DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Margulis gabber galil graph

Definition

For integer $m \geq 1$ , let $V = (Z / m Z)^{2}$ . The Margulis–Gabber–Galil graph has the following eight slots at $(x, y)$ , with all arithmetic modulo $m$ : $(x + 2 y, y), (x + 2 y + 1, y), (x - 2 y, y), (x - 2 y - 1, y), (x, y + 2 x), (x, y + 2 x + 1), (x, y - 2 x), (x, y - 2 x - 1) .$ Multiplicities and fixed points are retained under Regular multigraph and normalized adjacency. Write $T_{1} (x, y) = (x + 2 y, y)$ and $T_{2} (x, y) = (x, y + 2 x)$ . Pair each forward affine map with its inverse as reverse ports, even when their destinations coincide.

Depends on

Regular multigraph and normalized adjacency

Used by

Dependency tree · one level

1 result within one dependency step 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

Hoory–Linial–Wigderson, Expander Graphs and Their Applications, May 2006 draft; Chapter8 Construction8.1, p69. (standard reference, not scraped)