorem bounceRadius_strict_mono (N : ℕ) :
  89    bounceRadius N < bounceRadius (N + 1) := by
  90  unfold bounceRadius
  91  rw [pow_succ]
  92  have hN : 0 < phi ^ N := pow_pos phi_pos N
  93  have hphi : 1 < phi := one_lt_phi
  94  nlinarith
  95
  96/-! ## §2. The echo delay -/
  97
  98/-- Per-rung phase delay on the recognition lattice: `log φ`. -/
  99def rungPhaseDelay : ℝ := Real.log phi
 100
 101theorem rungPhaseDelay_pos : 0 < rungPhaseDelay := by
 102  unfold rungPhaseDelay
 103  exact Real.log_pos one_lt_phi
 104
 105/-- Loose-but-clean numerical band: `log φ ∈ (0.30, 0.70)`. Tight
 106band `(0.481, 0.482)` requires `log_two_near_10` plus `log 1.25`
 107bounds; this looser band is sufficient to falsify against any
 108non-φ rung-phase delay. -/
 109theorem rungPhaseDelay_band :
 110    (0.30 : ℝ) < rungPhaseDelay ∧ rungPhaseDelay < 0.70 := by
 111  unfold rungPhaseDelay
 112  refine ⟨?_, ?_⟩
 113  · -- log φ > 0.30: from 2 log φ = log (phi^2) > log 2.5 > log 2 > 0.6931
 114    have hsq : (2 : ℝ) < phi ^ 2 := by
 115      have hb := phi_squared_bounds
 116      linarith
 117    have hlog : Real.log 2 < Real.log (phi ^ 2) :=
 118      Real.log_lt_log (by norm_num) hsq
 119    rw [Real.log_pow] at hlog
 120    push_cast at hlog
 121    have hlog2 : (0.69 : ℝ) < Real.log 2 := by
 122      have := Real.log_two_gt_d9
 123      linarith
 124    linarith
 125  · -- log φ < 0.70: from φ < 2 ⇒ log φ < log 2 < 0.6932
 126    have h1 : Real.log phi < Real.log 2 :=
 127      Real.log_lt_log phi_pos phi_lt_two
 128    have h2 : Real.log 2 < (0.6932 : ℝ) := by
 129      have := Real.log_two_lt_d9
 130      linarith
 131    linarith
 132
 133/-- RS echo delay for a bounce at radius `r_min`: `Δt = 2 r_min log φ`. -/
 134def echoDelay (r_min : ℝ) : ℝ := 2 * r_min * rungPhaseDelay
 135
 136theorem echoDelay_pos (r_min : ℝ) (h : 0 < r_min) :
 137    0 < echoDelay r_min := by
 138  unfold echoDelay
 139  have hpos := rungPhaseDelay_pos
 140  positivity
 141
 142/-- The echo delay scales linearly in the bounce radius. -/
 143theorem echoDelay_scaling (r₁ r₂ : ℝ) (h : 0 < r₁) :
 144    echoDelay (r₁ * r₂) = r₂ * echoDelay r₁ := by
 145  unfold echoDelay
 146  ring
 147
 148/-- After two rung steps, the echo delay multiplies by `φ²`. -/
 149theorem echoDelay_two_step (N : ℕ) :
 150    echoDelay (bounceRadius (N + 2)) =
 151      echoDelay (bounceRadius N) * phi ^ 2 := by
 152  unfold echoDelay
 153  rw [bounceRadius_two_step]
 154  ring
 155
 156/-! ## §3. Echo amplitude damping (per-echo geometric ratio 1/φ) -/
 157
 158/-- Per-echo amplitude damping ratio: 1/φ. -/
 159def echoDampingRatio : ℝ := 1 / phi
 160
 161theorem echoDampingRatio_pos : 0 < echoDampingRatio := by
 162  unfold echoDampingRatio
 163  exact div_pos one_pos phi_pos
 164
 165theorem echoDampingRatio_lt_one : echoDampingRatio < 1 := by
 166  unfold echoDampingRatio
 167  rw [div_lt_one phi_pos]
 168  exact one_lt_phi
 169
 170theorem echoDampingRatio_band :
 171    (0.617 : ℝ) < echoDampingRatio ∧ echoDampingRatio < 0.622 := by
 172  unfold echoDampingRatio
 173  refine ⟨?_, ?_⟩
 174  · rw [lt_div_iff₀ phi_pos]
 175    have := phi_lt_onePointSixTwo
 176    nlinarith
 177  · rw [div_lt_iff₀ phi_pos]
 178    have := phi_gt_onePointSixOne
 179    nlinarith
 180
 181/-- The cumulative damping after `n` echoes: geometric series with
 182ratio `1/φ`. Each successive echo's amplitude is `(1/φ)^n` times the
 183initial echo. -/
 184def cumulativeEchoAmplitude (n : ℕ) : ℝ := echoDampingRatio ^ n
 185
 186theorem cumulativeEchoAmplitude_pos (n : ℕ) :
 187    0 < cumulativeEchoAmplitude n := by
 188  unfold cumulativeEchoAmplitude
 189  exact pow_pos echoDampingRatio_pos n
 190
 191theorem cumulativeEchoAmplitude_strictly_decreasing (n : ℕ) :
 192    cumulativeEchoAmplitude (n + 1) < cumulativeEchoAmplitude n := by
 193  unfold cumulativeEchoAmplitude
 194  rw [pow_succ]
 195  have hpos : 0 < echoDampingRatio ^ n :=
 196    pow_pos echoDampingRatio_pos n
 197  have hlt : echoDampingRatio < 1 := echoDampingRatio_lt_one
 198  nlinarith
 199
 200/-! ## §4. Master certificate -/
 201
 202structure BlackHoleEchoesCert where
 203  bounceRadius_pos : ∀ N : ℕ, 0 < bounceRadius N
 204  bounceRadius_two_step :
 205    ∀ N : ℕ, bounceRadius (N + 2) = bounceRadius N * phi ^ 2
 206  bounceRadius_strict_mono :
 207    ∀ N : ℕ, bounceRadius N < bounceRadius (N + 1)
 208  rungPhaseDelay_pos : 0 < rungPhaseDelay
 209  echoDelay_pos : ∀ r_min : ℝ, 0 < r_min → 0 < echoDelay r_min
 210  echoDelay_two_step :
 211    ∀ N : ℕ, echoDelay (bounceRadius (N + 2)) =
 212      echoDelay (bounceRadius N) * phi ^ 2
 213  echoDampingRatio_pos : 0 < echoDampingRatio
 214  echoDampingRatio_lt_one : echoDampingRatio < 1
 215  echoDampingRatio_band :
 216    (0.617 : ℝ) < echoDampingRatio ∧ echoDampingRatio < 0.622
 217  cumulativeEchoAmplitude_strictly_decreasing :
 218    ∀ n : ℕ,
 219      cumulativeEchoAmplitude (n + 1) < cumulativeEchoAmplitude n
 220
 221def blackHoleEchoesCert : BlackHoleEchoesCert where
 222  bounceRadius_pos := bounceRadius_pos
 223  bounceRadius_two_step := bounceRadius_two_step
 224  bounceRadius_strict_mono := bounceRadius_strict_mono
 225  rungPhaseDelay_pos := rungPhaseDelay_pos
 226  echoDelay_pos := echoDelay_pos
 227  echoDelay_two_step := echoDelay_two_step
 228  echoDampingRatio_pos := echoDampingRatio_pos
 229  echoDampingRatio_lt_one := echoDampingRatio_lt_one
 230  echoDampingRatio_band := echoDampingRatio_band
 231  cumulativeEchoAmplitude_strictly_decreasing :=
 232    cumulativeEchoAmplitude_strictly_decreasing
 233
 234/-- **BLACK-HOLE ECHOES ONE-STATEMENT.** RS predicts a bounce at
 235radius `r_min(N) = ℓ_P · φ^(N/2)` (here in RS-native units `ℓ_P = 1`,
 236parameterised by integer rung gap `N`); each ringdown event carries
 237an echo train at delay `Δt = 2 r_min log φ`, with per-echo
 238amplitude damping `1/φ ∈ (0.617, 0.622)` (geometric, strictly
 239decreasing). Two-rung-step identities: `r_min(N+2) = r_min(N) · φ²`
 240and `Δt(N+2) = Δt(N) · φ²`. -/
 241theorem black_hole_echoes_one_statement :
 242    (∀ N : ℕ, 0 < bounceRadius N) ∧
 243    (∀ N : ℕ, bounceRadius (N + 2) = bounceRadius N * phi ^ 2) ∧
 244    (∀ r_min : ℝ, 0 < r_min → 0 < echoDelay r_min) ∧
 245    (∀ N : ℕ, echoDelay (bounceRadius (N + 2)) =
 246        echoDelay (bounceRadius N) * phi ^ 2) ∧
 247    ((0.617 : ℝ) < echoDampingRatio ∧ echoDampingRatio < 0.622) :=
 248  ⟨bounceRadius_pos, bounceRadius_two_step, echoDelay_pos,
 249   echoDelay_two_step, echoDampingRatio_band⟩
 250
 251end
 252
 253end BlackHoleEchoesFromBounce
 254end Gravity
 255end IndisputableMonolith
 256