=== A. critical window: m = round(m_c + y sqrt(m_c)), increasing k ===
y=-2       d=            1891 k=            1000 m=      108 rho=  0.6836 y= -1.9794 p_exact=0.992810 p_mix=0.992810 |diff|=2.0e-14 p_AF=0.989742 Phi(-y/rt2)=0.919194
y=-2       d=           19626 k=           10000 m=      373 rho=  0.8147 y= -1.9801 p_exact=0.989777 p_mix=0.989777 |diff|=6.2e-14 p_AF=0.988152 Phi(-y/rt2)=0.919264
y=-2       d=          198764 k=          100000 m=     1235 rho=  0.8930 y= -1.9883 p_exact=0.982954 p_mix=0.982954 |diff|=2.4e-13 p_AF=0.981923 Phi(-y/rt2)=0.920127
y=-2       d=         1995995 k=         1000000 m=     4004 rho=  0.9387 y= -2.0024 p_exact=0.973184 p_mix=0.973184 |diff|=1.2e-12 p_AF=0.972544 Phi(-y/rt2)=0.921603
y=-2       d=       199959078 k=       100000000 m=    40921 rho=  0.9804 y= -1.9987 p_exact=0.950203 p_mix=0.950203 |diff|=4.3e-10 p_AF=0.949997 Phi(-y/rt2)=0.921213
y=-2       d=     19999588012 k=     10000000000 m=   411987 rho=  0.9938 y= -2.0006 p_exact=0.934556 p_mix=0.934556 |diff|=2.7e-10 p_AF=0.934496 Phi(-y/rt2)=0.921416
y=-2       d=   1999995871333 k=   1000000000000 m=  4128666 rho=  0.9980 y= -1.9998 p_exact=0.926651 p_mix=0.926651 |diff|=2.9e-09 p_AF=0.926633 Phi(-y/rt2)=0.921326
y=-2       d= 199999958685543 k= 100000000000000 m= 41314456 rho=  0.9994 y= -2.0000 p_exact=0.923382 p_mix=0.923382 |diff|=2.8e-08 p_AF=0.923376 Phi(-y/rt2)=0.921355

y=-1       d=            1880 k=            1000 m=      119 rho=  0.8300 y= -1.0170 p_exact=0.967107 p_mix=0.967107 |diff|=3.4e-14 p_AF=0.962262 Phi(-y/rt2)=0.763964
y=-1       d=           19606 k=           10000 m=      393 rho=  0.9044 y= -0.9963 p_exact=0.951293 p_mix=0.951293 |diff|=9.9e-14 p_AF=0.948993 Phi(-y/rt2)=0.759426
y=-1       d=          198728 k=          100000 m=     1271 rho=  0.9458 y= -0.9925 p_exact=0.923346 p_mix=0.923346 |diff|=3.4e-14 p_AF=0.922315 Phi(-y/rt2)=0.758588
y=-1       d=         1995931 k=         1000000 m=     4068 rho=  0.9689 y= -1.0069 p_exact=0.890556 p_mix=0.890556 |diff|=2.4e-13 p_AF=0.890191 Phi(-y/rt2)=0.761761
y=-1       d=       199958875 k=       100000000 m=    41124 rho=  0.9902 y= -1.0001 p_exact=0.826470 p_mix=0.826470 |diff|=9.3e-10 p_AF=0.826504 Phi(-y/rt2)=0.760274
y=-1       d=     19999587369 k=     10000000000 m=   412630 rho=  0.9969 y= -1.0004 p_exact=0.789004 p_mix=0.789004 |diff|=1.2e-09 p_AF=0.789042 Phi(-y/rt2)=0.760343
y=-1       d=   1999995869301 k=   1000000000000 m=  4130698 rho=  0.9990 y= -1.0002 p_exact=0.771668 p_mix=0.771668 |diff|=3.0e-10 p_AF=0.771684 Phi(-y/rt2)=0.760298
y=-1       d= 199999958679114 k= 100000000000000 m= 41320885 rho=  0.9997 y= -1.0000 p_exact=0.764543 p_mix=0.764543 |diff|=2.6e-08 p_AF=0.764549 Phi(-y/rt2)=0.760247

