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.

Stress-tensor Ward identity on the sphere

Statement

For a fixed-coordinate sphere Virasoro block family of def-cft-fixed-coordinate-sphere-virasoro-block-family, the stress-tensor insertion is $G (w) = \sum_{j = 1}^{n} (\frac{h_{j}}{(w - z_{j})^{2}} + \frac{1}{w - z_{j}} \partial_{z_{j}}) F (z) .$ The constant Laurent coefficient of this expression at $w = z_{i}$ , which is the block with $L_{- 2}$ inserted at slot $i$ , equals $\sum_{j \neq i} (\frac{h_{j}}{(z_{i} - z_{j})^{2}} + \frac{1}{z_{i} - z_{j}} \partial_{z_{j}}) F (z) .$

Statement references

Fixed-coordinate sphere Virasoro block family

Facts & Assumptions

Given: Distinct $z_{j}$ , primary weights $h_{j}$ , and a block family with stress insertion $G$ as in Fixed-coordinate sphere Virasoro block family.

[F1]

Fixed-coordinate sphere Virasoro block family supplies functionals $Φ_{z}$ on the algebraic tensor product, with $F (z)$ their evaluation on the chosen primary tuple; the Laurent coefficient of $G_{z}$ at $t_{i}^{0}$ is $Φ_{z} (v_{1} \otimes \dots \otimes L_{- 2} v_{i} \otimes \dots \otimes v_{n})$ . It also supplies the stated primary principal parts, absence of other finite poles, and $O (w^{- 4})$ behavior at infinity.

[F2]

The same definition supplies the three global Ward identities for translations, dilations, and special conformal transformations.

[F3]

Fixed-coordinate sphere Virasoro block family adopts the local Laurent indexing $T (t) = \sum_{m} L_{m} t^{- m - 2}$ for its stress-tensor matrix element. Thus its constant coefficient at $t = 0$ is the block with $L_{- 2}$ inserted, as specified in [F1].

Proof

1.1F1F2algebra

Set $R (w) = \sum_{j} (h_{j} F (w - z_{j})^{- 2} + (\partial_{z_{j}} F) (w - z_{j})^{- 1})$ . By [F1], $G - R$ has no principal part at any finite $z_{j}$ , hence is entire on $C$ . Expanding at infinity gives $R (w) = \frac{\sum_{j} \partial_{z_{j}} F}{w} + \frac{\sum_{j} (z_{j} \partial_{z_{j}} F + h_{j} F)}{w^{2}} + \frac{\sum_{j} (z_{j}^{2} \partial_{z_{j}} F + 2 z_{j} h_{j} F)}{w^{3}} + O (w^{- 4}) .$ Each of the first three coefficients vanishes by [F2], so $R (w) = O (w^{- 4})$ . Also $G (w) = O (w^{- 4})$ by [F1]. Thus the entire function $G - R$ tends to zero at infinity and is identically zero by Liouville's theorem.

2.1F1F3step 1.1algebra∎

Around $w = z_{i}$ , the $j = i$ terms of $R$ have only powers $(w - z_{i})^{- 2}$ and $(w - z_{i})^{- 1}$ , so contribute no constant Laurent coefficient. Every $j \neq i$ term is analytic at $z_{i}$ , with constant coefficient $h_{j} F / (z_{i} - z_{j})^{2} + (\partial_{z_{j}} F) / (z_{i} - z_{j})$ . By [F3], the constant coefficient of $G$ is the $L_{- 2}$ -insertion. Summing these terms proves the second formula.

Stress-tensor Ward identity on the sphere

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources