# `src/forms/integral/` editing guide
This directory joins integral lattices, finite quadratic modules, codes,
modular forms, and Clifford/Weyl constructions. Integral/rational arithmetic
and algebraic phase classifiers are exact unless an API explicitly returns a
checked failure; complex Gauss sums and Weil matrices are numerical and use
documented tolerances.
## Map
| `lattice/` | Gram matrices, duals, determinant, signature, minima, roots |
| `diagonal.rs` | integral diagonal and parity data |
| `discriminant/` | finite quadratic modules, Gauss phases, and Weil matrices |
| `fqm_witt.rs` | bounded Witt reduction for finite quadratic modules |
| `genus.rs` | canonical p-adic symbols and Nikulin criteria |
| `mass_formula.rs` | exact mass and Bernoulli/Eisenstein constants |
| `kneser.rs` | denominator-checked p-neighbors and mass closure |
| `codes.rs` | binary/p-ary codes and Constructions A, B, and D |
| `theta.rs`, `modular.rs` | exact theta coefficients and modular-form identities |
| `root_lattices.rs`, `niemeier.rs` | ADE and Niemeier catalogues |
| `clifford_lattices.rs` | Clifford/spinor certificates for Barnes--Wall data |
| `weyl_versors.rs` | roots and Weyl reflections realized by Pin versors |
## Invariants
- Integral Gram matrices are symmetric and dimension-checked at construction.
- Scaling conventions in Constructions A/B/D are part of the API. Return
`None` when the requested scaled Gram is not integral; do not round.
- Discriminant-form arithmetic and algebraic phase classification are exact.
Complex Gauss sums and `Complex64` Weil matrices are numerical realizations
with explicit tolerances. The Weil `S` prefactor uses the conjugate positive
Milgram phase, and verification checks the metaplectic relations rather than
the false shortcut `S^4 = I`.
- Odd lattices require the oddity correction. Do not apply the even-lattice
Milgram formula unchanged.
- Canonical 2-adic symbols require train, compartment, sign-walking, and oddity
normalization. Preserve equivalent-symbol tests when editing reduction.
- Kneser neighbors must validate denominators and isotropic lines before
constructing an integral Gram matrix.
- Catalogue data are standard/source-backed inputs. Separate catalogue checks
from generated theorem claims.
- Automorphism-group orders, counts, discriminants, and other fixed-width
mathematical data use `u128`/`i128`.
## Cross-pillar boundaries
- `IntegralForm` may export rational and mod-two Clifford metrics, but the
Clifford engine remains generic and owns geometric multiplication.
- Brown invariants of 2-elementary discriminant forms and real signatures meet
through Milgram's relation; they are computed by different exact paths.
- Code-to-lattice and Clifford-to-lattice constructors must retain explicit
witnesses rather than asserting an isomorphism from matching headline
invariants alone.
- Rank-24 Niemeier statements may be catalogue-backed even when lower-rank
genera have explicit neighbor enumeration. State that distinction.
## Verification
Use focused determinant, parity, theta, mass, and Weil-relation tests for the
changed object, then `cargo test --workspace` and Clippy. New catalogue rows
need a cited source and a test of every consumed field.