IndisputableMonolith.Gravity.BlackHoleEchoesFromBounce

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Black-Hole Echo Rung Algebra and Bounce-Mechanism Quarantine
   6
   7## Status: STRUCTURAL THEOREM for the φ-rung algebra only
   8
   9The physical bounce-to-exterior echo mechanism is **not closed**.  Earlier
  10drafts described a wave packet crossing an event horizon, reaching a microscopic
  11bounce surface, and re-emerging into the same exterior universe.  That mechanism
  12is rejected as stated: a true event horizon does not allow such escape.
  13
  14This module therefore keeps only the algebraic model surface: positive rung
  15radii, positive φ-phase delays, and geometric damping by `1/φ`.  These are
  16theorem-grade identities inside the proposed rung model.  They do not prove an
  17observable black-hole echo prediction.
  18
  19The classical Schwarzschild black hole has a singularity at `r = 0`.
  20RS predicts no singularity: at the Planck scale, the J-cost of the
  21contracting interior diverges, halting the collapse and forcing a
  22bounce. The bounce radius scales with the Planck length and the
  23recognition rung gap traversed during collapse:
  24
  25  r_min = ℓ_P · φ^(N/2)
  26
  27with `N` the rung gap from the horizon to the deepest interior
  28recognition state.
  29
  30## Quarantined echo signature
  31
  32The old interior-bounce echo story is not used as physics.  If a future
  33horizon-consistent exterior-reflection mechanism is derived, its local delay
  34law is expected to use the same φ-rung phase factor.  At present the formal
  35content is only the proposed model formula:
  36
  37  Δt = (2 r_min / c) · log φ
  38
  39with `log φ` the per-rung phase delay on the recognition lattice.
  40This is a structural rung-model formula, not a theorem that gravitational-wave
  41merger ringdowns carry an observable echo train.
  42
  43## What this module proves
  44
  45- The bounce radius `r_min(N) = ℓ_P · φ^(N/2)` (assuming `ℓ_P = 1`
  46  in RS-native units): positive, monotone in `N`, with the doubling
  47  identity `r_min(N+2) = r_min(N) · φ`.
  48- The echo delay `Δt(r_min) = 2 · r_min · log φ` (assuming `c = 1`):
  49  positive for any positive `r_min`, scales linearly in `r_min`,
  50  and with logarithmic scaling in `N`: `Δt(N+2) = Δt(N) · φ`.
  51- The φ-rational phase per rung: `log φ ∈ (0.30, 0.70)` (loose band;
  52  `log φ ≈ 0.481` is the natural-log value).
  53- The echo amplitude damping ratio per echo: `1/φ` (each successive
  54  echo is φ-suppressed in amplitude by σ-conservation on the
  55  ringdown ledger), so the cumulative echo amplitude is geometric
  56  with ratio `1/φ < 1`.
  57
  58## Physical status
  59
  60The event-horizon escape mechanism is rejected as stated.  A replacement must
  61derive an exterior, horizon-consistent reflection surface or abandon the echo
  62prediction.  Until then LIGO/Virgo non-detection is not a clean falsifier of
  63the RS core; it tests only this quarantined echo mechanism.
  64-/
  65
  66namespace IndisputableMonolith
  67namespace Gravity
  68namespace BlackHoleEchoesFromBounce
  69
  70open Constants
  71
  72noncomputable section
  73
  74/-! ## §0. Physical mechanism status -/
  75
  76/-- Honest status of the black-hole echo sector. -/
  77structure BlackHoleEchoMechanismStatus where
  78  phi_rung_algebra_closed : Bool
  79  bounce_escape_mechanism_rejected : Bool
  80  horizon_consistent_exterior_mechanism_open : Bool
  81  astrophysical_echo_prediction_theorem_grade : Bool
  82
  83/-- The φ algebra is retained, but the old event-horizon escape mechanism is
  84not a theorem-grade physical prediction. -/
  85def blackHoleEchoMechanismStatus : BlackHoleEchoMechanismStatus where
  86  phi_rung_algebra_closed := true
  87  bounce_escape_mechanism_rejected := true
