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 Hilbert polynomial of finite points on the projective line

Example

For any field $k$ , a finite closed subscheme $Z \subseteq P_{k}^{1}$ of length $d$ has the constant Hilbert polynomial $P (r) = d$ for the usual polarization, including nonreduced points and points with nontrivial residue field. A family of such points belongs to the constant-polynomial stratum only when it is flat and finitely presented as specified in Hilbert functor of flat finitely presented projective families.

Verification

Given: AC and DC, a field $k$ , and a finite closed subscheme $Z$ of length $d$ .

[F1] Finite schemes have the constant length polynomial (The Hilbert polynomial of a finite scheme is its length). The fixed-polynomial subfunctor and its universal stratum are Hilbert functor of flat finitely presented projective families, Universal family and open and closed Hilbert polynomial strata.

1.1F1algebra

By the finite-scheme supplier in [F1], every invertible twist restricted to $Z$ has $d$ -dimensional sections and zero higher cohomology. Thus its Euler characteristic is the constant $d$ for every integer twist. This includes nilpotent structure and residue-field degrees, since length is the full dimension of the finite coordinate algebra over $k$ .

2.1F1step 1.1algebra∎

The Hilbert polynomial is consequently $P (r) = d$ . For a flat finitely presented family of total fibre length $d$ , the classifying map lands in the corresponding open and closed stratum by [F1]. The finite-scheme supplier also gives $d = 0$ for the empty subscheme.

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