excited_jcost

formal source

  35theorem vacuum_jcost : Jcost 1 = 0 := Jcost_unit0
  36
  37/-- Excited sectors: J > 0. -/
  38theorem excited_jcost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  39    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  40
  41/-- Spectral gap exists on discretized lattice (structural claim). -/
  42def latticeSpacingGap (a : ℝ) (ha : 0 < a) : Prop := ∃ Δ > 0, Δ < Jcost (1 + a)
  43
  44theorem lattice_gap_witness (a : ℝ) (ha : 0 < a) : latticeSpacingGap a ha := by
  45  unfold latticeSpacingGap
  46  refine ⟨Jcost (1 + a) / 2, ?_, ?_⟩
  47  · apply div_pos
  48    · exact Jcost_pos_of_ne_one _ (by linarith) (by linarith)
  49    · norm_num
  50  · linarith [Jcost_pos_of_ne_one (1 + a) (by linarith) (by linarith)]
  51
  52structure HamiltonianSpectrumCert where
  53  five_sectors : Fintype.card SpectralSector = 5
  54  vacuum : Jcost 1 = 0
  55  excited : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  56  lattice_gap : ∀ (a : ℝ) (ha : 0 < a), latticeSpacingGap a ha
  57
  58def hamiltonianSpectrumCert : HamiltonianSpectrumCert where
  59  five_sectors := spectralSectorCount
  60  vacuum := vacuum_jcost
  61  excited := excited_jcost
  62  lattice_gap := lattice_gap_witness
  63
  64end IndisputableMonolith.Physics.RecognitionHamiltonianSpectrum