ExampleConstruction: AI-generatedVerification: Not suppliedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

Pattern inventory of square colourings by number of red vertices

Example

Give blue weight $1$ and red weight $r$ in the dihedral action on the square. Substituting

$s_{i} = 1 + r^{i}$

into the cycle index of The cycle index of D_8 yields

$1 + r + 2 r^{2} + r^{3} + r^{4} .$

So, up to dihedral symmetry, there is:

$1$ colouring with $0$ red vertices;
$1$ colouring with $1$ red vertex;
$2$ colourings with $2$ red vertices, namely adjacent and opposite pairs;
$1$ colouring with $3$ red vertices;
$1$ colouring with $4$ red vertices.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.