ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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 one-dimensional Clifford algebra for any quadratic coefficient

Example

Let $k$ be a field of characteristic not $2$ , let $V = k e$ and let $B (e, e) = a \in k$ . Then $C l (V, Q)$ has basis $1, \overset{ˉ}{e}$ and multiplication table determined by ${\overset{ˉ}{e}}^{2} = a 1$ . This remains true at $a = 0$ ; no matrix classification for $a \neq 0$ is asserted.

Facts & Assumptions

Given: The one-dimensional quadratic space $V = k e$ and the scalar $a = B (e, e)$ .

[F1]

For a one-dimensional quadratic space, the ordered Clifford monomials $1, \overset{ˉ}{e}$ form a basis (PBW basis and exterior associated graded for a Clifford algebra).

[F2]

The canonical generator satisfies ${\overset{ˉ}{e}}^{2} = Q (e) 1$ (Universal Clifford algebra of a quadratic space).

Verification

technique · direct

1.1givenF1F2

Since $(e)$ is an ordered basis and $Q (e) = B (e, e) = a$ , [F1] gives the basis $1, \overset{ˉ}{e}$ and [F2] gives ${\overset{ˉ}{e}}^{2} = a 1$ .

2.1step 1.1algebra∎

Bilinearity and the unit law determine every product of basis elements from this relation; when $a = 0$ , [F1] still ensures $\overset{ˉ}{e} \neq 0$ , while [F2] gives ${\overset{ˉ}{e}}^{2} = 0$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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