SLSQP maximisation of Phi(s)/(w(a)w(b)) over k-chains (value 1 is attained by Farey chains)
  gap (1, 1)  k=2  max Phi/(w(a)w(b)) - 1 =  2.220e-16   (min relative constraint slack there: -0.0e+00)
  gap (1, 1)  k=3  max Phi/(w(a)w(b)) - 1 =  8.882e-16   (min relative constraint slack there: -4.4e-15)
  gap (1, 1)  k=4  max Phi/(w(a)w(b)) - 1 =  9.992e-15   (min relative constraint slack there: -3.7e-14)
  gap (1, 1)  k=5  max Phi/(w(a)w(b)) - 1 =  8.882e-16   (min relative constraint slack there: -6.6e-15)
  gap (1, 1)  k=6  max Phi/(w(a)w(b)) - 1 =  5.329e-15   (min relative constraint slack there: -2.5e-14)
  gap (1, 0.1)  k=2  max Phi/(w(a)w(b)) - 1 =  2.220e-16   (min relative constraint slack there: -0.0e+00)
  gap (1, 0.1)  k=3  max Phi/(w(a)w(b)) - 1 =  1.205e-09   (min relative constraint slack there: -5.3e-10)
  gap (1, 0.1)  k=4  max Phi/(w(a)w(b)) - 1 =  2.248e-10   (min relative constraint slack there: -1.4e-10)
  gap (1, 0.1)  k=5  max Phi/(w(a)w(b)) - 1 =  5.563e-10   (min relative constraint slack there: -2.7e-10)
  gap (1, 0.1)  k=6  max Phi/(w(a)w(b)) - 1 =  1.050e-10   (min relative constraint slack there: -8.8e-11)
  gap (0.1, 0.13)  k=2  max Phi/(w(a)w(b)) - 1 = -1.554e-15   (min relative constraint slack there: -1.3e-16)
  gap (0.1, 0.13)  k=3  max Phi/(w(a)w(b)) - 1 =  9.748e-14   (min relative constraint slack there: -1.7e-10)
  gap (0.1, 0.13)  k=4  max Phi/(w(a)w(b)) - 1 = -1.332e-15   (min relative constraint slack there: -3.2e-14)
  gap (0.1, 0.13)  k=5  max Phi/(w(a)w(b)) - 1 = -1.821e-12   (min relative constraint slack there: -4.0e-14)
  gap (0.1, 0.13)  k=6  max Phi/(w(a)w(b)) - 1 = -1.257e-11   (min relative constraint slack there: -4.3e-10)
  gap (5, 3)  k=2  max Phi/(w(a)w(b)) - 1 =  5.773e-15   (min relative constraint slack there: -4.6e-15)
  gap (5, 3)  k=3  max Phi/(w(a)w(b)) - 1 =  7.550e-15   (min relative constraint slack there: -5.8e-15)
  gap (5, 3)  k=4  max Phi/(w(a)w(b)) - 1 =  2.887e-14   (min relative constraint slack there: -6.9e-14)
  gap (5, 3)  k=5  max Phi/(w(a)w(b)) - 1 =  2.220e-16   (min relative constraint slack there: -5.6e-16)
  gap (5, 3)  k=6  max Phi/(w(a)w(b)) - 1 =  2.220e-16   (min relative constraint slack there: -4.4e-16)
  gap (40, 0.7)  k=2  max Phi/(w(a)w(b)) - 1 =  1.088e-14   (min relative constraint slack there: -5.6e-15)
  gap (40, 0.7)  k=3  max Phi/(w(a)w(b)) - 1 =  4.441e-16   (min relative constraint slack there: -4.5e-14)
  gap (40, 0.7)  k=4  max Phi/(w(a)w(b)) - 1 =  1.332e-15   (min relative constraint slack there: -6.1e-15)
  gap (40, 0.7)  k=5  max Phi/(w(a)w(b)) - 1 =  6.661e-16   (min relative constraint slack there: -1.1e-14)
  gap (40, 0.7)  k=6  max Phi/(w(a)w(b)) - 1 =  1.554e-15   (min relative constraint slack there: -2.4e-14)
  gap (0.03, 0.02)  k=2  max Phi/(w(a)w(b)) - 1 = -7.105e-15   (min relative constraint slack there: -0.0e+00)
  gap (0.03, 0.02)  k=3  max Phi/(w(a)w(b)) - 1 = -1.918e-13   (min relative constraint slack there: -3.4e-13)
  gap (0.03, 0.02)  k=4  max Phi/(w(a)w(b)) - 1 = -7.105e-15   (min relative constraint slack there: -1.8e-16)
  gap (0.03, 0.02)  k=5  max Phi/(w(a)w(b)) - 1 = -7.105e-15   (min relative constraint slack there: -5.0e-13)
  gap (0.03, 0.02)  k=6  max Phi/(w(a)w(b)) - 1 = -2.128e-12   (min relative constraint slack there: -0.0e+00)
  overall max Phi/(w(a)w(b)) - 1 =  1.205e-09
  (SLSQP accepts constraint violations up to about 1e-8; positive excesses of order 1e-9 occur only at
   points that violate a constraint by a comparable amount, next to Farey chains, where Phi = w(a)w(b))
random recursive packings (chains inside chains, remaining gaps valued greedily)
  gap (1, 1): 300 packings, max total/(w(a)w(b)) - 1 =  2.220e-16, min ratio = 0.7740
  gap (2, 0.5): 300 packings, max total/(w(a)w(b)) - 1 =  0.000e+00, min ratio = 0.7170
  gap (0.2, 0.2): 300 packings, max total/(w(a)w(b)) - 1 =  1.554e-15, min ratio = 0.0375
