# cert_w0/ -- certified periods and \|W0\| on the curve through the vacuum (Section 5.2; SM Section S3.3)

Inputs: `../restrict/operator_LS.json` (the order-6 operator L_s in theta-form, exact integers; indicial polynomial
627750 theta^4 (theta-3)^2 at s = 0), `../restrict/tower_curve.json` (exact rho-jets of the Gamma-series to s^240),
`../restrict/series_curve.json`, `../restrict/gv_extracted.json`, `../restrict/s_vac.json` (the curve point s* as an exact
rational), and the flux data above.

- `run_cert.py stage1` solves for the integral symplectic frame at the point of maximal unipotent monodromy as an exact
  identity between the model prepotential (intersection numbers, the matrix a, the vector b, the zeta(3) chi term,
  the Gopakumar-Vafa numbers in the exactly known window) and the Gamma-series jets, over the ring Q[v^(+-1), zeta(3)],
  v = 2 pi i: fit on orders m <= 14, held out on orders 15..20 (exact match), and L_s annihilates every fitted tower to
  s^240. Flipping the sign of chi forces a spurious zeta(3) counterterm, which the correct frame does not need
  (`out_stage1.txt`). Writes `frame_solution.json`, `towers_ext.json`.
- `run_cert.py route R|C <dps>` transports the frame to s* along the real axis (R, 16 legs) or along a complex detour
  with a different base point (C, 54 legs), with the tail of every local expansion bounded by the lemma of
  `../majorant/`, contracts with the fluxes and writes `result_<route>_<dps>.json`.
- `run_cert.py checks` compares precisions and routes and writes `checks_cert.json`: dps 60 vs 150 agree to
  59 digits (the dps-60 radius), routes R and C overlap with 148 common
  digits, relative radius at dps 150 2.27e-158; it also compares the floating-point model at the same point
  with the certified midpoint, 4.489e-19 relative (the model value is read from `out_model_at_point.txt`;
  `kklt-evaluate.py` at the top of this download folder and `../cert_w0_vac/run_vac.py checks` recompute the same
  model-truncation offset at the vacuum).

Result (`result_R_150.json`): \|W0\|(s*, tau*) = 2.0366366850674102932936107130624826354921767707568221869698843593996978553049314586799159156499367194532045822982353423321750541081465890473316284356495857266e-8 +/- 4.62e-166 at the curve point s* = 0.0134681831826400193897483719453
and Im tau* = 68554572563187259046361432042141669/10000000000000000000000000000000000 (exact rationals). This is not the vacuum value: the true critical point lies off the
curve, and the relative offset 0.000230482 is certified in `../cert_w0_vac/`.
`cert_lib.py` holds the exact frame/series machinery and `cert_transport.py` the ball transport.