IndisputableMonolith.Gravity.BlackHoleEchoesSI

   1import Mathlib
   2import IndisputableMonolith.Foundation.SIBridgeClosure
   3import IndisputableMonolith.Gravity.BlackHoleEchoesFromBounce
   4
   5/-!
   6# Gravity Track 3.D: SI Lift of Quarantined Echo Rung Algebra
   7
   8## Status: STRUCTURAL THEOREM for SI conversion only
   9
  10## What this module closes
  11
  12This module converts the φ-rung algebra from
  13`Gravity.BlackHoleEchoesFromBounce` into SI units.  It does not close the
  14physical black-hole echo mechanism.  The imported native module now records the
  15old event-horizon escape mechanism as rejected and the horizon-consistent
  16exterior mechanism as open.
  17
  18The RS-native module proves the rung-model radius `r_min(N) = φ^N` (in
  19Planck units), the formal local delay `Δt = 2 r_min · log φ`, and the
  20algebraic damping ratio `1/φ`. The damping ratio is dimensionless and already
  21SI-invariant. The radius and formal delay are converted through the dimensional
  22bridge, but no observable merger-echo theorem is claimed here.
  23
  24## Substantive content
  25
  26* `planckTime_SI` and `planckLength_SI` — Planck time and length in SI,
  27  defined as `√(ℏ G / c⁵)` and `√(ℏ G / c³)` respectively. The squared
  28  identities are used as the primary algebraic content (sqrt-free).
  29
  30* `bounceRadius_SI N = planckLength_SI · φ^N` — bounce radius in meters
  31  at rung gap `N`.
  32
  33* `echoDelay_SI N = (2 · bounceRadius_SI N / c_SI) · log φ` — echo delay
  34  in seconds at rung gap `N`. Equivalent compact form:
  35  `echoDelay_SI N = 2 · planckTime_SI · φ^N · log φ` (proved as
  36  `echoDelay_SI_eq_planckTime_form`).
  37
  38* Positivity, monotonicity in `N`, and the two-step identity
  39  `echoDelay_SI (N+2) = echoDelay_SI N · φ²`.
  40
  41* Squared form `echoDelay_SI(N)² = 4 · (ℏG/c⁵) · φ^(2N) · (log φ)²`
  42  (sqrt-free; encodes the Planck-time-squared and the SI lift in clean
  43  algebraic form).
  44
  45* Master cert `BlackHoleEchoesSICert` bundling the above.
  46
  47## Anti-retreat principle satisfied
  48
  49The SI echo prediction is anchored on:
  50* `c_SI`, `hbar_SI` — SI-2019 exact (from `Foundation.SIBridgeClosure`).
  51* `G_SI` — the SINGLE CODATA measurement that anchors the bridge.
  52
  53No free dimensionless parameters; one dimensional anchor. The `1/φ`
  54damping ratio and the `log φ` per-rung phase delay are pure-φ-rational
  55content, dimensionless, and unaffected by the SI lift. The `2`
  56factor in `Δt = 2 r_min log φ` is the geometric two-way-traversal
  57factor for a bounce, NOT RS-forced.
  58
  59## Physical status
  60
  61These SI formulas are not a LIGO/Virgo falsifier until a horizon-consistent
  62exterior echo mechanism exists.  Sub-leading-log entropy remains a separate
  63black-hole discriminator; the echo mechanism is open or rejected as currently
  64stated.
  65
  66Zero `sorry`. Zero new RS-specific axioms.
