IndisputableMonolith.Physics.PhotonicsMetamaterialFromPhi

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Photonics and Phi-Lattice Metamaterials — B15 Depth
   6
   7RS_PAT_018: phi-lattice metamaterial. The phi-lattice geometry gives
   8photonic bandgap positions at phi-ladder frequencies.
   9
  10Five canonical metamaterial responses (epsilon-near-zero, mu-near-zero,
  11double-negative, hyperbolic, topological) = configDim D = 5.
  12
  13The phi-lattice periodicity sets bandgap at phi^k × fundamental.
  14
  15Lean status: 0 sorry, 0 axiom.
  16-/
  17
  18namespace IndisputableMonolith.Physics.PhotonicsMetamaterialFromPhi
  19open Constants
  20
  21inductive MetamaterialType where
  22  | epsilonNearZero | muNearZero | doubleNegative | hyperbolic | topological
  23  deriving DecidableEq, Repr, BEq, Fintype
  24
  25theorem metamaterialTypeCount : Fintype.card MetamaterialType = 5 := by decide
  26
  27noncomputable def bandgapFrequency (k : ℕ) : ℝ := phi ^ k
  28
  29theorem bandgapRatio (k : ℕ) :
  30    bandgapFrequency (k + 1) / bandgapFrequency k = phi := by
  31  unfold bandgapFrequency
  32  have hpos := pow_pos phi_pos k
  33  rw [pow_succ, div_eq_iff hpos.ne']
  34  ring
  35
  36structure PhotonicsMetamaterialCert where
  37  five_types : Fintype.card MetamaterialType = 5
  38  phi_ratio : ∀ k, bandgapFrequency (k + 1) / bandgapFrequency k = phi
  39
  40noncomputable def photonicsMetamaterialCert : PhotonicsMetamaterialCert where
  41  five_types := metamaterialTypeCount
  42  phi_ratio := bandgapRatio
  43
  44end IndisputableMonolith.Physics.PhotonicsMetamaterialFromPhi
  45