==================================================================================================== Proposition 3.1: lim_{eps->0} eps^-2 d_G(tau_{gamma_eps}, e) = ||F_x(X,Y)|| G = SU(2); = -tr(AB); d_G from the induced bi-invariant metric ==================================================================================================== CASE 1 -- curved connection predicted ||F|| = 1.693236545790 eps eps^-2 d_G(tau,e) |ratio - ||F|| | rel. err of log(tau)/(-eps^2) vs F ---------------------------------------------------------------------------------------------- 1.250e-01 1.705936374103 1.270e-02 1.102e-01 6.250e-02 1.700202006173 6.965e-03 5.465e-02 3.125e-02 1.696868878520 3.632e-03 2.719e-02 1.562e-02 1.695089534576 1.853e-03 1.356e-02 7.812e-03 1.694172170986 9.356e-04 6.771e-03 3.906e-03 1.693706631586 4.701e-04 3.383e-03 1.953e-03 1.693472155790 2.356e-04 1.691e-03 9.766e-04 1.693354492415 1.179e-04 8.454e-04 4.883e-04 1.693295554494 5.901e-05 4.226e-04 2.441e-04 1.693266058986 2.951e-05 2.113e-04 successive gap ratios (expect ~0.5 for an O(eps) residual): 0.548 0.521 0.510 0.505 0.502 0.501 0.501 0.500 0.500 -> finest gap 2.951e-05, matrix rel. err 2.113e-04, tolerance 1e-03: PASS CASE 2 -- flat control (A = h^-1 dh) predicted ||F|| = 0.000000000000 eps eps^-2 d_G(tau,e) |ratio - ||F|| | rel. err of log(tau)/(-eps^2) vs F ---------------------------------------------------------------------------------------------- 1.250e-01 0.000000000000 0.000e+00 0.000e+00 6.250e-02 0.000000000000 0.000e+00 0.000e+00 3.125e-02 0.000000000000 0.000e+00 0.000e+00 1.562e-02 0.000000000000 0.000e+00 0.000e+00 7.812e-03 0.000000000000 0.000e+00 0.000e+00 3.906e-03 0.000000000000 0.000e+00 0.000e+00 1.953e-03 0.000000000000 0.000e+00 0.000e+00 9.766e-04 0.000000000004 3.779e-12 3.779e-12 4.883e-04 0.000000000002 2.253e-12 2.253e-12 2.441e-04 0.000000000000 3.865e-13 3.865e-13 -> ratio at the finest eps is 3.865e-13, must vanish: PASS CASE 3 -- same connection, prediction missing [A_u, A_v] predicted ||F|| = 1.044030650891 eps eps^-2 d_G(tau,e) |ratio - ||F|| | rel. err of log(tau)/(-eps^2) vs F ---------------------------------------------------------------------------------------------- 1.250e-01 1.705936374103 6.619e-01 6.554e-01 6.250e-02 1.700202006173 6.562e-01 6.600e-01 3.125e-02 1.696868878520 6.528e-01 6.645e-01 1.562e-02 1.695089534576 6.511e-01 6.673e-01 7.812e-03 1.694172170986 6.501e-01 6.689e-01 3.906e-03 1.693706631586 6.497e-01 6.697e-01 1.953e-03 1.693472155790 6.494e-01 6.701e-01 9.766e-04 1.693354492415 6.493e-01 6.703e-01 4.883e-04 1.693295554494 6.493e-01 6.704e-01 2.441e-04 1.693266058986 6.492e-01 6.705e-01 successive gap ratios (expect ~0.5 for an O(eps) residual): 0.991 0.995 0.997 0.999 0.999 1.000 1.000 1.000 1.000 -> finest scalar gap 6.492e-01, matrix error 6.705e-01; neither may converge, so both above 1e-2 is the pass: PASS These plateau at two different quantities, which is worth keeping distinct. The scalar column measures lengths only, so its gap tends to | ||F|| - ||F_wrong|| | = 0.649205894899. The matrix column compares elements of su(2), so its error tends to the dropped commutator itself, ||F - F_wrong|| = ||[A_u, A_v]|| = 0.700035713375 (printed above relative to ||F_wrong||, hence 0.6705). CASE 3b -- the Case 3 gate, turned on itself A control that passes whatever it is shown is not a control. Case 3 passes only because its prediction omits the commutator; hand the same gate the CORRECT curvature and it has to go red. Both lines below come from the same code path that decides Case 3. wrong prediction (Case 3 as shipped): gap=6.492e-01 matrix=6.705e-01 gate=PASS correct prediction (mechanism removed): gap=2.951e-05 matrix=2.113e-04 gate=FAIL -> the gate must pass the first and fail the second: PASS ==================================================================================================== All cases pass. Proposition 3.1 is reproduced numerically, the flat control vanishes, the measurement is sensitive to the non-abelian term of the curvature, and the control that establishes that last point is itself falsifiable. ====================================================================================================