CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

Lower truncation alone does not imply locality

Statement refuted

Every pair of lower-truncated fields on a complex vector space is mutually local.

Facts & Assumptions

Given: The field and locality conventions of Formal distributions and fields on a vector space.

[F1]

Formal distributions and fields on a vector space: $A (z) = \sum_{n} A_{(n)} z^{- n - 1}$ is a field if for every vector $v$ , $A_{(n)} v = 0$ for all sufficiently large $n$ ; locality requires $(z - w)^{N} [A (z), B (w)] = 0$ for some $N \geq 0$ .

Proof

1.1F1givenconstruct

Let $V = C^{2}$ with basis $v_{1}, v_{2}$ . Define constant fields $A (z) = E_{12}$ and $B (w) = E_{21}$ , where $E_{12} v_{2} = v_{1}$ , $E_{12} v_{1} = 0$ , $E_{21} v_{1} = v_{2}$ , and $E_{21} v_{2} = 0$ . Their only nonzero modes are $A_{(- 1)} = E_{12}$ and $B_{(- 1)} = E_{21}$ , so both are fields by [F1].

2.1F1step 1.1algebra∎

Their commutator is $[A (z), B (w)] = E_{11} - E_{22} \neq 0$ . For every integer $N \geq 0$ , $(z - w)^{N} (E_{11} - E_{22})$ has nonzero coefficient $z^{N} (E_{11} - E_{22})$ , including $N = 0$ . Thus no locality exponent exists, although both fields are lower-truncated; the refuted implication fails.

Depends on

Formal distributions and fields on a vector space

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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