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.

The zero quadratic form gives the exterior algebra

Example

Let $B = 0$ on a finite-dimensional $k$ -vector space $V$ and hence $Q = 0$ . The two quotient presentations identify $C l (V, 0)$ canonically with $Λ V$ . If $e_{1}, \dots, e_{n}$ is an ordered basis, their common basis is $1$ and the increasing square-free monomials. The form is degenerate whenever $V \neq 0$ .

Facts & Assumptions

Given: A field $k$ of characteristic not $2$ , a finite-dimensional vector space $V$ with ordered basis $e_{1}, \dots, e_{n}$ , and $B = Q = 0$ .

[F1]

The Clifford algebra is the tensor-algebra quotient by the ideal generated by $v^{2} - Q (v) 1$ (Universal Clifford algebra of a quadratic space).

[F2]

The exterior algebra is the tensor-algebra quotient by the ideal generated by $v^{2}$ (Exterior Algebra Of A Finite Free Module).

[F3]

The ordered square-free Clifford monomials form a basis (PBW basis and exterior associated graded for a Clifford algebra).

[F4]

The increasing exterior monomials form a basis (Exterior Algebra Basis Monomials).

Verification

technique · direct

1.1givenF1F2construct

Since $Q (v) = 0$ for every $v$ , [F1] and [F2] quotient $T_{k} (V)$ by the same two-sided ideal. The map $[t]_{C l} \mapsto [t]_{Λ}$ is therefore well defined, preserves the induced algebra operations, and has the inverse $[t]_{Λ} \mapsto [t]_{C l}$ .

2.1givenF3F4step 1.1∎

By [F3] and [F4], the same ordered square-free monomials are bases on the two sides; for $n = 0$ both lists consist only of $1$ . Thus the canonical presentation isomorphism has the asserted basis description, including the degenerate case.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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.