Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

A disconnected graph has a Laplacian kernel spanned by its component indicators

Example

Let G have vertex set {1,2,3,4} and edge set {{1,2},{3,4}}. Then

ker⁡L(G)=span⁡{(1,1,0,0)T,(0,0,1,1)T}.

Facts & Assumptions

Given: The graph G with components {1,2} and {3,4}.

[L1]

The Laplacian kernel is spanned by the indicator vectors of the connected components (The multiplicity of the Laplacian eigenvalue 0 equals the number of connected components).

Verification

technique · direct
1.1

The graph G has exactly two connected components, namely {1,2} and {3,4}. Their indicator vectors are (1,1,0,0)T and (0,0,1,1)T.

given
2.1

By [L1], those two indicator vectors span the Laplacian kernel. They are linearly independent, so this displayed span is exactly ker⁡L(G).

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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