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.

A nontrivial virtual character on a two-point manifold

Example

Let $T = R / Z$ and let $M = {p, q}$ be a two-point smooth $0$ -manifold with trivial $T$ -action. For $k \in Z$ , let $C_{k}$ be the one-dimensional representation $[s] \mapsto e^{2 π i k s}$ . Take the zero-differential complex whose fibers are $E_{p}^{0} = C_{0} \oplus C_{1}$ , $E_{p}^{1} = C_{2}$ , $E_{q}^{0} = C_{1}$ , and $E_{q}^{1} = C_{0}$ , with all other fibers zero. Its equivariant index is $2 [C_{1}] - [C_{2}]$ , with character $2 z - z^{2}$ at $z = e^{2 π i s}$ . Every point is fixed, the tangent determinant is the empty determinant $1$ , and the fixed-point sum is also $2 z - z^{2}$ .

Facts & Assumptions

Given: The two-point compact smooth manifold, the trivial base action, the displayed fiber representations, and zero differential.

[F1]

The equivariant index is the alternating class of finite-dimensional cohomology representations and its character is the alternating trace. Equivariant elliptic-complex index and character

Verification

1.1givenF1algebra

Since $M$ is $0$ -dimensional, there is no nonzero cotangent vector at any point, so the symbol exactness condition is vacuous. The zero maps form a complex, and its cohomology in each degree is the displayed fiber representation. By [F1], its index is $[C_{0}] + [C_{1}] - [C_{2}] + [C_{1}] - [C_{0}] = 2 [C_{1}] - [C_{2}]$ , whose character is $2 z - z^{2}$ .

2.1givenF1step 1.1algebra∎

Every $g \in T$ fixes $p$ and $q$ . The tangent space at either point is the zero vector space, so $I - d g$ is its unique invertible endomorphism and has determinant $1$ . The local fiber supertraces are $1 + z - z^{2}$ at $p$ and $z - 1$ at $q$ . Their sum is $2 z - z^{2}$ , agreeing with the index character.

Depends on

Equivariant elliptic-complex index and character

A nontrivial virtual character on a two-point manifold

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 1 level