IndisputableMonolith.Bridge.DataCore

   1import Mathlib
   2
   3/-!
   4# Bridge Data Core (certified-surface friendly)
   5
   6This module contains only the **core** bridge-anchor data structure and the
   7dimensionless identity for the recognition length
   8\(\lambda_{\mathrm{rec}} = \sqrt{\hbar G / (\pi c^3)}\).
   9
  10It intentionally avoids importing `IndisputableMonolith.Core` (and other large
  11subsystems) so the certified import-closure stays small.
  12
  13The extended measurement/protocol helpers live in `IndisputableMonolith/Bridge/Data.lean`
  14and are intentionally kept out of the certified import-closure.
  15-/
  16
  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
  51    simpa using Real.sq_sqrt h_nonneg
  52  -- Now simplify the target expression
  53  calc
  54    (B.c ^ 3) * (Real.sqrt (B.hbar * B.G / (Real.pi * B.c ^ 3))) ^ 2 / (B.hbar * B.G)
  55        = (B.c ^ 3) * (B.hbar * B.G / (Real.pi * B.c ^ 3)) / (B.hbar * B.G) := by
  56          simp [hsq]
  57    _ = ((B.c ^ 3) * (B.hbar * B.G)) / ((Real.pi * B.c ^ 3) * (B.hbar * B.G)) := by
  58          field_simp
  59    _ = 1 / Real.pi := by
  60          field_simp [mul_comm, mul_left_comm, mul_assoc, pow_succ, pow_mul]
  61
  62/-- Dimensionless identity packaged with a physical-assumptions helper. -/
  63lemma lambda_rec_dimensionless_id_physical (B : BridgeData) (H : Physical B) :
  64  (B.c ^ 3) * (lambda_rec B) ^ 2 / (B.hbar * B.G) = 1 / Real.pi :=
  65  lambda_rec_dimensionless_id B H.c_pos H.hbar_pos H.G_pos
  66
  67/-- Positivity of λ_rec under physical assumptions. -/
  68lemma lambda_rec_pos (B : BridgeData) (H : Physical B) : 0 < lambda_rec B := by
  69  -- λ_rec = √(ħ G / (π c³)) > 0 since all components positive
  70  unfold lambda_rec
  71  apply Real.sqrt_pos.mpr
  72  apply div_pos
  73  · exact mul_pos H.hbar_pos H.G_pos
  74  · exact mul_pos Real.pi_pos (pow_pos H.c_pos 3)
  75
  76end IndisputableMonolith.BridgeData
  77