DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The E8 lattice from its Cartan Gram matrix

Definition

Let $L_{E_{8}} = Z^{8}$ with ordered basis $e_{1}, \dots, e_{8}$ and symmetric bilinear form whose Gram matrix $C = (c_{i j})$ has $c_{i i} = 2$ , has $c_{i j} = c_{j i} = - 1$ exactly when ${i, j}$ is one of ${1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {3, 8}$ , and has all other entries zero. The associated graph is the E8 Dynkin diagram; each edge records a Gram entry $- 1$ and each vertex records a diagonal entry $2$ .

Definition references

Integral lattice and Gram matrix conventions

Depends on

Integral intersection lattice of a simply connected four-manifold

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tim Perutz, Lectures on Gauge Theory and Symplectic Geometry (standard reference, not scraped)