gap_max_at_zero

formal source

 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
 155  unfold T_c_rung
 156  rcases hn.lt_or_eq with hn' | hn'
 157  · exact (one_lt_zpow₀ one_lt_phi hn').le
 158  · simp [hn'.symm]
 159
 160/-! ## §V. Coherence Condition: φ-Phonon Coupling -/
 161