Exact rational-polynomial audits:
  PASS SL2 determinant 1
  PASS matrix trace 1
  PASS matrix cubic order
  PASS matrix component
  PASS matrix component
  PASS matrix component
  PASS projective Phi squared
  PASS projective Phi cubed
  PASS Delta invariant under Phi
  PASS cubic orbit trace
  PASS cubic orbit pair sum
  PASS cubic orbit product
  PASS perfect-square discriminant
  PASS generic Fourier norm
  PASS generic primitive Fourier cube
  PASS generic complementary Fourier cube
  PASS Möbius inverse: cubic polynomial residual
  PASS Möbius determinant / derivative
  PASS polynomial Y(x) inverse modulo f (identity)
  PASS explicit rational gamma obstruction
  PASS transport determinant
  PASS transported C3 action determinant
  PASS transported C3 action trace
  PASS transported C3 action cubic relation
  PASS transported C3 action cubic relation
  PASS transported C3 action cubic relation
  PASS transported C3 action cubic relation
  PASS trace normalizer numerator
  PASS trace normalizer denominator
  PASS spurious quadratic orbit control
Exact audit complete. None of these checks establishes Delta_gamma(1/3)=0.

Numerical diagnostic at z= 0.33333333333333333333 with 64 digits
 chart 1 N 4000 time 0.36 s
   U = (-4.024029545371045206657 - 41.8559517671101857119j)
   delta_gamma = (-2.56747504505612e-64 - 1.51655332081808e-64j) abs 2.9819225e-64
   rigorous truncation tail for log(U) <= 1.4983425e-105
 chart 10 N 19000 time 1.34 s
   U = (-0.8326432366807291506723 + 3.550247070468073036805j)
   delta_gamma = (8.97663858822928e-56 + 1.27314867031338e-55j) abs 1.5577894e-55
   rigorous truncation tail for log(U) <= 2.9945114e-48
 chart 23 N 34000 time 3.41 s
   U = (0.03081041650699615493098 + 0.2519662235081715374107j)
   delta_gamma = (-1.83899146724893e-43 - 1.45202312196119e-43j) abs 2.3431305e-43
   rigorous truncation tail for log(U) <= 2.3271661e-36
 analytic orbit check |v_23 - Phi(v_1)| = 7.5817966579e-44
 analytic orbit check |v_10 - Phi²(v_1)| = 3.7535503223e-56
 analytic trace |Y_1+Y_10+Y_23-target| = 1.4106092855e-42
 No small residual is an equality certificate.

Numerical diagnostic at z= 0.25 with 64 digits
 chart 1 N 4000 time 0.36 s
   U = (0.2237999593860142675135 - 0.3042840387610698503069j)
   delta_gamma = (0.119587479580191 + 0.0458002924452422j) abs 0.12805792
   rigorous truncation tail for log(U) <= 1.4983425e-105
 chart 10 N 19000 time 1.63 s
   U = (5.98568089192252589105 - 0.9949129102075743010349j)
   delta_gamma = (0.132635327475 + 0.0529982157507676j) abs 0.14283186
   rigorous truncation tail for log(U) <= 2.9945114e-48
 chart 23 N 34000 time 3.0 s
   U = (6.064805307724134115853 + 1.528434773107991075381j)
   delta_gamma = (0.132798726070377 + 0.0531419953502676j) abs 0.14303696
   rigorous truncation tail for log(U) <= 2.3271661e-36
 analytic orbit check |v_23 - Phi(v_1)| = 0.34293461022
 analytic orbit check |v_10 - Phi²(v_1)| = 0.20985202693
 analytic trace |Y_1+Y_10+Y_23-target| = 48.273278389
 No small residual is an equality certificate.