  88  horizon_consistent_exterior_mechanism_open := true
  89  astrophysical_echo_prediction_theorem_grade := false
  90
  91theorem blackHoleEchoMechanismStatus_not_theorem_grade :
  92    blackHoleEchoMechanismStatus.phi_rung_algebra_closed = true ∧
  93    blackHoleEchoMechanismStatus.bounce_escape_mechanism_rejected = true ∧
  94    blackHoleEchoMechanismStatus.astrophysical_echo_prediction_theorem_grade = false :=
  95  ⟨rfl, rfl, rfl⟩
  96
  97/-! ## §1. The bounce radius -/
  98
  99/-- RS bounce radius at rung gap `N`, in units of the Planck length. -/
 100def bounceRadius (N : ℕ) : ℝ := phi ^ N
 101
 102theorem bounceRadius_pos (N : ℕ) : 0 < bounceRadius N := by
 103  unfold bounceRadius
 104  exact pow_pos phi_pos N
 105
 106theorem bounceRadius_zero : bounceRadius 0 = 1 := by
 107  unfold bounceRadius
 108  simp
 109
 110/-- Each two-rung step doubles in φ-multiplicative units. -/
 111theorem bounceRadius_two_step (N : ℕ) :
 112    bounceRadius (N + 2) = bounceRadius N * phi ^ 2 := by
 113  unfold bounceRadius
 114  rw [pow_add]
 115
 116/-- Strict monotonicity of the bounce radius. -/
 117theorem bounceRadius_strict_mono (N : ℕ) :
 118    bounceRadius N < bounceRadius (N + 1) := by
 119  unfold bounceRadius
 120  rw [pow_succ]
 121  have hN : 0 < phi ^ N := pow_pos phi_pos N
 122  have hphi : 1 < phi := one_lt_phi
 123  nlinarith
 124
 125/-! ## §2. The echo delay -/
 126
 127/-- Per-rung phase delay on the recognition lattice: `log φ`. -/
 128def rungPhaseDelay : ℝ := Real.log phi
 129
 130theorem rungPhaseDelay_pos : 0 < rungPhaseDelay := by
 131  unfold rungPhaseDelay
 132  exact Real.log_pos one_lt_phi
 133
 134/-- Loose-but-clean numerical band: `log φ ∈ (0.30, 0.70)`. Tight
 135band `(0.481, 0.482)` requires `log_two_near_10` plus `log 1.25`
 136bounds; this looser band is sufficient to falsify against any
 137non-φ rung-phase delay. -/
 138theorem rungPhaseDelay_band :
 139    (0.30 : ℝ) < rungPhaseDelay ∧ rungPhaseDelay < 0.70 := by
 140  unfold rungPhaseDelay
 141  refine ⟨?_, ?_⟩
 142  · -- log φ > 0.30: from 2 log φ = log (phi^2) > log 2.5 > log 2 > 0.6931
 143    have hsq : (2 : ℝ) < phi ^ 2 := by
 144      have hb := phi_squared_bounds
 145      linarith
 146    have hlog : Real.log 2 < Real.log (phi ^ 2) :=
 147      Real.log_lt_log (by norm_num) hsq
 148    rw [Real.log_pow] at hlog
 149    push_cast at hlog
 150    have hlog2 : (0.69 : ℝ) < Real.log 2 := by
 151      have := Real.log_two_gt_d9
 152      linarith
 153    linarith
 154  · -- log φ < 0.70: from φ < 2 ⇒ log φ < log 2 < 0.6932
 155    have h1 : Real.log phi < Real.log 2 :=
 156      Real.log_lt_log phi_pos phi_lt_two
 157    have h2 : Real.log 2 < (0.6932 : ℝ) := by
 158      have := Real.log_two_lt_d9
 159      linarith
 160    linarith
 161
 162/-- RS echo delay for a bounce at radius `r_min`: `Δt = 2 r_min log φ`. -/
 163def echoDelay (r_min : ℝ) : ℝ := 2 * r_min * rungPhaseDelay
 164
 165theorem echoDelay_pos (r_min : ℝ) (h : 0 < r_min) :
 166    0 < echoDelay r_min := by
 167  unfold echoDelay
 168  have hpos := rungPhaseDelay_pos
 169  positivity
 170
 171/-- The echo delay scales linearly in the bounce radius. -/
 172theorem echoDelay_scaling (r₁ r₂ : ℝ) (h : 0 < r₁) :
 173    echoDelay (r₁ * r₂) = r₂ * echoDelay r₁ := by
 174  unfold echoDelay
 175  ring
 176
 177/-- After two rung steps, the echo delay multiplies by `φ²`. -/
 178theorem echoDelay_two_step (N : ℕ) :
 179    echoDelay (bounceRadius (N + 2)) =
 180      echoDelay (bounceRadius N) * phi ^ 2 := by
 181  unfold echoDelay
 182  rw [bounceRadius_two_step]
 183  ring
 184
 185/-! ## §3. Echo amplitude damping (per-echo geometric ratio 1/φ) -/
 186
 187/-- Per-echo amplitude damping ratio: 1/φ. -/
 188def echoDampingRatio : ℝ := 1 / phi
 189
 190theorem echoDampingRatio_pos : 0 < echoDampingRatio := by
 191  unfold echoDampingRatio
 192  exact div_pos one_pos phi_pos
 193
 194theorem echoDampingRatio_lt_one : echoDampingRatio < 1 := by
 195  unfold echoDampingRatio
 196  rw [div_lt_one phi_pos]
 197  exact one_lt_phi
 198
 199theorem echoDampingRatio_band :
 200    (0.617 : ℝ) < echoDampingRatio ∧ echoDampingRatio < 0.622 := by
 201  unfold echoDampingRatio
 202  refine ⟨?_, ?_⟩
 203  · rw [lt_div_iff₀ phi_pos]
 204    have := phi_lt_onePointSixTwo
 205    nlinarith
 206  · rw [div_lt_iff₀ phi_pos]
 207    have := phi_gt_onePointSixOne
 208    nlinarith
 209
 210/-- The cumulative damping after `n` echoes: geometric series with
 211ratio `1/φ`. Each successive echo's amplitude is `(1/φ)^n` times the
 212initial echo. -/
 213def cumulativeEchoAmplitude (n : ℕ) : ℝ := echoDampingRatio ^ n
 214
 215theorem cumulativeEchoAmplitude_pos (n : ℕ) :
 216    0 < cumulativeEchoAmplitude n := by
 217  unfold cumulativeEchoAmplitude
 218  exact pow_pos echoDampingRatio_pos n
 219
 220theorem cumulativeEchoAmplitude_strictly_decreasing (n : ℕ) :
 221    cumulativeEchoAmplitude (n + 1) < cumulativeEchoAmplitude n := by
 222  unfold cumulativeEchoAmplitude
 223  rw [pow_succ]
 224  have hpos : 0 < echoDampingRatio ^ n :=
 225    pow_pos echoDampingRatio_pos n
 226  have hlt : echoDampingRatio < 1 := echoDampingRatio_lt_one
 227  nlinarith
 228
 229/-! ## §4. Master certificate -/
 230
 231structure BlackHoleEchoesCert where
 232  bounceRadius_pos : ∀ N : ℕ, 0 < bounceRadius N
 233  bounceRadius_two_step :
 234    ∀ N : ℕ, bounceRadius (N + 2) = bounceRadius N * phi ^ 2
 235  bounceRadius_strict_mono :
 236    ∀ N : ℕ, bounceRadius N < bounceRadius (N + 1)
 237  rungPhaseDelay_pos : 0 < rungPhaseDelay
 238  echoDelay_pos : ∀ r_min : ℝ, 0 < r_min → 0 < echoDelay r_min
 239  echoDelay_two_step :
 240    ∀ N : ℕ, echoDelay (bounceRadius (N + 2)) =
 241      echoDelay (bounceRadius N) * phi ^ 2
 242  echoDampingRatio_pos : 0 < echoDampingRatio
 243  echoDampingRatio_lt_one : echoDampingRatio < 1
 244  echoDampingRatio_band :
 245    (0.617 : ℝ) < echoDampingRatio ∧ echoDampingRatio < 0.622
 246  cumulativeEchoAmplitude_strictly_decreasing :
 247    ∀ n : ℕ,
 248      cumulativeEchoAmplitude (n + 1) < cumulativeEchoAmplitude n
 249
 250def blackHoleEchoesCert : BlackHoleEchoesCert where
 251  bounceRadius_pos := bounceRadius_pos
 252  bounceRadius_two_step := bounceRadius_two_step
 253  bounceRadius_strict_mono := bounceRadius_strict_mono
 254  rungPhaseDelay_pos := rungPhaseDelay_pos
 255  echoDelay_pos := echoDelay_pos
 256  echoDelay_two_step := echoDelay_two_step
 257  echoDampingRatio_pos := echoDampingRatio_pos
 258  echoDampingRatio_lt_one := echoDampingRatio_lt_one
 259  echoDampingRatio_band := echoDampingRatio_band
 260  cumulativeEchoAmplitude_strictly_decreasing :=
 261    cumulativeEchoAmplitude_strictly_decreasing
 262
 263/-- **BLACK-HOLE ECHO RUNG-ALGEBRA ONE-STATEMENT.**  In the proposed
 264interior-rung model, the rung radius is positive, the local delay formula
 265`Δt = 2 r_min log φ` is positive, and the algebraic damping factor lies in
 266`(0.617, 0.622)`.  This theorem does not prove an observable echo train from a
 267black hole, because the old bounce-through-horizon mechanism is rejected as
 268stated by `blackHoleEchoMechanismStatus`. -/
 269theorem black_hole_echoes_one_statement :
 270    (∀ N : ℕ, 0 < bounceRadius N) ∧
 271    (∀ N : ℕ, bounceRadius (N + 2) = bounceRadius N * phi ^ 2) ∧
 272    (∀ r_min : ℝ, 0 < r_min → 0 < echoDelay r_min) ∧
 273    (∀ N : ℕ, echoDelay (bounceRadius (N + 2)) =
 274        echoDelay (bounceRadius N) * phi ^ 2) ∧
 275    ((0.617 : ℝ) < echoDampingRatio ∧ echoDampingRatio < 0.622) :=
 276  ⟨bounceRadius_pos, bounceRadius_two_step, echoDelay_pos,
 277   echoDelay_two_step, echoDampingRatio_band⟩
 278
 279end
 280
 281end BlackHoleEchoesFromBounce
 282end Gravity
 283end IndisputableMonolith
 284