# The DKMM benchmark superpotential: exact inputs and the adjudication grid

Companion to [kklt-evaluate.py](kklt-evaluate.py), which recomputes everything
below from these inputs, and to [kklt-w0-ball.json](kklt-w0-ball.json), the
certification record of the proven enclosure. The setting is the vacuum of
Demirtas, Kim, McAllister and Moritz (DKMM), arXiv:1912.10047, on the mirror
of the degree-18 hypersurface in CP[1,1,1,6,9], restricted to the
G = Z6 x Z18 invariant locus (two effective moduli).

## Exact inputs (the polynomial layer; all rational)

| quantity | value | source |
|---|---|---|
| intersection numbers | kappa_111 = 9, kappa_112 = 3, kappa_122 = 1, kappa_222 = 0 | 1912.10047 eq. (12) |
| a | (1/2) [[9,3],[3,0]] | 1912.10047 eq. (12) |
| b | (17/4, 3/2) = c2.D/24 with c2.D = (102, 36) | 1912.10047 eq. (12); cross-checked against 2112.13863 model M_{2,39} |
| flux integers | M = (-16, 50), K = (3, -4) | 1912.10047 eq. (16) |
| N = kappa.M | [[6,2],[2,-16]], det N = -100 | derived, exact |
| flat direction p = N^-1 K | (2/5, 3/10), with K.p = 0 exactly | derived, exact |
| tadpole -M.K/2 | 124 <= Q_D3 = 138 | derived, exact |
| racetrack ratio A | -5/288 | derived from GV(1,0) = 540, GV(0,1) = 3 |
| racetrack constant c | sqrt(2/pi) 8640 / (2 pi i)^2 | derived; the letter's eq. (19) prints -sqrt(2/pi) 8640/(2 pi i)^3, a factor -1/(2 pi i) off, and the script's Layer 3 shows the printed number is consistent only with the derived constant |

Genus-0 Gopakumar-Vafa invariants for classes (d1, d2) with d1, d2 <= 4 are
taken from the independent table of Carta-Mininno-Shukla (arXiv:2112.13863,
Table "CY 39"); the degree <= 2 entries are cross-checked exactly against the
letter's own F_1, F_2 data, multicovers included.

## The zeta(3) term and its three settings

The prepotential constant is

    xi(chi_A) = zeta(3) chi_A / (2 (2 pi i)^3),

with chi_A the Euler characteristic of the A-side manifold, here
chi(X) = -540 for X = P[1,1,1,6,9][18]. The adjudication grid toggles exactly
this one term:

| variant | zeta(3) term | GV truncation | |W0| x 10^-8 | matches printed |
|---|---|---|---|---|
| V1 | kept (chi_A = -540) | deg <= 4 | 2.03710609 | DKMM 2.037 (arXiv:1912.10047) |
| V2 | dropped (xi = 0) | total degree <= 2 | 2.04819630 | Broeckel et al. 2.048 (arXiv:2108.04266) and Carta-Mininno-Shukla 2.0482 (arXiv:2112.13863) |
| V3 | sign reversed (chi_A = +540) | deg <= 4 | 2.05947161 | none of the three |

The nine-digit values are the paper's Table 6; the evaluator recomputes each
one from scratch (exact rational polynomial layer, mpmath dps 50 for the
instanton sums, Newton on the three F-term equations, every residual below
1e-45) and exits nonzero unless all three land on these strings and on the
printed roundings. It also verifies the flux sign-flip invariance
(M, K) -> (-M, -K), the two-precision control at dps 50 vs 90, the
instanton-shell hierarchy with its stated tail bound, and the racetrack
forensics on the letter's printed constant.

## The proven enclosure

[kklt-w0-ball.json](kklt-w0-ball.json) is a byte-exact copy of the
certification record behind Eq. (29) of the paper (SHA-256
28f3307f54fc8bc3543b50ed6bd75396761a6aef69843ac6aa15f038c5ca016a; the
evaluator refuses to run if the file's hash differs). Its enclosure is

    |W0| = 2.037106093311183419... x 10^-8,  half-width 4.36 x 10^-148,

together with certified enclosures of Im tau, Im U1, Im U2 and the F-term
residual box. The evaluator requires the midpoint to begin
2.037106093311183, the half-width to be below 10^-140, and its own kept-term
value V1 to agree with the midpoint to better than one part in 10^9 — the
level at which the truncation of the curve counts begins to matter (the
observed agreement is a few parts in 10^19).

## Controls and exit codes

- `python3 kklt-evaluate.py` — rc 0 only if every comparison passes.
- `python3 kklt-evaluate.py --mutate` — reverses the sign of chi in the V1
  branch; V1 then reproduces the V3 value and the run fails (rc 1) with named
  FAIL lines. This is the deliberate sanity control.
- Any edit to kklt-w0-ball.json makes the run refuse with rc 2 and both
  SHA-256 values printed.

## Sources

| tag | source | pristine-archive SHA-256 (first/last 8) |
|---|---|---|
| [L] | arXiv:1912.10047 v2 (DKMM PRL letter), LaTeX | 4c530d47...d8c81b |
| [B] | arXiv:2108.04266 (Broeckel, Cicoli, Maharana, Singh, Sinha), LaTeX | 7d08a878...9ed5c691 |
| [C] | arXiv:2112.13863 (Carta, Mininno, Shukla), LaTeX | eabaa17d...7cd35612 |
| [J] | arXiv:2107.09064 (the AdS4 companion; source file AdS4_v3.tex, the xi convention) | 03c7d9d7...8862e92b |