ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge 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 incidence variety of a point and a hyperplane

Example

In $P^{n} \times (P^{n})^{\lor}$ , write a point as $[x_{0} : \dots : x_{n}]$ and a hyperplane as $[a_{0} : \dots : a_{n}]$ . Incidence is the closed bilinear equation $\sum_{i = 0}^{n} a_{i} x_{i} = 0.$ For fixed $[a]$ its fibre is that hyperplane $P^{n - 1}$ ; for fixed $[x]$ its fibre is the projective space of hyperplanes through $[x]$ , also $P^{n - 1}$ when $n \geq 1$ . When $n = 0$ , the unique point of $P^{0}$ lies on no hyperplane: the incidence locus and both fibres are empty.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

MIT 18.725 Algebraic Geometry, Lecture 7, Lemma 16 (standard reference, not scraped)