  67-/
  68
  69namespace IndisputableMonolith
  70namespace Gravity
  71namespace BlackHoleEchoesSI
  72
  73open Constants
  74open IndisputableMonolith.Foundation.SIBridgeClosure
  75open IndisputableMonolith.Gravity.BlackHoleEchoesFromBounce
  76
  77noncomputable section
  78
  79/-! ## §1. Planck time and Planck length in SI -/
  80
  81/-- Planck time in SI: `t_Planck = √(ℏ_SI · G_SI / c_SI⁵)`. -/
  82def planckTime_SI : ℝ := Real.sqrt (hbar_SI * G_SI / c_SI ^ 5)
  83
  84/-- Planck length in SI: `ℓ_Planck = √(ℏ_SI · G_SI / c_SI³)`. -/
  85def planckLength_SI : ℝ := Real.sqrt (hbar_SI * G_SI / c_SI ^ 3)
  86
  87theorem planckTime_SI_pos : 0 < planckTime_SI := by
  88  unfold planckTime_SI
  89  rw [Real.sqrt_pos]
  90  exact div_pos (mul_pos hbar_SI_pos G_SI_pos) (pow_pos c_SI_pos 5)
  91
  92theorem planckLength_SI_pos : 0 < planckLength_SI := by
  93  unfold planckLength_SI
  94  rw [Real.sqrt_pos]
  95  exact div_pos (mul_pos hbar_SI_pos G_SI_pos) (pow_pos c_SI_pos 3)
  96
  97/-- Squared Planck time: `t_Planck² = ℏ G / c⁵`. -/
  98theorem planckTime_SI_sq :
  99    planckTime_SI ^ 2 = hbar_SI * G_SI / c_SI ^ 5 := by
 100  unfold planckTime_SI
 101  rw [Real.sq_sqrt]
 102  exact le_of_lt (div_pos (mul_pos hbar_SI_pos G_SI_pos) (pow_pos c_SI_pos 5))
 103
 104/-- Squared Planck length: `ℓ_Planck² = ℏ G / c³`. -/
 105theorem planckLength_SI_sq :
 106    planckLength_SI ^ 2 = hbar_SI * G_SI / c_SI ^ 3 := by
 107  unfold planckLength_SI
 108  rw [Real.sq_sqrt]
 109  exact le_of_lt (div_pos (mul_pos hbar_SI_pos G_SI_pos) (pow_pos c_SI_pos 3))
 110
 111/-- Geometric relation: `planckLength_SI = planckTime_SI · c_SI`. -/
 112theorem planckLength_SI_eq_planckTime_mul_c :
 113    planckLength_SI = planckTime_SI * c_SI := by
 114  unfold planckLength_SI planckTime_SI
 115  -- √(ℏG/c³) = √((ℏG/c⁵)·c²) = √(ℏG/c⁵) · √(c²) = √(ℏG/c⁵) · c (since c > 0)
 116  rw [show hbar_SI * G_SI / c_SI ^ 3 =
 117        (hbar_SI * G_SI / c_SI ^ 5) * c_SI ^ 2 by
 118        have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
 119        field_simp]
 120  rw [Real.sqrt_mul
 121        (le_of_lt
 122          (div_pos (mul_pos hbar_SI_pos G_SI_pos) (pow_pos c_SI_pos 5)))]
 123  rw [Real.sqrt_sq (le_of_lt c_SI_pos)]
 124
 125/-! ## §2. Bounce radius in SI -/
 126
 127/-- Bounce radius in SI at rung gap `N`: `r_min(N) = ℓ_Planck_SI · φ^N`.
 128This is the meter-scale lift of `BlackHoleEchoesFromBounce.bounceRadius N
 129= φ^N` (which is dimensionless in Planck units). -/
 130def bounceRadius_SI (N : ℕ) : ℝ := planckLength_SI * phi ^ N
 131
 132theorem bounceRadius_SI_pos (N : ℕ) : 0 < bounceRadius_SI N := by
 133  unfold bounceRadius_SI
 134  exact mul_pos planckLength_SI_pos (pow_pos phi_pos N)
 135
 136theorem bounceRadius_SI_two_step (N : ℕ) :
 137    bounceRadius_SI (N + 2) = bounceRadius_SI N * phi ^ 2 := by
 138  unfold bounceRadius_SI
 139  rw [pow_add]
 140  ring
 141
 142theorem bounceRadius_SI_strict_mono (N : ℕ) :
 143    bounceRadius_SI N < bounceRadius_SI (N + 1) := by
 144  unfold bounceRadius_SI
 145  rw [pow_succ]
 146  have hN : 0 < phi ^ N := pow_pos phi_pos N
 147  have hℓ : 0 < planckLength_SI := planckLength_SI_pos
 148  have hphi : 1 < phi := one_lt_phi
 149  nlinarith [mul_pos hℓ hN]
 150
 151/-! ## §3. Echo delay in SI -/
 152
 153/-- Echo delay in SI: `Δt = (2 · r_min / c) · log φ`. -/
 154def echoDelay_SI (N : ℕ) : ℝ :=
 155  (2 * bounceRadius_SI N / c_SI) * Real.log phi
 156
 157theorem echoDelay_SI_def (N : ℕ) :
 158    echoDelay_SI N = (2 * bounceRadius_SI N / c_SI) * Real.log phi := rfl
 159
 160/-- Compact form: `echoDelay_SI(N) = 2 · planckTime_SI · φ^N · log φ`.
 161Uses `planckLength_SI = planckTime_SI · c_SI`. -/
 162theorem echoDelay_SI_eq_planckTime_form (N : ℕ) :
 163    echoDelay_SI N = 2 * planckTime_SI * phi ^ N * Real.log phi := by
 164  unfold echoDelay_SI bounceRadius_SI
 165  rw [planckLength_SI_eq_planckTime_mul_c]
 166  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
 167  field_simp
 168
 169theorem echoDelay_SI_pos (N : ℕ) : 0 < echoDelay_SI N := by
 170  rw [echoDelay_SI_eq_planckTime_form]
 171  have h_log : 0 < Real.log phi := Real.log_pos one_lt_phi
 172  have h_phi_pow : 0 < phi ^ N := pow_pos phi_pos N
 173  have h_pt : 0 < planckTime_SI := planckTime_SI_pos
 174  have h2 : (0 : ℝ) < 2 := by norm_num
 175  positivity
 176
 177theorem echoDelay_SI_two_step (N : ℕ) :
 178    echoDelay_SI (N + 2) = echoDelay_SI N * phi ^ 2 := by
 179  rw [echoDelay_SI_eq_planckTime_form, echoDelay_SI_eq_planckTime_form,
 180      pow_add]
 181  ring
 182
 183theorem echoDelay_SI_strict_mono (N : ℕ) :
 184    echoDelay_SI N < echoDelay_SI (N + 1) := by
 185  rw [echoDelay_SI_eq_planckTime_form, echoDelay_SI_eq_planckTime_form,
 186      pow_succ]
 187  have h_log : 0 < Real.log phi := Real.log_pos one_lt_phi
 188  have h_phi_pow : 0 < phi ^ N := pow_pos phi_pos N
 189  have h_pt : 0 < planckTime_SI := planckTime_SI_pos
 190  have hphi : 1 < phi := one_lt_phi
 191  have h_phi_minus_one_pos : 0 < phi - 1 := by linarith
 192  have h_pt_phi_pow : 0 < planckTime_SI * phi ^ N :=
 193    mul_pos h_pt h_phi_pow
 194  have h_pt_phi_pow_log : 0 < planckTime_SI * phi ^ N * Real.log phi :=
 195    mul_pos h_pt_phi_pow h_log
 196  nlinarith [h_pt_phi_pow_log, h_phi_minus_one_pos]
 197
 198/-! ## §4. Squared form (sqrt-free Planck-units encoding)
 199
 200The squared echo delay encodes the dimensional bridge content
 201sqrt-free: `(Δt_SI)² = 4 · (ℏG/c⁵) · φ^(2N) · (log φ)²`. The factor
 202`ℏG/c⁵` is the Planck time squared; raising `φ^N` to the second power
 203gives `φ^(2N)`; the `(log φ)²` factor encodes the per-rung phase
 204delay.
 205-/
 206
 207theorem echoDelay_SI_sq (N : ℕ) :
 208    (echoDelay_SI N) ^ 2 =
 209      4 * (hbar_SI * G_SI / c_SI ^ 5) * phi ^ (2 * N) * (Real.log phi) ^ 2 := by
 210  rw [echoDelay_SI_eq_planckTime_form]
 211  have hphi_pow : phi ^ N * phi ^ N = phi ^ (2 * N) := by
 212    rw [show (2 * N : ℕ) = N + N from by omega, pow_add]
 213  have h_expand :
 214      (2 * planckTime_SI * phi ^ N * Real.log phi) ^ 2
 215        = 4 * planckTime_SI ^ 2 * (phi ^ N * phi ^ N) * (Real.log phi) ^ 2 := by
 216    ring
 217  rw [h_expand, hphi_pow, planckTime_SI_sq]
 218
 219/-! ## §5. Cumulative damping in SI (dimensionless, same as RS-native) -/
 220
 221/-- The per-echo amplitude damping ratio `1/φ` is dimensionless and
 222SI-invariant. We re-export it as `echoDampingRatio_SI` for cert-bundling
 223purposes. -/
 224def echoDampingRatio_SI : ℝ := echoDampingRatio
 225
 226theorem echoDampingRatio_SI_eq : echoDampingRatio_SI = 1 / phi := rfl
 227
 228theorem echoDampingRatio_SI_pos : 0 < echoDampingRatio_SI :=
 229  echoDampingRatio_pos
 230
 231theorem echoDampingRatio_SI_lt_one : echoDampingRatio_SI < 1 :=
 232  echoDampingRatio_lt_one
 233
 234theorem echoDampingRatio_SI_band :
 235    (0.617 : ℝ) < echoDampingRatio_SI ∧ echoDampingRatio_SI < 0.622 :=
 236  echoDampingRatio_band
 237
 238/-! ## §6. Master cert -/
 239
 240structure BlackHoleEchoesSICert where
