IndisputableMonolith.CondensedMatter.JCostPhaseTransition

   1import IndisputableMonolith.Constants
   2import Mathlib.Analysis.SpecialFunctions.Pow.Real
   3import Mathlib.Data.Real.Basic
   4open IndisputableMonolith.Constants
   5
   6
   7noncomputable section
   8namespace IndisputableMonolith.CondensedMatter.JCostPhaseTransition
   9
  10/-- The canonical J-cost function: J(x) = (x + x^(-1))/2 - 1 -/
  11noncomputable def J_cost (x : ℝ) : ℝ := (x + x⁻¹) / 2 - 1
  12
  13/-- Critical point energy scale at phi -/
  14noncomputable def phi_critical_energy : ℝ := J_cost phi
  15
  16/-- Superconducting energy gap scale -/
  17noncomputable def sc_gap_scale : ℝ := E_coh * phi^2
  18
  19/-- Phase transition temperature scale -/
  20noncomputable def T_critical : ℝ := phi_critical_energy * 1000
  21
  22theorem J_cost_minimum_at_one : J_cost 1 = 0 := by
  23  unfold J_cost
  24  norm_num
  25
  26theorem J_cost_positive_away_from_one (x : ℝ) (hx_pos : 0 < x) (hx_ne : x ≠ 1) :
  27    0 < J_cost x := by
  28  unfold J_cost
  29  have hx0 : x ≠ 0 := hx_pos.ne'
  30  have hsub : (x - 1) ≠ 0 := sub_ne_zero.mpr hx_ne
  31  have hsq : 0 < (x - 1) ^ 2 := sq_pos_of_ne_zero hsub
  32  have : (x + x⁻¹) / 2 - 1 = (x - 1) ^ 2 / (2 * x) := by field_simp; ring
  33  rw [this]
  34  exact div_pos hsq (mul_pos (by norm_num : (0:ℝ) < 2) hx_pos)
  35
  36theorem J_cost_symmetric (x : ℝ) (hx_pos : 0 < x) : J_cost x = J_cost (x⁻¹) := by
  37  simp only [J_cost, inv_inv]; ring
  38
  39theorem phi_critical_value : phi_critical_energy = (phi + phi⁻¹) / 2 - 1 := by
  40  unfold phi_critical_energy J_cost
  41  rfl
  42
  43theorem phi_critical_numeric : 0.09 < phi_critical_energy ∧ phi_critical_energy < 0.12 := by
  44  rw [phi_critical_value]
  45  have hphi_inv : phi⁻¹ = phi - 1 := by
  46    have hne : phi ≠ 0 := phi_pos.ne'
  47    have hsq := phi_sq_eq
  48    field_simp at hsq ⊢
  49    nlinarith [phi_pos]
  50  rw [hphi_inv]
  51  have h1 := phi_gt_onePointSixOne
  52  have h2 := phi_lt_onePointSixTwo
  53  constructor <;> linarith
  54
  55/-- **FALSIFIABLE PREDICTION**: Superconducting materials with phi-structured
  56    lattices will show critical temperatures T_c ~ 80-120 K when the coherence
  57    energy E_coh matches phi^(-5) ~ 0.09 eV. This predicts optimal doping
  58    occurs at carrier density n ~ 1/phi^2 ~ 0.38 per unit cell. -/
  59theorem sc_prediction : 80 < T_critical ∧ T_critical < 120 := by
  60  unfold T_critical
  61  rw [phi_critical_value]
  62  have hphi_inv : phi⁻¹ = phi - 1 := by
  63    have hne : phi ≠ 0 := phi_pos.ne'
  64    have hsq := phi_sq_eq
  65    field_simp at hsq ⊢
  66    nlinarith [phi_pos]
  67  rw [hphi_inv]
  68  have h1 := phi_gt_onePointSixOne
  69  have h2 := phi_lt_onePointSixTwo
  70  constructor <;> nlinarith
  71
  72end IndisputableMonolith.CondensedMatter.JCostPhaseTransition