BridgeData

formal source

  17namespace IndisputableMonolith.BridgeData
  18
  19/-- External bridge anchors provided as data (no axioms): G, ħ, c, plus display anchors. -/
  20structure BridgeData where
  21  G     : ℝ
  22  hbar  : ℝ
  23  c     : ℝ
  24  tau0  : ℝ
  25  ell0  : ℝ
  26
  27/-- Minimal physical assumptions on bridge anchors reused by analytical lemmas. -/
  28structure Physical (B : BridgeData) : Prop where
  29  c_pos    : 0 < B.c
  30  hbar_pos : 0 < B.hbar
  31  G_pos    : 0 < B.G
  32
  33/-- Recognition length from anchors: λ_rec = √(ħ G / c^3). -/
  34noncomputable def lambda_rec (B : BridgeData) : ℝ :=
  35  Real.sqrt (B.hbar * B.G / (Real.pi * (B.c ^ 3)))
  36
  37/-- Dimensionless identity for λ_rec (under mild physical positivity assumptions):
  38    (c^3 · λ_rec^2) / (ħ G) = 1/π. -/
  39lemma lambda_rec_dimensionless_id (B : BridgeData)
  40  (hc : 0 < B.c) (hh : 0 < B.hbar) (hG : 0 < B.G) :
  41  (B.c ^ 3) * (lambda_rec B) ^ 2 / (B.hbar * B.G) = 1 / Real.pi := by
  42  -- Expand λ_rec and simplify using sqrt and algebra
  43  unfold lambda_rec
  44  have h_pos : 0 < B.hbar * B.G / (Real.pi * B.c ^ 3) := by
  45    apply div_pos (mul_pos hh hG)
  46    exact mul_pos Real.pi_pos (pow_pos hc 3)
  47  -- Use (sqrt x)^2 = x for x ≥ 0
  48  have h_nonneg : 0 ≤ B.hbar * B.G / (Real.pi * B.c ^ 3) := le_of_lt h_pos
  49  have hsq : (Real.sqrt (B.hbar * B.G / (Real.pi * B.c ^ 3))) ^ 2 =
  50    B.hbar * B.G / (Real.pi * B.c ^ 3) := by