ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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 additive translation action embeds equivariantly as a parabola

Example

For $G = (C, +)$ acting on $A^{1}$ by $g \cdot x = x + g$ , the subspace $W = span (1, z, z^{2})$ is a generating rational coordinate submodule. Evaluation embeds $x$ as $(1, x, x^{2}) \in W^{*} ≅ A^{3}$ , with closed image $a = 1$ , $c = b^{2}$ . The ambient action is linear. Assume AC for the A-page embedding theorem.

Facts & Assumptions

Given: This translation action and AC (The Axiom of Choice).

[F1]

The coordinate action is rational and uses inverse pullback (Classical complex affine algebraic actions and rational modules, The coordinate ring of an affine algebraic action is a locally finite rational module).

[F2]

Evaluation in the dual of a generating rational submodule is an equivariant closed embedding (Every complex affine algebraic action has a finite-dimensional equivariant closed embedding).

Verification

1.1F1F2givenalgebra

Inverse pullback gives $g 1 = 1$ , $g z = z - g$ , and $g z^{2} = z^{2} - 2 g z + g^{2}$ . Therefore $W$ is stable with polynomial matrix entries in $g$ , and it generates $C [z]$ because it contains $z$ . Dualizing as in F2 gives $g (a, b, c) = (a, b + g a, c + 2 g b + g^{2} a)$ , a linear transformation for fixed $g$ with regular coefficients and the additive group law.

2.1F2step 1.1algebra∎

Evaluation is $(1, x, x^{2})$ , whose image is exactly ${a = 1, c = b^{2}}$ : each point there is uniquely obtained by $x = b$ , a regular inverse. The action in step 1.1 sends it to $(1, x + g, (x + g)^{2})$ , proving equivariance directly and displaying F2's construction.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Michel Brion, Introduction to actions of algebraic groups (2010) (standard reference, not scraped)
Philippe Gille, Introduction to reductive group schemes over rings, full notes retrieved 2026-10-02 (standard reference, not scraped)
J. S. Milne, Algebraic Groups (2022) (standard reference, not scraped)