theorem landau_at_minimizer (t : ℝ) :
landauPotential t (max 0 (-t)) = -(max 0 (-t)) ^ 2 / 2 := by
unfold landauPotential
by_cases ht : 0 ≤ t
· rw [max_eq_left (by linarith : -t ≤ 0)]; ring
· push_neg at ht
rw [max_eq_right (by linarith : 0 ≤ -t)]; ringPhase Transition at T* (Gap #8)