Referee 2 MC (r2_is): deviation d = r/e^{-y^2/2} - 1 at i = y n^{3/2}, versus the paper's 'next term' e^{-y^3/(3 sqrt n)} - 1
    n     y         r       SE         d      pred  d*sqrt(n)  (d-pred)*sqrt(n)
  100  0.25  0.991083  1.5e-04  +0.02254  -0.00052     +0.225            +0.231
  100  0.50  0.909469  1.4e-04  +0.03056  -0.00416     +0.306            +0.347
  100  1.00  0.616813  1.2e-04  +0.01695  -0.03278     +0.170            +0.497
  100  1.50  0.304511  7.5e-05  -0.06204  -0.10640     -0.620            +0.444
  100  2.00  0.105303  3.3e-05  -0.22191  -0.23407     -2.219            +0.122
  100  3.00  0.003533  1.8e-06  -0.68197  -0.59343     -6.820            -0.885
  400  0.25  0.977415  1.8e-04  +0.00844  -0.00026     +0.169            +0.174
  400  0.50  0.893331  1.8e-04  +0.01228  -0.00208     +0.246            +0.287
  400  1.00  0.611233  1.4e-04  +0.00775  -0.01653     +0.155            +0.486
  400  1.50  0.316495  9.1e-05  -0.02513  -0.05470     -0.503            +0.591
  400  2.00  0.122127  4.3e-05  -0.09760  -0.12483     -1.952            +0.545
  400  3.00  0.007031  3.6e-06  -0.36709  -0.36237     -7.342            -0.094
 1600  0.25  0.972504  1.7e-04  +0.00337  -0.00013     +0.135            +0.140
 1600  0.50  0.887263  1.7e-04  +0.00540  -0.00104     +0.216            +0.258
 1600  1.00  0.608989  1.2e-04  +0.00405  -0.00830     +0.162            +0.494
 1600  1.50  0.321070  8.1e-05  -0.01103  -0.02773     -0.441            +0.668
 1600  2.00  0.129248  3.7e-05  -0.04498  -0.06449     -1.799            +0.781
 1600  3.00  0.009042  3.6e-06  -0.18607  -0.20148     -7.443            +0.617
 6400  0.25  0.971079  4.6e-05  +0.00190  -0.00007     +0.152            +0.158
 6400  0.50  0.884385  3.6e-04  +0.00214  -0.00052     +0.171            +0.213
 6400  1.00  0.607627  2.0e-04  +0.00181  -0.00416     +0.145            +0.477
 6400  1.50  0.322971  1.1e-04  -0.00518  -0.01396     -0.414            +0.703
 6400  2.00  0.132389  4.8e-05  -0.02177  -0.03278     -1.742            +0.881
 6400  3.00  0.010062  6.2e-06  -0.09425  -0.10640     -7.540            +0.972

Sign check: for y <= 1 the observed deviation is POSITIVE at every n, while -y^3/(3 sqrt n) is negative;
the residual (d - pred)*sqrt(n) is roughly 0.5*y at n >= 400 (numerical fit only, not proved).
y=0.25: deviation ratios d(n)/d(4n): 2.67, 2.50, 1.77
y=0.5: deviation ratios d(n)/d(4n): 2.49, 2.27, 2.52
y=1.0: deviation ratios d(n)/d(4n): 2.19, 1.91, 2.24
y=1.5: deviation ratios d(n)/d(4n): 2.47, 2.28, 2.13
y=2.0: deviation ratios d(n)/d(4n): 2.27, 2.17, 2.07
y=3.0: deviation ratios d(n)/d(4n): 1.86, 1.97, 1.97
Correction (referee 2): the residual (d - pred)*sqrt(n) at n = 6400 is 0.16, 0.21, 0.48, 0.70, 0.88, 0.97 for
y = 0.25, 0.5, 1, 1.5, 2, 3, so "0.5*y" is only a rough description for y <= 2. The robust conclusions are:
(i) for y <= 1 the deviation has the opposite sign to -y^3/(3 sqrt n); (ii) at y = 1.5 that term overshoots the
deviation by a factor 2-3; (iii) only for y >= 2 does it account for most of the deviation.
