LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

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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.

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A local counterterm changes only the representative of a local anomaly class

Statement

Let $A_{U}^{g, p}$ be the finite-jet horizontal-form spaces on a smooth open set $U \subseteq R^{4}$ , with nilpotent BRST differential $s$ , horizontal differential $d_{H}$ , and $s d_{H} + d_{H} s = 0$ . In bidegree $(1, 4)$ , define $H^{1, 4} (s ∣ d_{H})$ as the ghost-number-one four-forms $a_{4}$ satisfying $s a_{4} + d_{H} b_{3} = 0$ for some $b_{3}$ , modulo the subspace ${s C_{4} + d_{H} r_{3} : C_{4} \in A_{U}^{0, 4}, r_{3} \in A_{U}^{1, 3}}$ . If a local counterterm changes an anomaly representative from $a_{4}$ to $a_{4}^{'} = a_{4} + s C_{4} + d_{H} r_{3}$ , then $[a_{4}^{'}] = [a_{4}]$ in $H^{1, 4} (s ∣ d_{H})$ .

Statement references

Bilal, §§9.2 and 9.4

Alvarez, §2.4

Facts & Assumptions

Given: The local BRST complex, the relative cohomology quotient, and a counterterm change $a_{4}^{'} - a_{4} = s C_{4} + d_{H} r_{3}$ .

[F1]

Local BRST Cohomology Modulo Total Derivatives: the denominator of $H^{1, 4} (s ∣ d_{H})$ is exactly ${s C_{4} + d_{H} r_{3} : C_{4} \in A_{U}^{0, 4}, r_{3} \in A_{U}^{1, 3}}$ .

Proof

1.1givenF1algebra

By the assumed transformation rule, $a_{4}^{'} - a_{4} = s C_{4} + d_{H} r_{3}$ , which is an element of the denominator subspace in [F1].

2.1F1step 1.1algebra∎

Two cocycles whose difference belongs to the denominator define the same quotient class. Therefore $[a_{4}^{'}] = [a_{4}]$ in the stated local BRST cohomology.

Depends on

Local BRST Cohomology Modulo Total Derivatives