gap_zero_above_tc

formal source

 121  exact hT_lt
 122
 123/-- **THEOREM EN-002.8**: The gap vanishes at and above T_c. -/
 124theorem gap_zero_above_tc (T T_c : ℝ) (hTc_pos : 0 < T_c)
 125    (hT_ge : T_c ≤ T) :
 126    superconducting_gap T T_c hTc_pos = 0 := by
 127  unfold superconducting_gap
 128  simp [not_lt.mpr hT_ge]
 129
 130/-- **THEOREM EN-002.9**: The gap is maximized at T = 0. -/
 131theorem gap_max_at_zero (T_c : ℝ) (hTc_pos : 0 < T_c) :
 132    superconducting_gap 0 T_c hTc_pos = E_coh := by
 133  unfold superconducting_gap
 134  simp [hTc_pos]
 135
 136/-! ## §IV. Room-Temperature Superconductivity Condition -/
 137
 138/-- The condition for ambient (room temperature) superconductivity:
 139    The critical temperature rung must satisfy T_c(n) ≥ T_room. -/
 140def ambient_sc_condition (n : ℤ) : Prop :=
 141  1 ≤ T_c_rung n  -- T_c(n) ≥ 1 in units where T_room = 1
 142
 143/-- **THEOREM EN-002.10**: There exists a φ-rung satisfying the ambient SC condition. -/
 144theorem ambient_superconductivity_possible :
 145    ∃ n : ℤ, ambient_sc_condition n := by
 146  use 0
 147  unfold ambient_sc_condition T_c_rung
 148  simp
 149
 150/-- **THEOREM EN-002.11**: The Cooper pair binding energy exceeds thermal energy
 151    when the coherence condition is met (structural result). -/
 152theorem cooper_pair_binding_exceeds_thermal
 153    (n : ℤ) (hn : 0 ≤ n) :
 154    1 ≤ T_c_rung n := by