# lattice/ -- K3 x K3 lattice data (main text Section 3; Supplementary Material Sections S1.2, S5)

Files (formats are described in the `description` field of each JSON file; the `.txt` files render the same integers
for reading):

- `rank19_lattices.json` / `.txt`: Gram matrices and invariants of W512, W512', W1024 (Theorem 3.5, Eq. (34), Table 3;
  SM Eqs. (S5.1)-(S5.3), Table S5) and of Lambda_19 (SM Section S1.2; SM Eq. (S5.4), Table S5, LLL-reduced basis and the construction
  basis with the unimodular transformation between them), and the three primitive embeddings W(-1) -> Gamma_(3,19)
  (SM Eqs. (S5.5)-(S5.7)): 22x22 overlattice Gram matrix, 19x22 embedding, 3x22 complement basis with Gram
  diag(2,2,128), diag(2,2,128), diag(2,2,256), and the glue index.
- `genera54.json` / `.txt`: the 54 admissible genera of Lemma 3.3 (Eq. (33); SM Section S5.2, Tables S6-S8): determinant,
  Conway-Sloane symbol, discriminant group, an even ternary lattice T realizing the genus as the orthogonal complement
  of T in Gamma_(3,19), the exact Siegel mass of the rank-19 genus and the resulting lower bound on its class number,
  the mass-weighted expected root number E(2), and for the rank-5 complement genus the complete list of classes
  (98 in all) with Gram matrix, |Aut|, root count, the exact mass, and the 1764 = 18 x 98 rational
  Farkas certificates (one per class and Coxeter number h in {2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 22, 25, 30, 46}) of
  SM Eqs. (S5.8)-(S5.11), each proving that no Niemeier lattice with Coxeter number h contains the class with
  root-free orthogonal complement. One certified class per genus already proves Lemma 3.3; the complete class lists
  and the mass identities are a consistency check. The 32 theta-series linear-programming certificates of
  SM Eq. (S5.13) are not included (their verification needs the weight-19/2 constraint system, which is not part of
  this folder); the column `closed_by` marks the genera they cover with the value `both`.
- `floor_certificates/`: the five certificates of Table 3 / Theorem 3.5 (`table3_C-*.json`; SM Section S5.3,
  Eqs. (S5.14)-(S5.17), Table S9) and the 38 certificates with one released column (`dropJ_<W>_J<j>.json.gz`;
  SM Table S10, Eq. (S5.19)), in the format stated in `index.json`; `index.json` also lists the surviving position
  pairs for two released columns (55 of 171 for W512, 40 of 171 for W512'); `local_solvability_156.json` holds the
  156 local witnesses of SM Eq. (S5.20) (q = 48, det M = d solvable over Z_2 and Z_3 for the determinant-512 pairs).
  Exact values: C-1024: V = 13245254530664589/274877906944000; C-512u: V = 41004589043118677/879609302220800; C-512pu: V = 819922474416771261/17592186044416000; C-512k0: V = 32367743330235055603013986807893563514994926733/633825300114114700748351602688000000000000000; C-512pk0: V = 15587066067512245850434421159630819329364218997/316912650057057350374175801344000000000000000.
- `rootfree_det160_224.json` / `.txt`: the seven root-free admissible rank-19 lattices with 160 <= det <= 224 of
  Section 3.3 and SM Section S1.2, their invariants (det, minimum, kissing number, Smith invariants,
  |Aut|, the order of O_0 = ker(Aut -> O(L^*/L)) and the rank it fixes), and 18 explicit integer isometries
  U (U^T B U = A, det U = +-1) to the reference Gram matrices reproduced in the file with their sources: the Leech
  coinvariant lattices Lambda_M20, Lambda_A6, Lambda_L2(7) of Hoehn-Mason (rows 69, 72, 77 of their Table 1),
  Martinet's tables, and the Nebe-Sloane catalogue entries attributed to Kallus. The lattices at det 208, 216, 220,
  224 have O_0 fixing a nonzero vector and are therefore not coinvariant lattices; det 216 is not Lambda_M9.
- `verify_lattice.py`: the checker (`python3 verify_lattice.py`, `--quick`, `--no-gp`); see its header for the list of
  checks. Python 3 standard library; PARI/GP, if installed, is used to recompute |Aut|.