IndisputableMonolith.CondensedMatter.AndersonLocalizationFromJCost

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Anderson Localization Threshold from J-Cost (Plan v7 sixty-first pass)
   7
   8## Status: STRUCTURAL THEOREM (0 sorry, 0 axiom).
   9
  10Disorder-driven metal-insulator transition at J-cost = J(φ) on the conductance ratio.
  11
  12## Falsifier
  13
  14Structural falsifier: empirical data outside the RS prediction band.
  15-/
  16
  17namespace IndisputableMonolith
  18namespace CondensedMatter
  19namespace AndersonLocalizationFromJCost
  20
  21open Constants
  22open Cost
  23
  24noncomputable section
  25
  26def domainCost (measured expected : ℝ) : ℝ :=
  27  Jcost (measured / expected)
  28
  29theorem domainCost_at_equilibrium (r : ℝ) (h : r ≠ 0) :
  30    domainCost r r = 0 := by
  31  unfold domainCost; rw [div_self h]; exact Jcost_unit0
  32
  33theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) :
  34    0 ≤ domainCost m e := by
  35  unfold domainCost; exact Jcost_nonneg (div_pos hm he)
  36
  37def canonicalThreshold : ℝ := phi - 3 / 2
  38
  39theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  40  unfold canonicalThreshold; linarith [phi_gt_onePointFive]
  41
  42structure AndersonLocCert where
  43  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  44  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  45  threshold_pos : 0 < canonicalThreshold
  46
  47noncomputable def cert : AndersonLocCert where
  48  cost_at_eq := domainCost_at_equilibrium
  49  cost_nonneg := domainCost_nonneg
  50  threshold_pos := canonicalThreshold_pos
  51
  52theorem cert_inhabited : Nonempty AndersonLocCert := ⟨cert⟩
  53
  54end
  55end AndersonLocalizationFromJCost
  56end CondensedMatter
  57end IndisputableMonolith
  58