=== check1: Dixon core facts (mpmath, 30 digits) ===
pi_3 = B(1/3,1/3) = 5.299916250856349871941068
h = sqrt(3)/2 * pi_3 = 4.589862111171578624739583

[C1] first integral sm^3+cm^3-1 at sample points (paths from 0 cross pole regions)
  u = (0.7 + 0.3j)  sm = (0.698482318184407 + 0.245697556469151j)  cm = (0.941233181522204 - 0.130562791585575j)  |sm^3+cm^3-1| = 2.03e-31
  u = (1.9 + 2.2j)  sm = (-0.291729568181582 - 0.859664702101876j)  cm = (-0.592701513825524 - 0.574202486892066j)  |sm^3+cm^3-1| = 5.26e-28
  u = (-2.4 + 4.1j)  sm = (0.25478063555388 - 0.504841971223467j)  cm = (1.05628171257346 - 0.00906873961180538j)  |sm^3+cm^3-1| = 9.53e-29
  u = (3.3 - 1.7j)  sm = (-0.439758313784647 - 0.475896475113483j)  cm = (-0.520360873422629 + 0.770662231102416j)  |sm^3+cm^3-1| = 4.59e-28
  u = (0.71 + 3.67j)  sm = (0.0594931755389375 - 0.998276338079646j)  cm = (-0.331549560784959 + 1.104822075584j)  |sm^3+cm^3-1| = 6.17e-28
  u = (1.0 + 4.5898621j)  sm = (-8.56813404720533 + 2.75781538151252e-14j)  cm = (8.57267216992921 - 2.75489634021843e-14j)  |sm^3+cm^3-1| = 2.19e-26
  u = (3.6 + 0.05j)  sm = (-9.5974216305698 + 7.19339875939944j)  cm = (9.59807226121273 - 7.1911746760842j)  |sm^3+cm^3-1| = 7.07e-26
  u = (5.1 + 4.4j)  sm = (1.08098510042324 - 0.0937478061086335j)  cm = (-0.702040787121929 + 0.229931693182317j)  |sm^3+cm^3-1| = 5.55e-28
  worst: 7.07e-26

[C2] sm(u(x)) - x with u(x) = x 2F1(1/3,2/3;4/3;x^3)
  x = 0.3  u(x) = 0.30136765441063822228  sm(u)-x = (0.0 + 0.0j)  cm(u)-(1-x^3)^(1/3) = (0.0 + 0.0j)
  x = 0.6  u(x) = 0.62417479894809192697  sm(u)-x = (0.0 + 0.0j)  cm(u)-(1-x^3)^(1/3) = (-9.86e-32 + 0.0j)
  x = 0.9  u(x) = 1.085721443912177336  sm(u)-x = (9.86e-32 + 0.0j)  cm(u)-(1-x^3)^(1/3) = (-2.96e-31 + 0.0j)
  x = 0.99  u(x) = 1.4553961441302365294  sm(u)-x = (0.0 + 0.0j)  cm(u)-(1-x^3)^(1/3) = (-3.94e-31 + 0.0j)
  u(1) - pi_3/3 = -3.94e-31

[C3] periods and symmetries
  u = pi_3               sm = (-4.88e-28 + 0.0j)   cm-1 = (6.41e-29 + 0.0j)
  u = pi_3*omega         sm = (3.01e-28 - 5.2e-28j)   cm-1 = (6.53e-29 - 1.03e-31j)
  u = 2ih                sm = (-4.3e-28 - 6.61e-28j)   cm-1 = (-1.92e-28 + 2.1e-28j)
  u = pi_3*omega^2       sm = (2.87e-28 + 5.01e-28j)   cm-1 = (-2.28e-29 + 1.61e-31j)
  u = pi_3 + pi_3*omega  sm = (-2.07e-28 - 3.6e-28j)   cm-1 = (1.72e-29 - 2.13e-32j)
  rotation: sm(omega u) - omega sm(u) = (0.0 + 0.0j)  cm(omega u) - cm(u) = (-1.97e-31 + 4.93e-32j)
  conjugation: sm(conj u) - conj sm(u) = (0.0 + 0.0j)  cm: (0.0 + 0.0j)
  sm(u+pi_3)-sm(u) = (-1.01e-28 - 2.51e-28j)  cm: (-3.26e-28 - 3.59e-29j)
  sm(u+pi_3 omega)-sm(u) = (-2.56e-28 + 1.01e-28j)  cm: (2.01e-28 + 9.99e-29j)
  sm(pi_3/3) - 1 = (-9.86e-32 + 0.0j)  cm(pi_3/3) = (-5.18e-31 + 0.0j)

[C4] reality on the line Im u = h (direct paths from 0, no periodicity used)
  x = 0.0137  sm = 1.27392337442195  cm = -1.02198822085189  |Im sm|,|Im cm| = 2.32e-28, 5.83e-28  |sm^3+cm^3-1| = 9.89e-28
  x = 0.67618953  sm = 4.83503845827463  cm = -4.82073753570623  |Im sm|,|Im cm| = 2.41e-26, 2.42e-26  |sm^3+cm^3-1| = 8.65e-27
  x = 1.3386791  sm = -2.16143013505919  cm = 2.23054680168889  |Im sm|,|Im cm| = 1.98e-27, 1.49e-27  |sm^3+cm^3-1| = 7.21e-27
  x = 2.0011686  sm = -0.679935354690894  cm = 1.09539183312425  |Im sm|,|Im cm| = 6.38e-28, 2.04e-28  |sm^3+cm^3-1| = 1.69e-28
  x = 2.6636581  sm = 0.0136999941287469  cm = 0.999999142882701  |Im sm|,|Im cm| = 2.06e-28, 1.38e-29  |sm^3+cm^3-1| = 5.25e-29
  x = 3.3261477  sm = 0.643289467697175  cm = 0.901968158222954  |Im sm|,|Im cm| = 2.0e-28, 1.34e-28  |sm^3+cm^3-1| = 1.25e-28
  x = 3.9886372  sm = 0.974209624912929  cm = 0.422451366027704  |Im sm|,|Im cm| = 1.55e-28, 1.12e-27  |sm^3+cm^3-1| = 7.79e-28
  x = 4.6511267  sm = 1.00430931288647  cm = -0.235035330235266  |Im sm|,|Im cm| = 7.63e-29, 7.36e-28  |sm^3+cm^3-1| = 3.72e-28
  x = 5.3136163  sm = 1.27392337442195  cm = -1.02198822085189  |Im sm|,|Im cm| = 5.14e-28, 1.2e-27  |sm^3+cm^3-1| = 1.75e-27
  worst |Im| on the line Im u = h: 2.42e-26

[C5] zeros of sm + i, sm - i, and poles of sm in R = [0,pi_3) x (0,h)
  sm = -i (zeros of sm+i) : 3 points in R (float grid 12x9, Newton), refined to 30 digits:
    w = (0.7497704548828558397944785 + 3.646997258195469354794754j)   |sm-target| = 8.61e-31   |cm| = 1.12246204831   dist to lines Im=0,h: 0.942864853
    w = (1.766638750285449957313689 + 1.885729705952218539889656j)   |sm-target| = 6.31e-31   |cm| = 1.12246204831   dist to lines Im=0,h: 1.885729706
    w = (2.7835070456880440748329 + 3.646997258195469354794754j)   |sm-target| = 2.2e-30   |cm| = 1.12246204831   dist to lines Im=0,h: 0.942864853
  sm = +i (zeros of sm-i) : 3 points in R (float grid 12x9, Newton), refined to 30 digits:
    w = (0.1335489202598691388623662 + 0.9428648529761092699448282j)   |sm-target| = 1.97e-31   |cm| = 1.12246204831   dist to lines Im=0,h: 0.942864853
    w = (3.399728580311030775765013 + 0.9428648529761092699448282j)   |sm-target| = 1.22e-30   |cm| = 1.12246204831   dist to lines Im=0,h: 0.942864853
    w = (4.416596875713624893284224 + 2.704132405219360084849926j)   |sm-target| = 1.25e-30   |cm| = 1.12246204831   dist to lines Im=0,h: 1.885729706
  reflection relation: for w in Z_S, conj(w) + pi_3 omega reduced mod pi_3 should be in Ztilde_S:
     (3.3997285803110307758 + 0.94286485297610926994j)  min dist to Ztilde_S = 3.46e-28
     (4.4165968757136248933 + 2.7041324052193600848j)  min dist to Ztilde_S = 5.2e-28
     (0.13354892025986913886 + 0.94286485297610926994j)  min dist to Ztilde_S = 8.99e-28
  mutual distances within Z_S (for the choice of rho): ['2.03374', '2.03374', '2.03374']
  poles of sm in the open rectangle (0,pi_3) x (0,h): 3
    w = (0.8833193751427249786568447 + 1.529954037057192874913194j)   |sm| at distance 1e-8: 1.0e+8   cm/sm there: (0.5 - 0.866025403784j)
    w = (3.533277500570899914627379 + 1.710645989802495622630158e-81j)   |sm| at distance 1e-8: 1.0e+8   cm/sm there: (-1.0 - 1.71064609083e-97j)
    w = (3.533277500570899914627379 + 3.059908074114385749826388j)   |sm| at distance 1e-8: 1.0e+8   cm/sm there: (0.5 + 0.866025403784j)
  real pole of sm near 2 pi_3/3: (3.533277500570899914627379 + 0.0j)   2 pi_3/3 = 3.533277500570899914627379

DONE check1
