LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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 Segre map is well defined and injective

Statement

The Segre point map is a well-defined injective map of point sets. This includes $m = 0$ or $n = 0$ .

Facts & Assumptions

Given: Representatives $x \neq 0$ and $y \neq 0$ .

Proof

1.1givenalgebra

Replacing $x, y$ by $λ x, μ y$ replaces every coordinate $x_{i} y_{j}$ by the single nonzero scalar $λ μ$ times it. Some product $x_{i_{0}} y_{j_{0}}$ is nonzero, so these coordinates define a well-defined projective point.

2.1step 1.1algebra

Choose $i_{0}, j_{0}$ with $x_{i_{0}} \neq 0$ and $y_{j_{0}} \neq 0$ . From a projective matrix $[z_{i j}]$ in the image, the ratios $z_{i j_{0}} : z_{i_{0} j_{0}}$ recover $[x_{i}]$ , and $z_{i_{0} j} : z_{i_{0} j_{0}}$ recover $[y_{j}]$ .

3.1step 2.1∎

Thus equal Segre images have equal two projective factors. The same calculation works when one factor has one homogeneous coordinate, proving the endpoint cases.

The Segre map is well defined and injective

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources