RemarkRemark: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (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.

Formal and affine vertex-algebra input boundaries

Remark

Use the ordinary, non-super complex convention in which a formal distribution is a coefficient family and a field additionally satisfies vectorwise lower truncation. An ordinary vertex algebra uses such fields in a state–field map with a nonzero vacuum, translation operator, vacuum and creation, translation covariance, and locality. Those algebraic axioms alone impose no specified Virasoro vector or associated weight decomposition. They require neither that the only subspaces stable under translation and all state modes are zero and the whole space, nor a positive-definite conjugate-symmetric sesquilinear form with any prescribed compatibility with the modes. They assert no analytic convergence of operator products.

For this workspace, “conformal vertex algebra of CFT type” means the selected stronger convention: a specified Virasoro vector with $L (- 1) = T$ , diagonalizable $L (0)$ , a nonnegative integer grading with finite-dimensional weight spaces, and $V_{0} = C 1$ . A vector-space grading that may accompany an algebraic construction does not supply this specified conformal vector or the CFT-type vacuum-space condition.

For a finite-dimensional complex simple Lie algebra $g$ , the normalized invariant form $B$ used in the affine central extension, and each $k \in C$ , the universal affine theorem gives a unique ordinary algebraic vertex-algebra structure on the induced affine vacuum module $V^{k} (g)$ with the prescribed current fields. Fact [F4] records the theorem's exact hypotheses and conclusion; the supplier's proof works at every complex level and uses no division by a level-dependent scalar. It does not choose the conformal vector or grading data specified in [F3], make any further assertion about the module's affine action, or select a quotient and assert properties of that quotient. It neither supplies a conjugate-symmetric sesquilinear form $⟨ \cdot, \cdot ⟩$ on $V^{k} (g)$ with $⟨ v, v ⟩ > 0$ for nonzero $v$ , nor asserts that the affine action preserves such a form. The formal identities in [F1]–[F2] assert no analytic convergence of operator products or correlation functions. These algebraic results therefore do not establish existence of an analytic or physical CFT.

Remark references

Christophe Nozaradan, Introduction to Vertex Algebras, §§1.3–1.4 and 4.2–4.3

Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras, §§2.2–2.3, 2.5, 2.7, and §3.1, Proposition 3.1

Facts & Assumptions

Given: Work over $C$ in the ordinary non-super formal convention. Use the current pinned definitions of formal distributions and vertex algebras, the selected CFT-type convention, and the published universal affine theorem with exactly its stated Lie-algebra, form, and level hypotheses.

[F1]

Formal distributions and fields on a vector space: a formal distribution is an arbitrary coefficient family; a field adds vectorwise lower truncation, and locality and products use formal coefficient and residue conventions without an analytic convergence assertion.

[F2]

Vertex algebra in the state–field convention: a vertex algebra consists of the ordinary state–field data and lower-truncated fields satisfying vacuum and creation, translation covariance, and locality; these axioms choose no Virasoro vector or associated weight decomposition and introduce no positive-definite sesquilinear form or analytic-convergence assertion.

[F3]

Conformal vertex algebra of CFT type: the selected CFT-type convention requires a specified Virasoro vector, $L (- 1) = T$ , diagonalizable $L (0)$ , a nonnegative integer grading with finite-dimensional weight spaces, and $V_{0} = C 1$ .

[F4]

Universal affine vertex algebras exist at every complex level: for finite-dimensional complex simple $g$ , its normalized invariant form, and every $k \in C$ , the induced affine vacuum module has a unique vertex-algebra structure with the prescribed current fields; its level hypothesis is exactly $k \in C$ , with no excluded value. Its conclusion supplies the state–field structure on that induced module: it does not choose the Virasoro vector or grading in [F3], select a quotient of the module, impose an additional condition on its affine-mode action, or supply a positive-definite sesquilinear form as described in the Remark.

Verification

1.1givenF1F2algebra

By [F1], a formal distribution need not be a field: the latter must be lower-truncated on each vector. By [F2], an ordinary vertex algebra uses fields and adds the vacuum, translation, state–field, creation, covariance, and locality data. Thus the formal coefficient-family convention, the field condition, and the ordinary vertex-algebra structure are distinct algebraic interfaces.

2.1F2F3step 1.1algebra

The ordinary axioms in [F2] do not supply a conformal vector or grading. The selected convention in [F3] separately requires the Virasoro vector, its translation-mode condition, the stated grading data, and $V_{0} = C 1$ . Hence a separate vector-space grading, if present, does not by itself meet [F3].

3.1F1F2F3F4step 1.1step 2.1algebra∎

The exact hypotheses and conclusion of [F4] give the ordinary algebraic vertex-algebra structure on $V^{k} (g)$ for every $k \in C$ , without excluding any complex level. This selects neither the additional Virasoro vector and weight data in [F3] nor a quotient of $V^{k} (g)$ , and it introduces no additional affine-action condition or positive-definite sesquilinear form. The formal scope in [F1]–[F2] supplies no analytic convergence or positivity premise. Thus these inputs support the stated algebraic conclusions only, not analytic correlation-function convergence or physical CFT existence.

Formal and affine vertex-algebra input boundaries

Remark

Remark references

Facts & Assumptions

Verification

Depends on

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Sources