# pfaffian_ads581/ -- the AdS vacuum 5-81-3213 (Section 5.4, Eq. (69); SM Eqs. (S3.5)-(S3.7))

Target: the vacuum labeled 5-81-3213 in the ancillary files of arXiv:2107.09064 (h^{2,1} = 5, h^{1,1} = 81), with
fluxes M = [1, -1, -5, 6, -3], K = [0, -4, 9, -5, 0] in the frame described in `card_ads-5-81-3213.json` (`comment`, `charge_vectors`,
`kappa`, `a_mat`, `c2D`, `chi_A`); the perturbatively flat direction, the
one-parameter curve s -> z(s) through the vacuum and the racetrack data are recorded in the card's `comment`; the evaluation point is s_vac = -0.00555555555555555555555555555556 (= -1/180) and tau* = 12 + i 19.8356340499703866831702370891 (Re tau from the flux frame, Im tau from the racetrack
solution; `verdict.py`). The
tadpole of these fluxes is half-integral, -M.K/2 = 71/2, which the source accounts for with one half D3-brane; the
computation is carried out under that convention (`allow_half_tadpole` in the card) and every result inherits the
condition.

Route. The mirror family has five moduli; restricted to the curve, the D-module generated by the fundamental period
has rank 17 (the 12 monomials of `module_basis` do not close). `a1_perprime.py` and `a1_lift.py` reconstructed the exact
17x17 connection matrix A(s) over Q from evaluations modulo eight primes (Chinese remaindering and rational
reconstruction; 123,824 coefficients, common denominator of degree 603): `A_exact.json`. These two
scripts are included as the record of the lift; they need a modular-arithmetic helper module (`pmodp`) and the per-prime
interpolants, which are not part of this folder, so they do not run from it. The folder's checks of `A_exact.json` are
the next two.
`check_pa1.py P` re-evaluates A(s) at a prime P never used in the lift and compares (512 nodes, no
differences: `check_PA1.json`); `jets_series.py 3000` writes the analytic 17-jet of the fundamental period on the curve
to order 3000 (`jets_3000.json`, 6.5 MB, about two minutes, not shipped) and `check_pa2.py` verifies that A annihilates it
exactly through s^3000 (`check_PA2.json`). `check_PA3_amended.json` records that A has a pole of order 19 at
s = 0 in this basis although the monodromy there is maximally unipotent (an apparent singularity of the basis), so
Frobenius seeding at s = 0 is not used. Instead `verdict.py` evaluates every period directly: the restricted rho-jet
Gamma-series towers (`tower_120.json`, M = 120) converge on \|s\| < 1/2 by the coefficient bound c(n) <= 2^m, and
\|s_vac\| = 1/180 lies well inside, so each period is a finite sum plus an explicit tail ball; the integral frame is fixed
by an exact fit of the Gamma-jets to the model prepotential on orders m <= 50 with orders 51..55 held out
(`check_PA4.json`, `f3_frame.json`); `check_PA5_substitute.json` records the convergence-radius argument that replaces
transport; `check_PA6.json` the two-precision comparison.

Result (`verdict_raw.json`): undressed \|W\| = sqrt(2/pi) \|(F - tau H)^T Sigma Pi\| at (s_vac, tau*) in the gauge X^0 = varpi_0(s):
2.0377822475698608540440976364927797294098195325121369695249464740504e-23 +/- 7.72e-91 (90 digits; the 60-digit ball overlaps), relative radius 3.14e-68; published
2.03778e-23, relative deviation 1.103e-06, inside the half-unit of the last published digit. Scope, verbatim from
`check_PA6.json`: "CONDITIONAL on C2: source papers' flux-quantization convention (odd integer quanta; one stuck half D3-brane, Q_flux=71/2 in Z+1/2)"

`pipe/` holds the library modules (`family.py` card parsing, `geff_series.py` Gamma-series and jets for toric
hypersurface families including non-simplicial effective cones, `pipe_lib.py` frame fit and contraction) and, under
`pipe/cards/`, the card and the data bank read by the checks (`w0_series_3000.json.gz`, the per-prime connection
samples `dmodule/conn_samples_<p>.npz`, `check_g2.json`).