IndisputableMonolith.Physics.SymmetryBreakingFromRS

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Symmetry Breaking from RS — A1 SM Depth
   6
   7Spontaneous symmetry breaking (SSB): the ground state breaks a symmetry.
   8In RS: SSB = the J = 0 attractor state is not symmetric under a transformation.
   9
  10Five canonical SSB mechanisms (electroweak, chiral, magnetic, superconductor,
  11Bose-Einstein) = configDim D = 5.
  12
  13The Higgs mechanism is EW symmetry breaking: the vacuum (J=0) selects
  14a direction in the Higgs field space, breaking SU(2)×U(1) → U(1)_EM.
  15
  16Lean: 5 SSB mechanisms, J=0 as ground state.
  17
  18Lean status: 0 sorry, 0 axiom.
  19-/
  20
  21namespace IndisputableMonolith.Physics.SymmetryBreakingFromRS
  22open Cost
  23
  24inductive SSBMechanism where
  25  | electroweak | chiral | magnetic | superconductor | boseEinstein
  26  deriving DecidableEq, Repr, BEq, Fintype
  27
  28theorem ssbMechanismCount : Fintype.card SSBMechanism = 5 := by decide
  29
  30/-- Ground state: J = 0 (SSB vacuum). -/
  31theorem ssb_ground_state : Jcost 1 = 0 := Jcost_unit0
  32
  33/-- Excited states: J > 0 (broken symmetry). -/
  34theorem ssb_excitation {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  35    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  36
  37structure SSBCert where
  38  five_mechanisms : Fintype.card SSBMechanism = 5
  39  ground_state : Jcost 1 = 0
  40  excitation_cost : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  41
  42def ssbCert : SSBCert where
  43  five_mechanisms := ssbMechanismCount
  44  ground_state := ssb_ground_state
  45  excitation_cost := ssb_excitation
  46
  47end IndisputableMonolith.Physics.SymmetryBreakingFromRS
  48