potential_confining

formal source

  60
  61/-- **THEOREM**: The potential is confining (grows with r).
  62    Proof: V(r₂) - V(r₁) = (r₂ - r₁)(α/(r₁r₂) + σ) > 0 since r₂ > r₁, α ≥ 0, σ > 0. -/
  63theorem potential_confining (alpha sigma r₁ r₂ : ℝ) (ha : alpha ≥ 0) (hs : sigma > 0)
  64    (hr₁ : r₁ > 0) (hr₂ : r₂ > r₁) :
  65    cornellPotential alpha sigma r₂ (lt_trans hr₁ hr₂)
  66    > cornellPotential alpha sigma r₁ hr₁ := by
  67  unfold cornellPotential
  68  have hr₂_pos : r₂ > 0 := lt_trans hr₁ hr₂
  69  have hr₁_ne : r₁ ≠ 0 := ne_of_gt hr₁
  70  have hr₂_ne : r₂ ≠ 0 := ne_of_gt hr₂_pos
  71  have hdiff : r₂ - r₁ > 0 := sub_pos.mpr hr₂
  72  have hr₁r₂_pos : r₁ * r₂ > 0 := mul_pos hr₁ hr₂_pos
  73  rw [show (-alpha / r₂ + sigma * r₂ > -alpha / r₁ + sigma * r₁) ↔
  74          (-alpha / r₂ + sigma * r₂ - (-alpha / r₁ + sigma * r₁) > 0) from by
  75          constructor <;> intro h <;> linarith]
  76  have h : -alpha / r₂ + sigma * r₂ - (-alpha / r₁ + sigma * r₁)
  77         = (r₂ - r₁) * (alpha / (r₁ * r₂) + sigma) := by field_simp; ring
  78  rw [h]
  79  exact mul_pos hdiff (add_pos_of_nonneg_of_pos (div_nonneg ha (le_of_lt hr₁r₂_pos)) hs)
  80
  81/-! ## J-Cost Origin of Confinement -/
  82
  83/-- In RS, the confining potential comes from J-cost of maintaining color separation:
  84
  85    1. Color charge is a "ledger imbalance"
  86    2. Separating charges stretches the ledger connection
  87    3. The cost of stretching grows with distance
  88    4. This creates the linear σr term -/
  89noncomputable def jcostColorPotential (r : ℝ) (hr : r > 0) : ℝ :=
  90  -- Schematic: J-cost for color separation
  91  -- Short range: recognition events give -α/r
  92  -- Long range: ledger tension gives σr
  93  cornellPotential alphaSshort stringTension r hr