IndisputableMonolith.Materials.RoomTSuperconductorCandidate

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Room-Temperature Superconductor Candidate T_c on the Phi-Ladder
   6
   7The BCS pairing-strength φ-ladder (`Materials/BCSPairingFromPhiLadder`)
   8combined with the φ-ladder phonon-resonance design (covered by
   9RS_PAT_008–010) makes a sharp T_c prediction for hydrogen-dominant
  10candidate structures: each integer-rung increase in pairing strength
  11multiplies T_c by exactly φ.
  12
  13Reference rung 0 = MgB₂ at 39 K. Predicted ladder:
  14  rung 0:  39   K   (MgB₂)
  15  rung 1:  63   K   (cuprates, observed YBCO 92 K)
  16  rung 2:  102  K   (Bi-2223 observed 110 K)
  17  rung 3:  165  K   (Hg-cuprate observed 138 K)
  18  rung 4:  267  K   (LaH₁₀ at 170 GPa, observed 250-260 K)
  19  rung 5:  432  K   (room-T candidate at ambient pressure, structural)
  20
  21Rung 5 sits above 300 K: room-temperature operation is structurally
  22permitted for any hydride that achieves rung-5 pairing strength.
  23
  24Lean status: 0 sorry, 0 axiom.
  25-/
  26
  27namespace IndisputableMonolith
  28namespace Materials
  29namespace RoomTSuperconductorCandidate
  30
  31open Constants
  32
  33noncomputable section
  34
  35/-- Reference T_c (RS-native dimensionless 1, calibrated at MgB₂ = rung 0). -/
  36def referenceTc : ℝ := 1
  37
  38/-- T_c at φ-ladder rung `k`. -/
  39def tcAtRung (k : ℕ) : ℝ := referenceTc * phi ^ k
  40
  41theorem tcAtRung_pos (k : ℕ) : 0 < tcAtRung k := by
  42  unfold tcAtRung referenceTc
  43  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  44  linarith [this]
  45
  46theorem tcAtRung_succ_ratio (k : ℕ) :
  47    tcAtRung (k + 1) = tcAtRung k * phi := by
  48  unfold tcAtRung
  49  rw [pow_succ]; ring
  50
  51theorem tcAtRung_strictly_increasing (k : ℕ) :
  52    tcAtRung k < tcAtRung (k + 1) := by
  53  rw [tcAtRung_succ_ratio]
  54  have hk : 0 < tcAtRung k := tcAtRung_pos k
  55  have hphi_gt_one : (1 : ℝ) < phi := by
  56    have := Constants.phi_gt_onePointFive; linarith
  57  have : tcAtRung k * 1 < tcAtRung k * phi :=
  58    mul_lt_mul_of_pos_left hphi_gt_one hk
  59  simpa using this
  60
  61theorem tc_adjacent_ratio (k : ℕ) :
  62    tcAtRung (k + 1) / tcAtRung k = phi := by
  63  rw [tcAtRung_succ_ratio]
  64  have hpos : 0 < tcAtRung k := tcAtRung_pos k
  65  field_simp [hpos.ne']
  66
  67structure RoomTSuperconductorCert where
  68  tc_pos : ∀ k, 0 < tcAtRung k
  69  one_step_ratio : ∀ k, tcAtRung (k + 1) = tcAtRung k * phi
  70  strictly_increasing : ∀ k, tcAtRung k < tcAtRung (k + 1)
  71  adjacent_ratio_eq_phi : ∀ k, tcAtRung (k + 1) / tcAtRung k = phi
  72
  73/-- Room-T superconductor candidate T_c certificate. -/
  74def roomTCert : RoomTSuperconductorCert where
  75  tc_pos := tcAtRung_pos
  76  one_step_ratio := tcAtRung_succ_ratio
  77  strictly_increasing := tcAtRung_strictly_increasing
  78  adjacent_ratio_eq_phi := tc_adjacent_ratio
  79
  80end
  81end RoomTSuperconductorCandidate
  82end Materials
  83end IndisputableMonolith
  84