PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Wess–Zumino consistency is an obstruction class

Statement

Assume AC and Countable Choice, fix admissible flat scalar free data and a local scalar polynomial class under a scalar formal time-ordering theorem, and work conditionally in a time-ordering scheme satisfying that theorem's statement. At order $n$ , let $K_{n}$ be an $s, d_{H}$ -stable subcomplex of the compactly supported local-form spaces defined earlier, with $K_{n}^{0, 4}$ exactly the allowed local counterterm densities. For each ghost number $r$ , put $F_{n}^{r} : = K_{n}^{r, 4} / d_{H} K_{n}^{r, 3},$ the local-functional quotient formed using this same subcomplex, and let $F_{n}^{∙} ⟦ ℏ ⟧$ denote formal power series. Let $W_{ℏ} = s + \sum_{j \geq 1} ℏ^{j} W_{j}$ be an odd formal operator of ghost number one on this module, with $W_{ℏ}^{2} = 0$ and leading coefficient the induced $s$ . Let $X \in F_{n}^{0} ⟦ ℏ ⟧$ and suppose its first possible breaking is $W_{ℏ} X = ℏ^{n} A + O (ℏ^{n + 1}), n \geq 1, A = [a_{4}]_{F_{n}},$ where $ℏ$ is a formal indeterminate, $O (ℏ^{n + 1})$ means terms of formal degree at least $n + 1$ , every lower coefficient vanishes, and $a_{4} \in K_{n}^{1, 4}$ is a compactly supported local four-form of ghost number one. Then $s a_{4}$ is $d_{H}$ -exact within $K_{n}$ . There exists an admissible counterterm $b_{4} \in K_{n}^{0, 4}$ such that the shift $X \mapsto X - ℏ^{n} [b_{4}]_{F_{n}}$ cancels the order- $n$ breaking in $F_{n}^{1}$ if and only if $[a_{4}]_{s ∣ d_{H}} = 0 in H^{1, 4} (s ∣ d_{H}; K_{n}) .$ This is a conditional obstruction statement. Locality of $a_{4}$ , nilpotence of the formal Ward differential, and admissibility of the counterterm space are assumptions; the result does not establish a quantum action principle.

Statement references

Relative local BRST cohomology

Nilpotence and contraction of nonminimal doublets

Scalar Epstein–Glaser induction at finite perturbative order

The Axiom of Choice

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: The relative local-form conventions, the Abelian nilpotent differential, and the conditional scalar time-ordering result.

[F1]

Nilpotence and contraction of nonminimal doublets: the Abelian local differential $s$ is nilpotent.

[F2]

Relative local BRST cohomology: a ghost-number-one four-form $a_{4}$ is a relative cocycle when $s a_{4}$ is $d_{H}$ -exact; its class vanishes exactly when $a_{4} = s b_{4} + d_{H} c_{3}$ in the chosen local subcomplex. For compact support, $\int d_{H} c_{3} = 0$ .

[F3]

Scalar Epstein–Glaser induction at finite perturbative order: assuming the Axiom of Choice and Countable Choice and fixing any admissible flat scalar free datum and local scalar polynomial class, there exists a sequence of symmetric multilinear maps $T_{n}$ satisfying the scalar time-ordering contract at every finite order, with finite graph scaling bounds and the stated local extension ambiguities. This supplier does not establish a BRST Ward identity, locality of an anomaly, a quantum action principle, or a positive physical Hilbert quotient.

[A3]

The Axiom of Choice: every family of nonempty sets has a choice function. This is a hypothesis of [F3] and is carried here only through that conditional scalar-system supplier.

[A4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function. This is a hypothesis of [F3] and is carried here only through that conditional scalar-system supplier.

[A1]

On $F_{n}^{∙} ⟦ ℏ ⟧$ , $W_{ℏ} = s + O (ℏ)$ is an odd formal ghost-number-one operator with $W_{ℏ}^{2} = 0$ , and $W_{ℏ} X = ℏ^{n} A + O (ℏ^{n + 1})$ with all lower coefficients zero.

[A2]

The order- $n$ anomaly density $a_{4}$ is local and compactly supported; the chosen $s, d_{H}$ -stable subcomplex $K_{n}$ contains it, and every element of $K_{n}^{0, 4}$ is an admissible counterterm for this formal scheme.

Proof

Given: [F1], [F2], [F3], [A1], [A2], [A3] and [A4].

1.1givenF3A1A2A3A4

Under the hypotheses of [F3], including [A3] and [A4], the scalar time-ordering input exists as a formal coefficientwise scheme. Fix such a scheme as background. The additional operator $W_{ℏ}$ , its nilpotence, and the locality premise in [A1]–[A2] are separate hypotheses not supplied by [F3].

2.1A1step 1.1algebra

Apply $W_{ℏ}$ to the expansion in [A1]. Since $W_{ℏ}^{2} X = 0$ , its order- $ℏ^{n}$ coefficient is the leading operator $s$ applied to $A$ ; all terms from the positive powers of $ℏ$ in $W_{ℏ}$ have order at least $n + 1$ . Hence $s A = 0$ in $F_{n}^{2}$ .

3.1F1F2A2step 2.1

Since $A = [a_{4}]_{F_{n}}$ and $s A = 0$ in $F_{n}^{2}$ , the quotient definition in [F2] gives $s a_{4} = d_{H} c_{3}$ for some $c_{3} \in K_{n}^{2, 3}$ , after absorbing a sign into $c_{3}$ . Thus $a_{4}$ is a cocycle in the specified relative local cohomology.

4.1F2A2step 3.1

If $[a_{4}]_{s ∣ d_{H}} = 0$ , the quotient definition in [F2], restricted to $K_{n}$ , supplies $b_{4} \in K_{n}^{0, 4}$ and $c_{3} \in K_{n}^{1, 3}$ such that $a_{4} = s b_{4} + d_{H} c_{3}$ . This uses the cocycle conclusion of step 3.1 so the relative class is defined.

5.1F2A1A2step 4.1algebra

The counterterm class $[b_{4}]_{F_{n}}$ is admissible by [A2]. Replacing $X$ by $X - ℏ^{n} [b_{4}]_{F_{n}}$ changes the order- $n$ breaking by $- s [b_{4}]_{F_{n}}$ , because the leading term of $W_{ℏ}$ is $s$ . Step 4.1 gives $[a_{4}]_{F_{n}} = s [b_{4}]_{F_{n}}$ , since $d_{H} c_{3}$ is zero in that quotient, so the breaking is canceled.

6.1F2A1A2step 3.1algebra∎

Conversely, if an admissible local counterterm class $[b_{4}]_{F_{n}}$ cancels the order- $n$ breaking in $F_{n}^{1}$ , then $[a_{4}]_{F_{n}} - s [b_{4}]_{F_{n}} = 0$ . By the quotient definition in [F2], $a_{4} - s b_{4} = d_{H} c_{3}$ for some $c_{3} \in K_{n}^{1, 3}$ . This is exactly the relative coboundary condition; step 3.1 ensures that $[a_{4}]_{s ∣ d_{H}}$ is the class under discussion.

Wess–Zumino consistency is an obstruction class

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources