ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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.

The cycle $C_{4}$ has adjacency spectrum ${2, 0, 0, - 2}$

Example

The cycle graph $C_{4}$ has adjacency spectrum ${2, 0, 0, - 2}$ .

Facts & Assumptions

Given: The cycle graph $C_{4}$ .

[L1]

The cycle graph $C_{n}$ has adjacency eigenvalues $2 \cos (2 π j / n)$ for $0 \leq j < n$ (The cycle graph $C_{n}$ has adjacency spectrum ${2 \cos (2 π j / n) : 0 \leq j < n}$ ).

Verification

technique · direct

1.1

Applying [L1] with $n = 4$ gives the eigenvalues $2 \cos (0) = 2$ , $2 \cos (π / 2) = 0$ , $2 \cos (π) = - 2$ , and $2 \cos (3 π / 2) = 0$ .

L1algebra

2.1

Reordering these values from largest to smallest yields ${2, 0, 0, - 2}$ .

step 1.1∎

Depends on

The cycle graph $C_n$ has adjacency spectrum $\{2\cos(2\pi j/n):0\le j<n\}$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Steve Butler, Spectral Graph Theory course notes, lecture 3 (standard reference, not scraped)

The cycle C4C4​ has adjacency spectrum {2,0,0,−2}{2,0,0,−2}