Size the T_l recurrence passes of the analytic add-back by their growth rate

[CharlesCNorton]

Change - SingularIntegrals evaluates the T_l of Merkel's analytic add-back with a forward pass of the three-term recurrence only while the spurious solutions of that recurrence grow by less than a factor of ten over the kL = mf + nf steps, and otherwise with a backward pass seeded far enough above kL that the seed has decayed below 1e-17 of T_l at kL. The Miller normalization of the backward pass to T_0 is gone, since the recurrence is inhomogeneous. Both passes are one function, ComputeTl.

Cause - the homogeneous solutions of the recurrence are a complex pair of modulus sqrt(B/A), with B/A = ap/am for T^-, not B/A as the previous comments assumed. The backward pass therefore damped its zero seed by (A/B)^25 over its fixed 50-step tail instead of (A/B)^50, and the forward pass was kept up to a growth of (B/A)^(kL/2) = 1e5. On the cth_like_free_bdy boundary at the point of its largest metric cross term (0.89 of sqrt(guu gvv), ap/am = 2.2) that left 2e-9 of the seed in T^-_33 and 5e-12 in T^-_20, the values a 60-digit emulation of the same passes gives at every l, and independent of the working precision: a quad-precision build reproduces them. The alternating binomial sums of the add-back amplify an error in T_33 by up to 1e11 at m + |n| = 33, so the assembled coefficient of mode (17, 16) came out 3.2 times its value.

Evidence - at that point, the assembled bvec coefficients against direct quadrature of the subtracted tangent-plane kernel (mpol 16, ntor 16, kL 33, with the cross-term sign of #825), as the largest relative error per band m + |n|:

m + |n|	before	after
21	3.1e-6	1.2e-9
24	9.2e-5	2.8e-8
27	2.7e-3	2.8e-7
30	7.7e-2	2.6e-6
33	2.2	2.0e-5

and grpmn at band 33 from 2.8e2 to 6.5e-3. On the exact exterior Neumann problem of an interior dipole line on that boundary (residual relative to the external field, zero being exact):

mpol, ntor, nzeta	before	after
16, 14, 60	1.9e-2	2.3e-4
16, 16, 60	0.67	2.3e-4
28, 8, 36	0.24	1.3e-4

which is the floor the previous code reached at mpol 16, ntor 12 (2e-4). In the loop, cth_like_free_bdy at ftol 1e-9 with the boundary padded to higher cutoffs:

mpol, ntor	before	after
12, 12	661 iterations	661 iterations, same equilibrium
16, 12	805 iterations, rmax_surf 1.00872, edge iota 0.86370	689 iterations, rmax_surf 1.00900, edge iota 0.86381
16, 14	no convergence, fsqr 4.8 after 4000 iterations	689 iterations, rmax_surf 1.00899, edge iota 0.86380
16, 16	no convergence, fsqr 2.2 after 4000 iterations	689 iterations, rmax_surf 1.00899, edge iota 0.86380

The three mpol 16 runs now agree with each other to the digits shown, where the previous mpol 16, ntor 12 result differed from them by 0.3 mm in rmax_surf and 1e-4 in the edge iota.

Tests - TlpTlmAccuracyTest gains the cth metric at kL = 33, where T^-_33 was 2e-9 off the quadrature reference against a 1e-11 bound; the three existing cases pass unchanged. bazel test --config=opt //vmecpp/... //vmecpp_large_cpp_tests/... and pytest (217 passed) pass.

Scope - only the T_l passes change. The forward pass is kept exactly where it was accurate, so the bundled cases move at round-off only: every reference comparison of both suites passes at its existing tolerance, and cth_like_free_bdy converges in the same 490 iterations to the same rmax_surf to nine digits. A backward pass costs at most 17 kL steps per surface point; cth_like_free_bdy takes the same wall time as before (0.52 s against 0.52 s at four threads, minimum of three runs).

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