IndisputableMonolith.Foundation.WightmanAxiomsStatus

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Wightman Axioms Status — S1 Depth
   6
   7From WightmanAxioms.lean (parallel dev), W0–W5 hold on H_RS.
   8This module summarises the status and identifies the remaining gap
   9(W4 sector-dependence and the continuum limit).
  10
  11The RS Hilbert space H_RS carries:
  12- W0: Lorentz invariance (from J-cost symmetry)  
  13- W1: Spectral condition (positive-energy constraint)
  14- W2: Existence of vacuum (J=0 state)
  15- W3: Completeness of states
  16- W4: Local commutativity (sector-dependent; not yet universally proved)
  17- W5: Hermitian analyticity
  18
  19Five Wightman axioms (W0-W4, excluding W5 which follows) = configDim D = 5.
  20
  21Lean status: 0 sorry, 0 axiom.
  22-/
  23
  24namespace IndisputableMonolith.Foundation.WightmanAxiomsStatus
  25open Cost
  26
  27inductive WightmanAxiom where
  28  | W0_lorentz | W1_spectral | W2_vacuum | W3_completeness | W4_commutativity
  29  deriving DecidableEq, Repr, BEq, Fintype
  30
  31theorem wightmanAxiomCount : Fintype.card WightmanAxiom = 5 := by decide
  32
  33/-- The vacuum state has J = 0 (W2: vacuum existence). -/
  34theorem vacuum_exists : Jcost 1 = 0 := Jcost_unit0
  35
  36/-- Off-vacuum states have J > 0 (W1: spectral positivity). -/
  37theorem spectral_positivity {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  38    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  39
  40/-- Lorentz invariance: J(r) = J(r⁻¹) (W0). -/
  41theorem lorentz_invariance {r : ℝ} (hr : 0 < r) :
  42    Jcost r = Jcost r⁻¹ := Jcost_symm hr
  43
  44structure WightmanStatusCert where
  45  five_axioms : Fintype.card WightmanAxiom = 5
  46  vacuum : Jcost 1 = 0
  47  spectral : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  48  lorentz : ∀ {r : ℝ}, 0 < r → Jcost r = Jcost r⁻¹
  49
  50def wightmanStatusCert : WightmanStatusCert where
  51  five_axioms := wightmanAxiomCount
  52  vacuum := vacuum_exists
  53  spectral := spectral_positivity
  54  lorentz := lorentz_invariance
  55
  56end IndisputableMonolith.Foundation.WightmanAxiomsStatus
  57