T_c_optimization_finite_search

formal source

 108
 109/-- **THEOREM.** On any finite candidate range, an optimal rung
 110exists. This is the single-parameter optimization claim of RS_PAT_010. -/
 111theorem T_c_optimization_finite_search
 112    (omega_0 lam : ℝ) (n : ℕ) (h : 0 < n) :
 113    ∃ k_opt ∈ Finset.range n,
 114      ∀ k ∈ Finset.range n, T_c_phi_rung omega_0 lam k ≤ T_c_phi_rung omega_0 lam k_opt := by
 115  have hne : (Finset.range n).Nonempty := ⟨0, by simp [Finset.mem_range]; exact h⟩
 116  exact Finset.exists_max_image (Finset.range n) (T_c_phi_rung omega_0 lam) hne
 117
 118/-! ## §3. Single-parameter collapse -/
 119
 120/-- **THEOREM.** The optimization landscape collapses from
 121multi-parameter to a single integer parameter (the φ-rung): the
 122optimal T_c on a finite rung range is achieved at exactly one integer
 123`k_opt`. -/
 124theorem phi_ladder_optimization_collapses
 125    (omega_0 lam : ℝ) (n : ℕ) (h : 0 < n) :
 126    ∃ k_opt : ℕ, k_opt ∈ Finset.range n ∧
 127      T_c_phi_rung omega_0 lam k_opt =
 128        Finset.sup' (Finset.range n)
 129          ⟨0, by simp [Finset.mem_range]; exact h⟩
 130          (T_c_phi_rung omega_0 lam) := by
 131  have hne : (Finset.range n).Nonempty := ⟨0, by simp [Finset.mem_range]; exact h⟩
 132  obtain ⟨k_opt, hmem, h_eq⟩ :=
 133    Finset.exists_mem_eq_sup' hne (T_c_phi_rung omega_0 lam)
 134  exact ⟨k_opt, hmem, h_eq.symm⟩
 135
 136/-! ## §4. Master certificate -/
 137
 138/-- **HYDRIDE SC OPTIMIZATION MASTER CERTIFICATE.** Five clauses:
 139
 1401. `mu_star_in_band`: μ* ∈ (0, 1).
 1412. `lambda_pos`: e-ph coupling positive.