y=+0       d=            1868 k=            1000 m=      131 rho=  1.0058 y= +0.0330 p_exact=0.886722 p_mix=0.886722 |diff|=2.5e-14 p_AF=0.884187 Phi(-y/rt2)=0.490698
y=+0       d=           19586 k=           10000 m=      413 rho=  0.9988 y= -0.0124 p_exact=0.843836 p_mix=0.843836 |diff|=2.9e-14 p_AF=0.843674 Phi(-y/rt2)=0.503503
y=+0       d=          198692 k=          100000 m=     1307 rho=  1.0002 y= +0.0034 p_exact=0.771521 p_mix=0.771521 |diff|=1.3e-13 p_AF=0.772385 Phi(-y/rt2)=0.499047
y=+0       d=         1995866 k=         1000000 m=     4133 rho=  1.0001 y= +0.0042 p_exact=0.699198 p_mix=0.699198 |diff|=7.8e-16 p_AF=0.700122 Phi(-y/rt2)=0.498812
y=+0       d=       199958672 k=       100000000 m=    41327 rho=  1.0000 y= -0.0015 p_exact=0.592878 p_mix=0.592878 |diff|=1.3e-09 p_AF=0.593303 Phi(-y/rt2)=0.500435
y=+0       d=     19999586726 k=     10000000000 m=   413273 rho=  1.0000 y= -0.0002 p_exact=0.538121 p_mix=0.538121 |diff|=1.4e-09 p_AF=0.538266 Phi(-y/rt2)=0.500059
y=+0       d=   1999995867268 k=   1000000000000 m=  4132731 rho=  1.0000 y= -0.0002 p_exact=0.514828 p_mix=0.514828 |diff|=5.3e-10 p_AF=0.514875 Phi(-y/rt2)=0.500049
y=+0       d= 199999958672685 k= 100000000000000 m= 41327314 rho=  1.0000 y= +0.0001 p_exact=0.505517 p_mix=0.505517 |diff|=1.2e-08 p_AF=0.505532 Phi(-y/rt2)=0.499980

y=+1       d=            1857 k=            1000 m=      142 rho=  1.1818 y= +0.9954 p_exact=0.744794 p_mix=0.744794 |diff|=3.4e-15 p_AF=0.748874 Phi(-y/rt2)=0.240753
y=+1       d=           19565 k=           10000 m=      434 rho=  1.1029 y= +1.0206 p_exact=0.634715 p_mix=0.634715 |diff|=2.8e-14 p_AF=0.638641 Phi(-y/rt2)=0.235246
y=+1       d=          198656 k=          100000 m=     1343 rho=  1.0560 y= +0.9992 p_exact=0.526523 p_mix=0.526523 |diff|=9.0e-14 p_AF=0.529118 Phi(-y/rt2)=0.239924
y=+1       d=         1995802 k=         1000000 m=     4197 rho=  1.0313 y= +0.9998 p_exact=0.432230 p_mix=0.432230 |diff|=2.2e-13 p_AF=0.433632 Phi(-y/rt2)=0.239803
y=+1       d=       199958468 k=       100000000 m=    41531 rho=  1.0099 y= +1.0019 p_exact=0.318211 p_mix=0.318211 |diff|=1.1e-09 p_AF=0.318533 Phi(-y/rt2)=0.239323
y=+1       d=     19999586083 k=     10000000000 m=   413916 rho=  1.0031 y= +1.0000 p_exact=0.270533 p_mix=0.270533 |diff|=1.2e-09 p_AF=0.270609 Phi(-y/rt2)=0.239750
y=+1       d=   1999995865235 k=   1000000000000 m=  4134764 rho=  1.0010 y= +0.9999 p_exact=0.251467 p_mix=0.251467 |diff|=3.3e-10 p_AF=0.251487 Phi(-y/rt2)=0.239779
y=+1       d= 199999958666257 k= 100000000000000 m= 41333742 rho=  1.0003 y= +1.0000 p_exact=0.244096 p_mix=0.244096 |diff|=7.4e-09 p_AF=0.244102 Phi(-y/rt2)=0.239756

