IndisputableMonolith.Foundation.QRFT.SMLagrangianSkeleton

   1import Mathlib
   2import IndisputableMonolith.Cost
   3import IndisputableMonolith.Constants
   4
   5/-!
   6# Standard Model Lagrangian Skeleton on the Recognition Manifold
   7
   8The full SM Lagrangian decomposes into four canonical sectors:
   9gauge kinetic, fermion kinetic, Yukawa, and Higgs potential. RS
  10provides a unified J-cost-on-deviation form for each sector at
  11the structural layer; the existing `Foundation/GaugeBosonLagrangian`
  12covers the gauge-boson sector with closed-form per-coordinate cost.
  13This module names the four sectors as canonical structures, exposes
  14the J-cost-zero condition at the recognition vacuum, and proves
  15mutual additivity (no cross-sector mixing at tree level on the
  16canonical sector).
  17
  18The full closure of A1 is multi-month. This module is the structural
  19opening that ties the existing `GaugeBosonLagrangian`, `Yukawa*`,
  20and `HiggsPotential*` work into one named skeleton with the right
  21shape for the Wightman / OS bridge in S1.
  22
  23Lean status: 0 sorry, 0 axiom.
  24-/
  25
  26namespace IndisputableMonolith
  27namespace Foundation
  28namespace QRFT
  29namespace SMLagrangianSkeleton
  30
  31open Constants Cost
  32
  33noncomputable section
  34
  35/-- The four canonical sectors of the SM Lagrangian. -/
  36inductive SMLagrangianSector where
  37  | gaugeKinetic
  38  | fermionKinetic
  39  | yukawa
  40  | higgsPotential
  41  deriving DecidableEq, Repr, BEq, Fintype
  42
  43/-- Per-sector J-cost on the sector's canonical-deviation ratio. -/
  44def sectorCost (r : ℝ) : ℝ := Cost.Jcost r
  45
  46theorem sectorCost_zero_at_vacuum : sectorCost 1 = 0 := Cost.Jcost_unit0
  47
  48theorem sectorCost_reciprocal_symm {r : ℝ} (hr : 0 < r) :
  49    sectorCost r = sectorCost r⁻¹ := Cost.Jcost_symm hr
  50
  51theorem sectorCost_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ sectorCost r :=
  52  Cost.Jcost_nonneg hr
  53
  54theorem sectorCost_pos_off_vacuum {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  55    0 < sectorCost r := Cost.Jcost_pos_of_ne_one r hr hne
  56
  57/-- Total Lagrangian cost is the sum of per-sector costs (no
  58cross-sector mixing at tree level on the canonical sector). -/
  59def totalCost (r : SMLagrangianSector → ℝ) : ℝ :=
  60  sectorCost (r .gaugeKinetic) + sectorCost (r .fermionKinetic) +
  61    sectorCost (r .yukawa) + sectorCost (r .higgsPotential)
  62
  63/-- Total cost vanishes when every sector sits at unity. -/
  64theorem totalCost_zero_at_vacuum :
  65    totalCost (fun _ => 1) = 0 := by
  66  unfold totalCost
  67  simp [sectorCost_zero_at_vacuum]
  68
  69/-- Total cost is nonnegative when every sector is in the physical
  70domain `r > 0`. -/
  71theorem totalCost_nonneg (r : SMLagrangianSector → ℝ)
  72    (h : ∀ s, 0 < r s) : 0 ≤ totalCost r := by
  73  unfold totalCost
  74  have h1 := sectorCost_nonneg (h .gaugeKinetic)
  75  have h2 := sectorCost_nonneg (h .fermionKinetic)
  76  have h3 := sectorCost_nonneg (h .yukawa)
  77  have h4 := sectorCost_nonneg (h .higgsPotential)
  78  linarith
  79
  80/-- Sector count = 4 (matches the canonical SM Lagrangian decomposition). -/
  81theorem sector_count : Fintype.card SMLagrangianSector = 4 := by decide
  82
  83structure SMLagrangianCert where
  84  vacuum_zero : sectorCost 1 = 0
  85  reciprocal_symm : ∀ {r : ℝ}, 0 < r → sectorCost r = sectorCost r⁻¹
  86  cost_nonneg : ∀ {r : ℝ}, 0 < r → 0 ≤ sectorCost r
  87  total_vacuum_zero : totalCost (fun _ => 1) = 0
  88  total_nonneg :
  89    ∀ (r : SMLagrangianSector → ℝ), (∀ s, 0 < r s) → 0 ≤ totalCost r
  90  sector_count : Fintype.card SMLagrangianSector = 4
  91
  92/-- SM-Lagrangian-skeleton certificate. -/
  93def smLagrangianCert : SMLagrangianCert where
  94  vacuum_zero := sectorCost_zero_at_vacuum
  95  reciprocal_symm := sectorCost_reciprocal_symm
  96  cost_nonneg := sectorCost_nonneg
  97  total_vacuum_zero := totalCost_zero_at_vacuum
  98  total_nonneg := totalCost_nonneg
  99  sector_count := sector_count
 100
 101end
 102end SMLagrangianSkeleton
 103end QRFT
 104end Foundation
 105end IndisputableMonolith
 106