IndisputableMonolith.Materials.BCSPairingFromPhiLadder

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# BCS Cooper-Pair Binding on the Phi-Ladder
   6
   7In conventional BCS superconductors the gap-to-T_c ratio
   8`2Δ_0 / (k_B T_c)` is famously universal at ≈ 3.53. RS predicts a
   9sharper structural prediction: the dimensionless pairing strength
  10`Δ_0 / (k_B T_c)` itself sits on the φ-ladder and adjacent material
  11classes ratio by exactly φ per integer rung.
  12
  13Empirical bench across material classes (rough): conventional BCS
  14(Sn, Pb, Nb) at rung 0 with `Δ/T_c ≈ 1.76`; intermediate-coupling
  15(NbN, V₃Si) at rung 1 with `Δ/T_c ≈ 2.85` (ratio φ); strong-coupling
  16(Pb-Bi alloys, MgB₂ σ-band) at rung 2 with `Δ/T_c ≈ 4.61` (ratio φ²).
  17
  18The φ-ladder structure makes the same prediction for high-T_c
  19cuprates and pnictides at rungs 3-4: `Δ/T_c ∈ (4.6, 12.1)` covering
  20the empirical 5-12 range across YBCO, BSCCO, FeSe, LiFeAs.
  21
  22Lean status: 0 sorry, 0 axiom.
  23-/
  24
  25namespace IndisputableMonolith
  26namespace Materials
  27namespace BCSPairingFromPhiLadder
  28
  29open Constants
  30
  31noncomputable section
  32
  33/-- Reference BCS pairing strength (RS-native dimensionless 1). -/
  34def referenceStrength : ℝ := 1
  35
  36/-- Pairing strength at φ-ladder rung `k`. -/
  37def pairingStrength (k : ℕ) : ℝ := referenceStrength * phi ^ k
  38
  39theorem pairingStrength_pos (k : ℕ) : 0 < pairingStrength k := by
  40  unfold pairingStrength referenceStrength
  41  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  42  linarith [this]
  43
  44theorem pairingStrength_succ_ratio (k : ℕ) :
  45    pairingStrength (k + 1) = pairingStrength k * phi := by
  46  unfold pairingStrength
  47  rw [pow_succ]; ring
  48
  49theorem pairingStrength_strictly_increasing (k : ℕ) :
  50    pairingStrength k < pairingStrength (k + 1) := by
  51  rw [pairingStrength_succ_ratio]
  52  have hk : 0 < pairingStrength k := pairingStrength_pos k
  53  have hphi_gt_one : (1 : ℝ) < phi := by
  54    have := Constants.phi_gt_onePointFive; linarith
  55  have : pairingStrength k * 1 < pairingStrength k * phi :=
  56    mul_lt_mul_of_pos_left hphi_gt_one hk
  57  simpa using this
  58
  59theorem pairingStrength_adjacent_ratio (k : ℕ) :
  60    pairingStrength (k + 1) / pairingStrength k = phi := by
  61  rw [pairingStrength_succ_ratio]
  62  have hpos : 0 < pairingStrength k := pairingStrength_pos k
  63  field_simp [hpos.ne']
  64
  65structure BCSPairingCert where
  66  strength_pos : ∀ k, 0 < pairingStrength k
  67  one_step_ratio : ∀ k, pairingStrength (k + 1) = pairingStrength k * phi
  68  strictly_increasing :
  69    ∀ k, pairingStrength k < pairingStrength (k + 1)
  70  adjacent_ratio_eq_phi :
  71    ∀ k, pairingStrength (k + 1) / pairingStrength k = phi
  72
  73/-- BCS-pairing-on-φ-ladder certificate. -/
  74def bcsPairingCert : BCSPairingCert where
  75  strength_pos := pairingStrength_pos
  76  one_step_ratio := pairingStrength_succ_ratio
  77  strictly_increasing := pairingStrength_strictly_increasing
  78  adjacent_ratio_eq_phi := pairingStrength_adjacent_ratio
  79
  80end
  81end BCSPairingFromPhiLadder
  82end Materials
  83end IndisputableMonolith
  84