y=+2       d=            1846 k=            1000 m=      153 rho=  1.3720 y= +1.9579 p_exact=0.552024 p_mix=0.552024 |diff|=2.1e-15 p_AF=0.561201 Phi(-y/rt2)=0.083111
y=+2       d=           19545 k=           10000 m=      454 rho=  1.2069 y= +2.0044 p_exact=0.390620 p_mix=0.390620 |diff|=5.6e-14 p_AF=0.394872 Phi(-y/rt2)=0.078190
y=+2       d=          198620 k=          100000 m=     1379 rho=  1.1134 y= +1.9950 p_exact=0.272829 p_mix=0.272829 |diff|=7.8e-15 p_AF=0.274228 Phi(-y/rt2)=0.079166
y=+2       d=         1995738 k=         1000000 m=     4261 rho=  1.0630 y= +1.9953 p_exact=0.194930 p_mix=0.194930 |diff|=3.1e-13 p_AF=0.195201 Phi(-y/rt2)=0.079138
y=+2       d=       199958265 k=       100000000 m=    41734 rho=  1.0198 y= +2.0005 p_exact=0.119848 p_mix=0.119848 |diff|=6.0e-10 p_AF=0.119761 Phi(-y/rt2)=0.078596
y=+2       d=     19999585440 k=     10000000000 m=   414559 rho=  1.0062 y= +2.0002 p_exact=0.093796 p_mix=0.093796 |diff|=6.2e-10 p_AF=0.093751 Phi(-y/rt2)=0.078627
y=+2       d=   1999995863202 k=   1000000000000 m=  4136797 rho=  1.0020 y= +1.9999 p_exact=0.084283 p_mix=0.084283 |diff|=5.9e-11 p_AF=0.084267 Phi(-y/rt2)=0.078659
y=+2       d= 199999958659828 k= 100000000000000 m= 41340171 rho=  1.0006 y= +2.0000 p_exact=0.080715 p_mix=0.080715 |diff|=2.4e-09 p_AF=0.080710 Phi(-y/rt2)=0.078646

=== B. BHT critical rate (ECP 31 (2026) Rem. 1.2): k = ceil(d/2 + gamma sqrt d), 2 gamma = sqrt(e pi); predicted limit p -> 1/2 ===
BHT-crit   d=            1000 k=             547 m=       93 rho=  0.9275 y= -0.3631 p_exact=0.929392 p_mix=0.929392 |diff|=1.6e-14 p_AF=0.924006 Phi(-y/rt2)=0.601312
BHT-crit   d=           10000 k=            5147 m=      293 rho=  0.9768 y= -0.2012 p_exact=0.885675 p_mix=0.885675 |diff|=6.5e-14 p_AF=0.884247 Phi(-y/rt2)=0.556563
BHT-crit   d=          100000 k=           50463 m=      925 rho=  0.9928 y= -0.1105 p_exact=0.814606 p_mix=0.814606 |diff|=1.2e-13 p_AF=0.814983 Phi(-y/rt2)=0.531136
BHT-crit   d=         1000000 k=          501462 m=     2923 rho=  0.9976 y= -0.0656 p_exact=0.736209 p_mix=0.736209 |diff|=6.6e-13 p_AF=0.737058 Phi(-y/rt2)=0.518498
BHT-crit   d=       100000000 k=        50014612 m=    29223 rho=  0.9997 y= -0.0239 p_exact=0.611331 p_mix=0.611331 |diff|=8.6e-10 p_AF=0.611816 Phi(-y/rt2)=0.506754
BHT-crit   d=     10000000000 k=      5000146115 m=   292229 rho=  1.0000 y= -0.0065 p_exact=0.545519 p_mix=0.545519 |diff|=1.7e-09 p_AF=0.545689 Phi(-y/rt2)=0.501830
BHT-crit   d=   1000000000000 k=    500001461142 m=  2922283 rho=  1.0000 y= -0.0021 p_exact=0.517685 p_mix=0.517685 |diff|=1.6e-09 p_AF=0.517740 Phi(-y/rt2)=0.500600
BHT-crit   d= 100000000000000 k=  50000014611412 m= 29222823 rho=  1.0000 y= -0.0009 p_exact=0.506687 p_mix=0.506687 |diff|=7.5e-09 p_AF=0.506705 Phi(-y/rt2)=0.500257
BHT-crit   d=10000000000000000 k=5000000146114118 m=292228235 rho=  1.0000 y= -0.0003 p_exact=0.502454 p_mix=0.502454 |diff|=9.9e-08 p_AF=0.502459 Phi(-y/rt2)=0.500096