 241  planckTime_SI_pos : 0 < planckTime_SI
 242  planckLength_SI_pos : 0 < planckLength_SI
 243  planckTime_SI_sq :
 244    planckTime_SI ^ 2 = hbar_SI * G_SI / c_SI ^ 5
 245  planckLength_SI_sq :
 246    planckLength_SI ^ 2 = hbar_SI * G_SI / c_SI ^ 3
 247  planckLength_SI_eq_planckTime_mul_c :
 248    planckLength_SI = planckTime_SI * c_SI
 249  bounceRadius_SI_pos : ∀ N : ℕ, 0 < bounceRadius_SI N
 250  bounceRadius_SI_two_step :
 251    ∀ N : ℕ, bounceRadius_SI (N + 2) = bounceRadius_SI N * phi ^ 2
 252  bounceRadius_SI_strict_mono :
 253    ∀ N : ℕ, bounceRadius_SI N < bounceRadius_SI (N + 1)
 254  echoDelay_SI_def :
 255    ∀ N : ℕ, echoDelay_SI N = (2 * bounceRadius_SI N / c_SI) * Real.log phi
 256  echoDelay_SI_eq_planckTime_form :
 257    ∀ N : ℕ, echoDelay_SI N = 2 * planckTime_SI * phi ^ N * Real.log phi
 258  echoDelay_SI_pos : ∀ N : ℕ, 0 < echoDelay_SI N
 259  echoDelay_SI_two_step :
 260    ∀ N : ℕ, echoDelay_SI (N + 2) = echoDelay_SI N * phi ^ 2
 261  echoDelay_SI_strict_mono :
 262    ∀ N : ℕ, echoDelay_SI N < echoDelay_SI (N + 1)
 263  echoDelay_SI_sq :
 264    ∀ N : ℕ, (echoDelay_SI N) ^ 2 =
 265      4 * (hbar_SI * G_SI / c_SI ^ 5) * phi ^ (2 * N) * (Real.log phi) ^ 2
 266  echoDampingRatio_SI_band :
 267    (0.617 : ℝ) < echoDampingRatio_SI ∧ echoDampingRatio_SI < 0.622
 268
 269def blackHoleEchoesSICert : BlackHoleEchoesSICert where
 270  planckTime_SI_pos := planckTime_SI_pos
 271  planckLength_SI_pos := planckLength_SI_pos
 272  planckTime_SI_sq := planckTime_SI_sq
 273  planckLength_SI_sq := planckLength_SI_sq
 274  planckLength_SI_eq_planckTime_mul_c := planckLength_SI_eq_planckTime_mul_c
 275  bounceRadius_SI_pos := bounceRadius_SI_pos
 276  bounceRadius_SI_two_step := bounceRadius_SI_two_step
 277  bounceRadius_SI_strict_mono := bounceRadius_SI_strict_mono
 278  echoDelay_SI_def := echoDelay_SI_def
 279  echoDelay_SI_eq_planckTime_form := echoDelay_SI_eq_planckTime_form
 280  echoDelay_SI_pos := echoDelay_SI_pos
 281  echoDelay_SI_two_step := echoDelay_SI_two_step
 282  echoDelay_SI_strict_mono := echoDelay_SI_strict_mono
 283  echoDelay_SI_sq := echoDelay_SI_sq
 284  echoDampingRatio_SI_band := echoDampingRatio_SI_band
 285
 286theorem blackHoleEchoesSICert_inhabited : Nonempty BlackHoleEchoesSICert :=
 287  ⟨blackHoleEchoesSICert⟩
 288
 289/-- **BLACK-HOLE ECHO SI RUNG-ALGEBRA ONE-STATEMENT.**  The SI lift of the
 290quarantined rung model has positive radius, positive formal delay, the
 291two-step φ² scaling law, and dimensionless damping ratio
 292`1/φ ∈ (0.617, 0.622)`.  This theorem does not assert an observable echo on
 293BH-BH merger ringdowns. -/
 294theorem black_hole_echoes_SI_one_statement :
 295    (∀ N : ℕ, 0 < bounceRadius_SI N) ∧
 296    (∀ N : ℕ, bounceRadius_SI (N + 2) = bounceRadius_SI N * phi ^ 2) ∧
 297    (∀ N : ℕ, 0 < echoDelay_SI N) ∧
 298    (∀ N : ℕ, echoDelay_SI (N + 2) = echoDelay_SI N * phi ^ 2) ∧
 299    (∀ N : ℕ, (echoDelay_SI N) ^ 2 =
 300        4 * (hbar_SI * G_SI / c_SI ^ 5) * phi ^ (2 * N) * (Real.log phi) ^ 2) ∧
 301    ((0.617 : ℝ) < echoDampingRatio_SI ∧ echoDampingRatio_SI < 0.622) :=
 302  ⟨bounceRadius_SI_pos, bounceRadius_SI_two_step, echoDelay_SI_pos,
 303   echoDelay_SI_two_step, echoDelay_SI_sq, echoDampingRatio_SI_band⟩
 304
 305end
 306
 307end BlackHoleEchoesSI
 308end Gravity
 309end IndisputableMonolith
 310