# cert_w0_vac/ -- the certified DKMM critical point (main text Eqs. (66)-(67), Table 6; SM Section S3.3)

Inputs: `../restrict/` (the two-parameter Picard-Fuchs ideal restricted to the curve z_1 = s^4, z_2 = s^3 and its
jet tower), `../cert_w0/` (certified on-curve period vector, integral frame, transport library). Fluxes
F = (7,3,-24,0,-16,50), H = (0,3,-4,0,0,0) in the slot order (F_0,F_1,F_2,X^0,X^1,X^2); W = sqrt(2/pi) (F - tau H)^T Sigma Pi;
e^{-K} = Re(-i Pi^dagger Sigma Pi) * 2 Im tau; \|W0\| = \|W\| / sqrt(e^{-K}).

What the scripts do:

- `jet_ideal.py` derives, by exact D-module identities over Q (sympy), the 6x6 connection along the curve and the
  transverse-derivative operators, and writes `connection_z.json`, `curve_T.json`; `jet_checks.py` re-verifies these
  identities in exact rational arithmetic (`out_jet_checks.txt`).
- `run_vac.py solve <dps> <route>` transports the certified period jets to the fixed curve point, solves the three
  F-term equations D_tau W = D_(U^1) W = D_(U^2) W = 0 by Newton iteration in the coordinates
  x = (Re w_1, Im w_1, Re w_2, Im w_2, Re tau, Im tau), w_a := z_a / z_a^* - 1, and then certifies the zero with the
  Krawczyk operator K(X) = y - Y F(y) + (I - Y J(X))(X - y) evaluated in ball arithmetic over the box X: containment
  K(X) in int(X) proves existence and uniqueness in X. The first containing box has radius 1e-10; seven contractions
  bring the radii to about 5e-149 (the six final radii are in `box_rad`).
- `run_vac.py checks` compares the dps-60 and dps-150 runs, route R with route C, the vacuum value with the on-curve
  companion (`penalty_rel`), and the certified value with the floating-point model of `../pilot/` evaluated at the
  same point, and records the verdict on the published 2.037e-8 (`checks_vac.json`).

Results (route R, 150 digits; `result_vac_R_150.json`, full digit strings there):

- \|W0\|_vac = 2.0371060933111834191228615593319846787338892373526324726660609722933741523879816788352864219194388563789736419637602168325205746924380676300e-8 +/- 4.36e-148
- Im tau_vac = 6.8554572563187259039156861046681776294147790681460856004327242213796069793833849860207836062604891703496482555325773458985318315717717357518791965797 +/- 6.76e-149; Re tau_vac = [+/- 1.64e-150]
- Im U^1 = 2.74217698020852099220720053021178472801..., Im U^2 = 2.05663277310473392972944546537015494378...
- w_1, w_2 (real parts, stored to 25 digits): [7.944427495308857534704976e-5 +/- 1.45e-30], [7.944617279031411801840431e-5 +/- 1.58e-30]; imaginary parts [+/- 4.12e-150], [+/- 3.09e-150]
- offset of the vacuum value from the on-curve companion: [0.000230482072338 +/- 3.96e-16]
- e^{-K_cs} in the homogeneous gauge X^0 = varpi_0: [484.648180749750442098438725739 +/- 1.80e-28]
- route C (independent complex path): \|W0\|_vac = 2.03710609331118341912286155933198467873...e-8 +/- 7.52e-138; the two balls overlap (129 common digits).

The published value \|W0\| = 2.037e-8 of arXiv:1912.10047 lies inside its four-digit rounding band of the certified ball.
The file `result_vac_R_150.json` is byte-identical to `kklt-w0-ball.json` at the top of this download folder, which
`kklt-evaluate.py` identifies by SHA-256.