CONTROL 1: odd N (every row must FAIL by O(1))
FAMILY a=1.0 b=0.5 N=3 tau=1 : a''=0.8685170918213297643730738 b''=0.06574145408933511781346312  closed-form check |da''|=1.14794e-40
  t=0.3 rot=1.0 closure=5.61151e-40 |C0*'|=1.05674 k607=0.632288 k609=0.921603 k610=0.938027 k905=0.420197 k906=1.49129 k907b_dual=1.71952
  t=1.2 rot=1.0 closure=2.66804e-39 |C0*'|=1.37059 k607=0.418571 k609=0.657077 k610=0.67449 k905=0.0143857 k906=0.577372 k907b_dual=0.7199
  t=2.6 rot=1.0 closure=1.12311e-39 |C0*'|=0.752644 k607=2.58142 k609=50.068 k610=0.346863 k905=4.8255 k906=0.0718998 k907b_dual=0.720781
FAMILY a=1.0 b=0.5 N=5 tau=1 : a''=0.9227001422795942723511322 b''=0.3183952772306516384451984
  t=0.3 rot=1.0 closure=2.39715e-40 |C0*'|=0.0632077 k607=3.35817 k609=0.33842 k610=0.269648 k905=387.789 k906=0.0906353 k907b_dual=0.0398725
  t=1.2 rot=1.0 closure=2.16137e-40 |C0*'|=0.0492359 k607=0.841271 k609=0.287157 k610=0.332071 k905=1.06451 k906=0.116401 k907b_dual=0.0442837
  t=2.6 rot=1.0 closure=7.00622e-41 |C0*'|=0.0735703 k607=0.685226 k609=0.214836 k610=0.0405887 k905=0.911243 k906=0.0821766 k907b_dual=0.0319885
FAMILY a=1.0 b=0.8 N=7 tau=1 : a''=0.9224828288467989384488491 b''=0.7006957752956646839060139
  t=0.3 rot=1.0 closure=7.525e-40 |C0*'|=0.000360964 k607=0.00102965 k609=0.000424056 k610=0.00152078 k905=0.00170387 k906=0.000332134 k907b_dual=4.39985e-5
  t=1.2 rot=1.0 closure=1.97916e-40 |C0*'|=0.000358776 k607=0.00105037 k609=0.000432598 k610=0.00155143 k905=0.00173821 k906=0.000338822 k907b_dual=4.48846e-5
  t=2.6 rot=1.0 closure=5.6483e-40 |C0*'|=0.000392211 k607=0.000645231 k609=0.000265422 k610=0.000951027 k905=0.00106537 k906=0.000208015 k907b_dual=2.75477e-5
FAMILY a=1.0 b=0.5 N=5 tau=2 : a''=0.866038617765429330159655 b''=0.00478408413966629471633939
  t=0.3 rot=2.0 closure=1.50012e-38 |C0*'|=93.7085 k607=0.580338 k609=0.00409932 k610=7.37801 k905=0.330937 k906=0.745318 k907b_dual=1.64221
  t=1.2 rot=2.0 closure=8.01981e-38 |C0*'|=56.8388 k607=0.386419 k609=0.00239946 k610=1.70984 k905=0.0278747 k906=14.8242 k907b_dual=0.717909
  t=2.6 rot=2.0 closure=3.12823e-38 |C0*'|=115.701 k607=2.06592 k609=0.00507199 k610=0.988824 k905=7.28909 k906=0.0798276 k907b_dual=0.718303
CONTROL 2: non-primitive even list (3-periodic of (1, 0.5) traversed twice, N=6 steps on the N=3 caustic)
  t=0.3 rot=2.0 closure=1.04529e-39 censym=5.98354 |C0*'|=1.05674 k607=0.632288 k609=0.921603 k610=0.938027 k905=0.420197 k906=1.49129 k907b_dual=1.71952
  t=1.2 rot=2.0 closure=5.11077e-39 censym=7.13761 |C0*'|=1.37059 k607=0.418571 k609=0.657077 k610=0.67449 k905=0.0143857 k906=0.577372 k907b_dual=0.7199
CONTROL 3: non-confocal 'caustic' (b'' scaled by 1+1e-3): closure and reflection cross-check must flag it
  t=0.3 closure=0.00253233 reflect_defect=0.00136845 censym=0.00468485 k607=1.08768 k609=0.00652824 k610=0.00141594 k905=0.0123975 k906=0.00903111 k907b_dual=0.00358634
  t=1.2 closure=0.00841636 reflect_defect=0.00385504 censym=0.00445301 k607=1.41841 k609=0.000495325 k610=0.0062114 k905=0.00368557 k906=0.00547302 k907b_dual=0.00199378
CONTROL 4: one vertex displaced along e by 1e-6 (rows built from P and P' only)
  k607 |A*1/A*2-1| = 1.11394 ; k406a |C0*'| = 2.64221e-7 ; k903b |A1/A2-1| = 2.81034e-6 ; k604b = 2.80135e-7
CONTROL 5: wrong partner point M=(-c, b/10) instead of f2 for the P'' pedal (k609): ratio-1 per family
  N=4 b/a=0.5: k609_1/k609_ctrlM - 1 = ['-2.29589e-41', '-1.03315e-40', '3.44383e-40']
  N=6 b/a=0.5: k609_1/k609_ctrlM - 1 = ['-0.00115252', '-0.00115252', '-0.00115252']
  N=8 b/a=0.8: k609_1/k609_ctrlM - 1 = ['-0.00311249', '-0.00311249', '-0.00311249']
  N=10 b/a=0.1765: k609_1/k609_ctrlM - 1 = ['-0.0001551', '-0.0001551', '-0.0001551']
