ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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 full Hilbert functor need not be quasi-compact

Example

For every field $k$ , the full Hilbert scheme of $P_{k}^{1}$ is not quasi-compact. Its fixed-polynomial pieces are projective, while there are infinitely many nonempty open and closed pieces.

Verification

Given: AC and DC and a field $k$ .

[F1] The full Hilbert scheme is the disjoint union of fixed-polynomial projective representatives (Projective Hilbert schemes represent all flat finitely presented families).

[F2] A length- $d$ finite subscheme has constant polynomial $d$ (The Hilbert polynomial of a finite scheme is its length).

1.1F2algebra

For every $d \geq 1$ , the subscheme on the affine chart given by $x^{d} = 0$ , viewed as a closed subscheme of $P_{k}^{1}$ supported at $[0 : 1]$ , is finitely presented and has length $d$ . Being over a field it is flat. Thus the polynomial- $d$ stratum is nonempty by [F2]. These are distinct strata for distinct $d$ .

2.1F1step 1.1algebra∎

All strata are open and closed by [F1], and they form an open cover of the full Hilbert scheme. No finite subfamily of this cover contains the nonempty strata for all $d$ . This cover has no finite subcover, proving failure of quasi-compactness and therefore of finite type or properness over $k$ .

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