== (a) exact checks (rational arithmetic): 2 Var_mu(|x|) = (1/2)[d + d(d-1) a2 - d^2 a1^2] and E_mu|x| = (d/2)(1 - a1)
d=1: identities hold exactly at the 60 rational points p = k/61: True
d=2: identities hold exactly at the 60 rational points p = k/61: True
d=3: identities hold exactly at the 60 rational points p = k/61: True
d=4: identities hold exactly at the 60 rational points p = k/61: True
d=5: identities hold exactly at the 60 rational points p = k/61: True
2Var at p=1/d minus closed form 2d(d^4+7d^3-20d^2+16d-4)/(45d^3-96d^2+68d-16): 0
45(d^4+7d^3-20d^2+16d-4) - d(45d^3-96d^2+68d-16) = 411*d**3 - 968*d**2 + 736*d - 180 ; values at d=2..6: [708, 4413, 13580, 30675, 58164]
denominator 45d^3-96d^2+68d-16 at d=2..6: [96, 539, 1600, 3549, 6656]

== (b) full chain vs lumped chain: max over p-grid of |tau_full - tau_lumped| (0 means lambda_2 is attained on level functions)
d= 2: max|tau_full - tau_lumped| = 8.882e-16;  max tau_full/tau_lumped = 1.000000
d= 3: max|tau_full - tau_lumped| = 7.105e-15;  max tau_full/tau_lumped = 1.000000
d= 4: max|tau_full - tau_lumped| = 7.994e-15;  max tau_full/tau_lumped = 1.000000
d= 5: max|tau_full - tau_lumped| = 2.132e-14;  max tau_full/tau_lumped = 1.000000
d= 6: max|tau_full - tau_lumped| = 7.550e-14;  max tau_full/tau_lumped = 1.000000
d= 7: max|tau_full - tau_lumped| = 7.816e-14;  max tau_full/tau_lumped = 1.000000
d= 8: max|tau_full - tau_lumped| = 1.368e-13;  max tau_full/tau_lumped = 1.000000
d= 9: max|tau_full - tau_lumped| = 1.119e-13;  max tau_full/tau_lumped = 1.000000
d=10: max|tau_full - tau_lumped| = 5.294e-13;  max tau_full/tau_lumped = 1.000000

== (c) large d (lumped chain, a lower bound for tau): tau(0)=d/2, tau(1-)=d
   d  tau_l(1/d)  2Var(1/d)   2d^2/45 dVar/E|x| at 1/d sup_p tau_l argmax p*d sup/(d/2)  sup/d^2
   4       3.874      2.220     0.711            3.700       4.442      2.400     2.221   0.2776
   8      11.108      5.991     2.844            9.414      11.993      2.100     2.998   0.1874
  12      22.951     11.174     6.400           17.269      23.665      1.700     3.944   0.1643
  16      39.765     17.777    11.378           27.259      40.078      1.350     5.010   0.1566
  24      88.563     35.249    25.600           53.639      88.656      0.900     7.388   0.1539
  32     157.663     58.408    45.511           88.554     159.544      0.650     9.972   0.1558
  48     356.879    121.793   102.400          183.985     370.925      0.400    15.455   0.1610
  64     637.530    207.932   182.044          313.549     676.960      0.250    21.155   0.1653
  96    1443.344    448.478   409.600          675.077    1577.081      0.150    32.856   0.1711
 128    2575.343    780.046   728.178         1173.139    2864.118      0.100    44.752   0.1748
 192    5818.297   1716.247  1638.400         2578.864    6603.927      0.100    68.791   0.1791
 256   10366.839   3016.538  2912.711         4530.721   11918.570      0.050    93.114   0.1819
 384   23381.386   6709.385  6553.600        10072.837   27232.172      0.050   141.834   0.1847
 512   41619.765  11858.588 11650.844        17799.486   48794.404      0.050   190.603   0.1861

== (c') fixed p, growing d: tau_lumped(p) * p / d  (heuristic: tau ~ d/p for fixed p)
p=0.5: d=16: 0.8116  d=64: 0.9424  d=256: 0.9847  d=1024: 0.9961
p=0.2: d=16: 0.4575  d=64: 0.7646  d=256: 0.9262  d=1024: 0.9802
p=0.1: d=16: 0.2496  d=64: 0.5778  d=256: 0.8402  d=1024: 0.9536
p=0.05: d=16: 0.1223  d=64: 0.3876  d=256: 0.7107  d=1024: 0.9046
