ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

Why a nonproper fixed locus needs another theory

Example

Let $T = C^{\times}$ act trivially on $X = C^{\times} = Spec C [z, z^{- 1}]$ . Then the fixed locus is $F = X$ , and $F \to p t$ is not proper. Here proper means finite-type, separated, and universally closed; after base change to $A^{1}$ with coordinate $s$ , the closed subset ${(z, s) : z s = 1} \subseteq C^{\times} \times A^{1}$ projects to the nonclosed subset $A^{1} ∖ {0}$ . Consequently the intrinsic localized pushforward defined using a proper fixed-locus trace has no value on $[F]_{T}$ ; this example assigns no fixed-point integral or reciprocal-Euler sum.

Example references

[Borel–Moore proper pushforward] Ginzburg, Section 1

[Fixed-locus localized pushforward and its proper-trace hypothesis] Edidin–Graham, Section 4

Facts & Assumptions

Given: The trivial action of $T = C^{\times}$ on $X = Spec C [z, z^{- 1}]$ , the algebraic definition of properness in the Example, and rational coefficients.

[F1]

The localized pushforward in this library is defined by the proper trace $p_{F *}$ after inverting the closed-inclusion Borel–Moore map, and requires $F \to p t$ proper. Intrinsic localized pushforward for a proper fixed locus

[F2]

Borel–Moore fundamental classes and proper pushforwards use the normalized equivariant construction. Localized torus-equivariant Borel–Moore classes

[A1]

AC is inherited through the Borel–Moore construction; the failure of properness below is choice-free. Localized torus-equivariant Borel–Moore classes

Verification

Given: The data in the Example.

1.1givenconstruct

The action is trivial, so every point is fixed and $F = X$ ; the single affine chart $X$ is a finite invariant affine atlas. The morphism $X \to p t$ is finite type and separated. After base change to $A_{s}^{1}$ , the equation $z s = 1$ defines a closed subset of $X \times A_{s}^{1}$ , but its projection is $A_{s}^{1} ∖ {0}$ , which is not closed. Thus $X \to p t$ is not universally closed and is not proper.

2.1A1F1F2step 1.1∎

By [F1], the construction $p_{*}^{l o c}$ requires the proper Borel–Moore trace $p_{F *}$ . Step 1.1 shows that this input map is not proper, so the definition supplies no such map on $H_{*}^{B M, T} (F; Q)$ and therefore no value on $[F]_{T}$ . Any different integration theory would require additional data and is not assigned a value here.

Why a nonproper fixed locus needs another theory

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources