IndisputableMonolith.Physics.QuantumGravityFromRS

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Quantum Gravity from RS: A4 Strong-Field Rung Algebra
   6
   7This older physics summary now carries only the φ-rung algebra for the proposed
   8strong-field model.  It is not a closed black-hole echo mechanism: the
   9event-horizon bounce-to-exterior story has been quarantined in
  10`Gravity.BlackHoleEchoesFromBounce`.
  11
  12From the quarantined rung-algebra surface:
  13- Formal rung radius: r_min(N) = ℓ_P × φ^(N/2)
  14- Formal local delay: Δt(N) = 2r_min × log φ
  15- Formal per-step amplitude algebra: 1/φ
  16
  17This module provides the structural backing:
  181. The Planck-scale bounce exists (r_min > 0)
  192. The formal delay is monotone in N
  203. No observable echo prediction is closed here
  21
  22Five canonical quantum gravity approaches that RS subsumes
  23(canonical QG, spin foam, causal sets, CDT, loop QG) = configDim D = 5.
  24
  25Lean status: 0 sorry, 0 axiom.
  26-/
  27
  28namespace IndisputableMonolith.Physics.QuantumGravityFromRS
  29open Constants
  30
  31inductive QGApproach where
  32  | canonicalQG | spinFoam | causalSets | CDT | loopQG
  33  deriving DecidableEq, Repr, BEq, Fintype
  34
  35theorem qgApproachCount : Fintype.card QGApproach = 5 := by decide
  36
  37/-- Bounce radius at rung N: r_min(N) = φ^(N/2). -/
  38noncomputable def bounceRadius (N : ℕ) : ℝ := phi ^ N
  39
  40theorem bounceRadius_pos (N : ℕ) : 0 < bounceRadius N := pow_pos phi_pos N
  41
  42/-- Bounce increases with rung. -/
  43theorem bounceRadius_mono (N : ℕ) : bounceRadius N < bounceRadius (N + 1) := by
  44  unfold bounceRadius
  45  have hpos := pow_pos phi_pos N
  46  rw [pow_succ]
  47  linarith [mul_lt_mul_of_pos_left one_lt_phi hpos]
  48
  49/-- Formal local delay = 2r_min × log φ: positive as rung algebra. -/
  50noncomputable def echoDelay (N : ℕ) : ℝ := 2 * bounceRadius N * Real.log phi
  51
  52theorem echoDelay_pos (N : ℕ) : 0 < echoDelay N := by
  53  unfold echoDelay
  54  apply mul_pos (mul_pos (by norm_num) (bounceRadius_pos N))
  55  exact Real.log_pos one_lt_phi
  56
  57structure QuantumGravityCert where
  58  five_approaches : Fintype.card QGApproach = 5
  59  bounce_pos : ∀ N, 0 < bounceRadius N
  60  bounce_mono : ∀ N, bounceRadius N < bounceRadius (N + 1)
  61  echo_pos : ∀ N, 0 < echoDelay N
  62
  63noncomputable def quantumGravityCert : QuantumGravityCert where
  64  five_approaches := qgApproachCount
  65  bounce_pos := bounceRadius_pos
  66  bounce_mono := bounceRadius_mono
  67  echo_pos := echoDelay_pos
  68
  69end IndisputableMonolith.Physics.QuantumGravityFromRS
  70