LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (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.

Bundle-gerbe three-curvature descends and is closed

Statement

For a bundle gerbe with product-compatible connection and curving $B$ satisfying ${pr}_{2}^{*} B - {pr}_{1}^{*} B = F_{\nabla} / (2 π i)$ , the form $d B$ is the pullback of a unique closed three-form $H$ on the base.

Statement references

U(1)-bundle gerbe with connection and curving

Basic forms descend along surjective submersions

Facts & Assumptions

Given: A surjective submersion $π : Y \to M$ , a bundle-gerbe connection $\nabla$ preserving the product, and a curving $B$ with the stated normalized curvature equation.

[F1]

U(1)-bundle gerbe with connection and curving: on $Y^{[2]}$ , $δ B = {pr}_{2}^{*} B - {pr}_{1}^{*} B = F_{\nabla} / (2 π i)$ ; the product connection makes curvature additive on $Y^{[3]}$ . For a unitary line connection, a local unit frame has connection form $A \in i Ω^{1}$ and curvature $F_{\nabla} = d A$ .

[F2]

Basic forms descend along surjective submersions: an equalizer form on $Y \times_{M} Y$ descends uniquely to $M$ .

Proof

1.1F1givenalgebra

On $Y^{[3]}$ , compatibility of the connection with the gerbe product gives ${pr}_{13}^{*} F_{\nabla} = {pr}_{12}^{*} F_{\nabla} + {pr}_{23}^{*} F_{\nabla}$ , so $δ F_{\nabla} = 0$ . In each local unit frame of the line bundle, $F_{\nabla} = d A$ by [F1], hence $d F_{\nabla} = d^{2} A = 0$ ; these local equalities agree on overlaps because curvature is globally defined. Therefore $d (δ B) = d F_{\nabla} / (2 π i) = 0$ . Since pullback commutes with $d$ , $δ (d B) = d (δ B) = 0$ , so the two pullbacks of $d B$ to $Y^{[2]}$ agree.

2.1F2step 1.1algebra∎

Apply [F2] to $α = d B$ to obtain a unique $H \in Ω^{3} (M)$ with $π^{*} H = d B$ . Then $π^{*} (d H) = d (π^{*} H) = d^{2} B = 0$ . Pullback by a surjective submersion is injective on forms because local sections recover the form, so $d H = 0$ .

Bundle-gerbe three-curvature descends and is closed

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources