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 compact symbol at a point retains the full torus character

Example

Let $T = R / Z$ , let $M = {p}$ , and put $C_{k} = C$ with $[s] \in T$ acting by $e^{2 π i k s}$ . Take the two-term complex $0 \to E^{0} \overset{0}{\to} E^{1} \to 0$ , where $E^{0} = C_{0} \oplus C_{1}$ and $E^{1} = C_{2}$ . Its cohomology is $H^{0} = \ker (0 : E^{0} \to E^{1}) = E^{0}$ and $H^{1} = \ker (0 : E^{1} \to 0) / im (0 : E^{0} \to E^{1}) = E^{1}$ . The compact symbol triple on $T^{*} M = {p}$ is $(E^{0}, E^{1}, 0)$ . Its virtual bundle is $[C_{0}] + [C_{1}] - [C_{2}]$ , whose character at $g = [s]$ is $1 + z - z^{2}$ , $z = e^{2 π i s}$ . The cohomological index character is defined here as the alternating cohomology trace $\sum_{j = 0}^{1} (- 1)^{j} tr (g ∣ H^{j})$ , which is also $1 + z - z^{2}$ . The normal Koszul denominator is $1$ because the normal bundle has rank zero.

Facts & Assumptions

Given: The one-point manifold, the displayed fiber representations, and the zero differential.

[F1]

The compact symbol is the even-to-odd principal-symbol triple, and a rank-zero normal bundle has exterior algebra consisting of one even trivial line. Equivariant compact-support symbol classes and the normal Koszul class

[F2]

For a finite-dimensional equivariant linear map $A : E^{+} \to E^{-}$ , its analytic equivariant index is $[\ker A] - [\ker A^{*}]$ . Equivariant compact-support symbol classes and the normal Koszul class

Verification

1.1givenF1F2algebra

There is no nonzero cotangent vector at $p$ , so ellipticity is vacuous. The differential is zero, hence $H^{0} = C_{0} \oplus C_{1}$ and $H^{1} = C_{2}$ . Its principal symbol is zero; by [F1] its symbol triple has exactly these even and odd bundles. The virtual fiber and the index in [F2] are both $[C_{0}] + [C_{1}] - [C_{2}]$ . Taking traces gives $1 + z - z^{2}$ .

2.1givenF1step 1.1algebra∎

Every $g$ fixes $p$ , and the cotangent inclusion of its fixed locus is the identity of the point. The normal bundle is zero. Thus $λ_{- 1} (0) = 1$ , and its character and the empty determinant both equal $1$ . Dividing the virtual fiber character from step 1.1 by this denominator gives $1 + z - z^{2}$ , agreeing with the index character. This verifies the point calculation directly and uses no general analytic/topological comparison theorem.

Depends on

Equivariant compact-support symbol classes and the normal Koszul class

A compact symbol at a point retains the full torus character

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels