=== check3: Abel-integral cross-checks (mpmath, 30 digits) ===
pi_3 = 5.299916250856349871941068   h = 4.589862111171578624739583

[A1] sm(u(x)) - x, u(x) by quadrature, sm by the ODE stepper
  x = (0.5 + 0.4j)  u(x) = (0.475148038033307423 + 0.408799243769196869j)  |sm(u)-x| = 0.0  |cm(u)-(1-x^3)^(1/3)| = 2.47e-32
  x = (-0.7 + 0.2j)  u(x) = (-0.677639635315918035 + 0.165135511516066478j)  |sm(u)-x| = 9.86e-32  |cm(u)-(1-x^3)^(1/3)| = 1.23e-32
  x = (0.1 - 0.8j)  u(x) = (0.148253967645222821 - 0.759893490641403817j)  |sm(u)-x| = 0.0  |cm(u)-(1-x^3)^(1/3)| = 2.47e-32
  x = (-0.3 - 0.9j)  u(x) = (-0.181003119753976471 - 1.03965444518845371j)  |sm(u)-x| = 1.41e-28  |cm(u)-(1-x^3)^(1/3)| = 1.25e-28
  x = (0.95 + 0.0j)  u(x) = (1.23081764314149927 + 0.0j)  |sm(u)-x| = 0.0  |cm(u)-(1-x^3)^(1/3)| = 1.97e-31
  x = (0.0 - 1.0j)  u(x) = (0.133548920259869139 - 0.94286485297610927j)  |sm(u)-x| = 1.78e-28  |cm(u)-(1-x^3)^(1/3)| = 1.7e-28
  x = (0.0 + 1.0j)  u(x) = (0.133548920259869139 + 0.94286485297610927j)  |sm(u)-x| = 1.78e-28  |cm(u)-(1-x^3)^(1/3)| = 1.7e-28
  x = (-1.5 + 0.5j)  u(x) = (-1.17567476886381386 + 0.176870206945620517j)  |sm(u)-x| = 6.3e-28  |cm(u)-(1-x^3)^(1/3)| = 6.1e-28
  worst: 6.3e-28

[A4] real period: u(1) = pi_3/3 ?  and the two other real pieces
  int_0^1 - pi_3/3 = -1.97e-31   int_1^inf - pi_3/3 = -1.97e-31   int_{-inf}^0 - pi_3/3 = -1.97e-31   total - pi_3 = 0.0

[A2/A3] Abel candidates for sm = -i, reduced into the strip S
  k = 0 : u_k = (0.133548920259869139 - 0.94286485297610927j)  reduced = (2.783507045688044074833 + 3.646997258195469354795j)   stepper: |sm - target| = 9.61e-28,  cm = (1.08421508149135 - 0.290514555507251j)
  k = 1 : u_k = (3.39972858031103078 - 0.94286485297610927j)  reduced = (0.7497704548828558397945 + 3.646997258195469354795j)   stepper: |sm - target| = 1.44e-28,  cm = (-0.290514555507251 + 1.08421508149135j)
  k = 2 : u_k = (1.76663875028544996 + 1.88572970595221854j)  reduced = (1.766638750285449957314 + 1.88572970595221853989j)   stepper: |sm - target| = 2.33e-28,  cm = (-0.7937005259841 - 0.7937005259841j)
  cube roots of 1 - x0^3 = 1 - i : ['(1.08421508149 - 0.290514555507j)', '(-0.290514555507 + 1.08421508149j)', '(-0.793700525984 - 0.793700525984j)']
  Im parts (must be strictly between 0 and h = 4.58986211117): ['3.6469972582', '3.6469972582', '1.88572970595']
  min mutual distance: 2.0337366

[A2/A3] Abel candidates for sm = +i, reduced into the strip S
  k = 0 : u_k = (0.133548920259869139 + 0.94286485297610927j)  reduced = (0.1335489202598691388624 + 0.9428648529761092699448j)   stepper: |sm - target| = 1.78e-28,  cm = (1.08421508149135 + 0.290514555507251j)
  k = 1 : u_k = (1.76663875028544996 - 1.88572970595221854j)  reduced = (4.416596875713624893284 + 2.70413240521936008485j)   stepper: |sm - target| = 4.24e-28,  cm = (-0.7937005259841 + 0.7937005259841j)
  k = 2 : u_k = (3.39972858031103078 + 0.94286485297610927j)  reduced = (3.399728580311030775765 + 0.9428648529761092699448j)   stepper: |sm - target| = 3.67e-28,  cm = (-0.290514555507251 - 1.08421508149135j)
  cube roots of 1 - x0^3 = 1 + i : ['(1.08421508149 + 0.290514555507j)', '(-0.793700525984 + 0.793700525984j)', '(-0.290514555507 - 1.08421508149j)']
  Im parts (must be strictly between 0 and h = 4.58986211117): ['0.942864852976', '2.70413240522', '0.942864852976']
  min mutual distance: 2.0337366

DONE check3