=== C. other regimes ===
m=1        d=            1000 k=             501 m=        1 rho=  0.0001 y= -9.5090 p_exact=1.000000 p_mix=1.000000 |diff|=0.0e+00 p_AF=0.995489 Phi(-y/rt2)=1.000000
m~d^0.4    d=            1000 k=             508 m=       15 rho=  0.0260 y= -8.0916 p_exact=1.000000 p_mix=1.000000 |diff|=6.7e-16 p_AF=1.000000 Phi(-y/rt2)=1.000000
rho~0.8    d=            1000 k=             541 m=       81 rho=  0.7114 y= -1.5343 p_exact=0.985099 p_mix=0.985099 |diff|=1.0e-14 p_AF=0.979890 Phi(-y/rt2)=0.861023
rho~1.2    d=            1000 k=             551 m=      101 rho=  1.0859 y= +0.4143 p_exact=0.849271 p_mix=0.849271 |diff|=7.5e-15 p_AF=0.848292 Phi(-y/rt2)=0.384775
m~d^0.6    d=            1000 k=             532 m=       63 rho=  0.4376 y= -3.3029 p_exact=0.999587 p_mix=0.999587 |diff|=1.8e-14 p_AF=0.998661 Phi(-y/rt2)=0.990242
k/d=0.6    d=            1000 k=             600 m=      199 rho=  3.8708 y= +9.7297 p_exact=0.001360 p_mix=0.001360 |diff|=7.1e-17 p_AF=0.000644 Phi(-y/rt2)=0.000000
d-k=1      d=            1000 k=             999 m=      997 rho= 58.3157 y=+75.8295 p_exact=0.111744 p_mix=0.111744 |diff|=1.4e-14 p_AF=0.139752 Phi(-y/rt2)=0.000000
d-k=4      d=            1000 k=             996 m=      991 rho= 57.7896 y=+75.3783 p_exact=0.038887 p_mix=0.038887 |diff|=9.6e-15 p_AF=0.015610 Phi(-y/rt2)=0.000000

m=1        d=           10000 k=            5001 m=        1 rho=  0.0000 y=-17.0362 p_exact=1.000000 p_mix=1.000000 |diff|=0.0e+00 p_AF=0.999120 Phi(-y/rt2)=1.000000
m~d^0.4    d=           10000 k=            5020 m=       39 rho=  0.0177 y=-14.8317 p_exact=1.000000 p_mix=1.000000 |diff|=3.8e-15 p_AF=1.000000 Phi(-y/rt2)=1.000000
rho~0.8    d=           10000 k=            5131 m=      261 rho=  0.7775 y= -2.0345 p_exact=0.991912 p_mix=0.991912 |diff|=2.8e-14 p_AF=0.990108 Phi(-y/rt2)=0.924868
rho~1.2    d=           10000 k=            5160 m=      319 rho=  1.1549 y= +1.2863 p_exact=0.606201 p_mix=0.606201 |diff|=2.6e-14 p_AF=0.611279 Phi(-y/rt2)=0.181529
m~d^0.6    d=           10000 k=            5126 m=      251 rho=  0.7197 y= -2.6080 p_exact=0.997388 p_mix=0.997388 |diff|=1.9e-15 p_AF=0.996387 Phi(-y/rt2)=0.967417
k/d=0.6    d=           10000 k=            6000 m=     1999 rho= 39.0007 y=+93.8401 p_exact=0.000000 p_mix=0.000000 |diff|=2.8e-222 p_AF=0.000000 Phi(-y/rt2)=0.000000
d-k=1      d=           10000 k=            9999 m=     9997 rho=585.2641 y=+471.4541 p_exact=0.054887 p_mix=0.054887 |diff|=4.6e-13 p_AF=0.028786 Phi(-y/rt2)=0.000000
d-k=4      d=           10000 k=            9996 m=     9991 rho=584.7373 y=+471.1974 p_exact=0.005275 p_mix=0.005275 |diff|=1.3e-15 p_AF=0.000073 Phi(-y/rt2)=0.000000

m=1        d=         1000000 k=          500001 m=        1 rho=  0.0000 y=-54.0396 p_exact=1.000000 p_mix=1.000000 |diff|=0.0e+00 p_AF=0.999984 Phi(-y/rt2)=1.000000
m~d^0.4    d=         1000000 k=          500126 m=      251 rho=  0.0074 y=-49.4187 p_exact=1.000000 p_mix=1.000000 |diff|=1.7e-15 p_AF=1.000000 Phi(-y/rt2)=1.000000
rho~0.8    d=         1000000 k=          501307 m=     2613 rho=  0.7974 y= -5.7881 p_exact=0.999999 p_mix=0.999999 |diff|=3.5e-13 p_AF=0.999998 Phi(-y/rt2)=0.999979
rho~1.2    d=         1000000 k=          501601 m=     3201 rho=  1.1960 y= +5.0654 p_exact=0.002002 p_mix=0.002002 |diff|=2.0e-16 p_AF=0.001814 Phi(-y/rt2)=0.000171
m~d^0.6    d=         1000000 k=          501991 m=     3981 rho=  1.8485 y=+19.4580 p_exact=0.000000 p_mix=0.000000 |diff|=6.3e-38 p_AF=0.000000 Phi(-y/rt2)=0.000000
k/d=0.6    d=         1000000 k=          600000 m=   199999 rho=3903.2896 y=+3478.2759 p_exact=0.000000 p_mix=0.000000 |diff|=0.0e+00 p_AF=0.000000 Phi(-y/rt2)=0.000000
d-k=1      d=         1000000 k=          999999 m=   999997 rho=58549.5973 y=+15491.0830 p_exact=0.014397 p_mix=0.014397 |diff|=8.7e-12 p_AF=0.000210 Phi(-y/rt2)=0.000000
d-k=4      d=         1000000 k=          999996 m=   999991 rho=58549.0704 y=+15491.0014 p_exact=0.000082 p_mix=0.000082 |diff|=1.1e-14 p_AF=0.000000 Phi(-y/rt2)=0.000000

m=1        d=       100000000 k=        50000001 m=        1 rho=  0.0000 y=-170.9410 p_exact=1.000000 p_mix=1.000000 |diff|=0.0e+00 p_AF=1.000000 Phi(-y/rt2)=1.000000
m~d^0.4    d=       100000000 k=        50000792 m=     1583 rho=  0.0029 y=-161.6874 p_exact=1.000000 p_mix=1.000000 |diff|=2.0e-13 p_AF=1.000000 Phi(-y/rt2)=1.000000
rho~0.8    d=       100000000 k=        50013069 m=    26137 rho=  0.7997 y=-18.0725 p_exact=1.000000 p_mix=1.000000 |diff|=2.2e-11 p_AF=1.000000 Phi(-y/rt2)=1.000000
rho~1.2    d=       100000000 k=        50016006 m=    32011 rho=  1.1995 y=+16.2815 p_exact=0.000000 p_mix=0.000000 |diff|=2.6e-29 p_AF=0.000000 Phi(-y/rt2)=0.000000
m~d^0.6    d=       100000000 k=        50031548 m=    63095 rho=  4.6588 y=+198.0593 p_exact=0.000000 p_mix=0.000000 |diff|=0.0e+00 p_AF=0.000000 Phi(-y/rt2)=0.000000
k/d=0.6    d=       100000000 k=        60000000 m= 19999999 rho=390332.1776 y=+111603.5186 p_exact=0.000000 p_mix=0.000000 |diff|=0.0e+00 p_AF=0.000000 Phi(-y/rt2)=0.000000
d-k=1      d=       100000000 k=        99999999 m= 99999997 rho=5854982.9182 y=+491701.8952 p_exact=0.003983 p_mix=0.003983 |diff|=6.1e-11 p_AF=0.000000 Phi(-y/rt2)=0.000000
d-k=4      d=       100000000 k=        99999996 m= 99999991 rho=5854982.3913 y=+491701.8694 p_exact=0.000001 p_mix=0.000001 |diff|=3.9e-15 p_AF=0.000000 Phi(-y/rt2)=0.000000

m=1        d=     10000000000 k=      5000000001 m=        1 rho=  0.0000 y=-540.5795 p_exact=1.000000 p_mix=1.000000 |diff|=0.0e+00 p_AF=1.000000 Phi(-y/rt2)=1.000000
m~d^0.4    d=     10000000000 k=      5000005000 m=     9999 rho=  0.0012 y=-522.0848 p_exact=1.000000 p_mix=1.000000 |diff|=1.8e-12 p_AF=1.000000 Phi(-y/rt2)=1.000000
rho~0.8    d=     10000000000 k=      5000130688 m=   261375 rho=  0.8000 y=-57.0809 p_exact=1.000000 p_mix=1.000000 |diff|=1.9e-10 p_AF=1.000000 Phi(-y/rt2)=1.000000
rho~1.2    d=     10000000000 k=      5000160060 m=   320119 rho=  1.2000 y=+51.5849 p_exact=0.000000 p_mix=0.000000 |diff|=5.1e-281 p_AF=0.000000 Phi(-y/rt2)=0.000000
m~d^0.6    d=     10000000000 k=      5000500000 m=   999999 rho= 11.7088 y=+1309.2172 p_exact=0.000000 p_mix=0.000000 |diff|=0.0e+00 p_AF=0.000000 Phi(-y/rt2)=0.000000
k/d=0.6    d=     10000000000 k=      6000000000 m=1999999999 rho=39033220.9837 y=+3534305.4188 p_exact=0.000000 p_mix=0.000000 |diff|=0.0e+00 p_AF=0.000000 Phi(-y/rt2)=0.000000
d-k=1      d=     10000000000 k=      9999999999 m=9999999997 rho=585498315.0090 y=+15554765.3409 p_exact=0.001135 p_mix=0.001135 |diff|=1.0e-08 p_AF=0.000000 Phi(-y/rt2)=0.000000
d-k=4      d=     10000000000 k=      9999999996 m=9999999991 rho=585498314.4820 y=+15554765.3328 p_exact=0.000000 p_mix=0.000000 |diff|=1.6e-13 p_AF=0.000000 Phi(-y/rt2)=0.000000

=== D. second-order (heuristic) correction in the window: A/F ~ y/sqrt2 - (pi e)^{3/4} log(N/(pi e)) / (4 N^{1/4}) ===
y=-1 k=1e4  p_exact=0.951293  Phi(-(y/rt2+corr))=0.950620  Phi(-y/rt2)=0.759426
y=-1 k=1e6  p_exact=0.890556  Phi(-(y/rt2+corr))=0.890192  Phi(-y/rt2)=0.761761
y=-1 k=1e8  p_exact=0.826470  Phi(-(y/rt2+corr))=0.826036  Phi(-y/rt2)=0.760274
y=-1 k=1e10 p_exact=0.789004  Phi(-(y/rt2+corr))=0.788787  Phi(-y/rt2)=0.760343
y=-1 k=1e12 p_exact=0.771668  Phi(-(y/rt2+corr))=0.771587  Phi(-y/rt2)=0.760298
y=-1 k=1e14 p_exact=0.764543  Phi(-(y/rt2+corr))=0.764516  Phi(-y/rt2)=0.760247
y=+0 k=1e4  p_exact=0.843836  Phi(-(y/rt2+corr))=0.830267  Phi(-y/rt2)=0.503503
y=+0 k=1e6  p_exact=0.699198  Phi(-(y/rt2+corr))=0.695883  Phi(-y/rt2)=0.498812
y=+0 k=1e8  p_exact=0.592878  Phi(-(y/rt2+corr))=0.591934  Phi(-y/rt2)=0.500435
y=+0 k=1e10 p_exact=0.538121  Phi(-(y/rt2+corr))=0.537828  Phi(-y/rt2)=0.500059
y=+0 k=1e12 p_exact=0.514828  Phi(-(y/rt2+corr))=0.514736  Phi(-y/rt2)=0.500049
y=+0 k=1e14 p_exact=0.505517  Phi(-(y/rt2+corr))=0.505488  Phi(-y/rt2)=0.499980
y=+1 k=1e4  p_exact=0.634715  Phi(-(y/rt2+corr))=0.588917  Phi(-y/rt2)=0.235246
y=+1 k=1e6  p_exact=0.432230  Phi(-(y/rt2+corr))=0.424121  Phi(-y/rt2)=0.239803
y=+1 k=1e8  p_exact=0.318211  Phi(-(y/rt2+corr))=0.316664  Phi(-y/rt2)=0.239323
y=+1 k=1e10 p_exact=0.270533  Phi(-(y/rt2+corr))=0.270172  Phi(-y/rt2)=0.239750
y=+1 k=1e12 p_exact=0.251467  Phi(-(y/rt2+corr))=0.251368  Phi(-y/rt2)=0.239779
y=+1 k=1e14 p_exact=0.244096  Phi(-(y/rt2+corr))=0.244067  Phi(-y/rt2)=0.239756
