ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pending

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.

Compactly supported background with a nonzero heavy-fermion anomaly variation

Physical scope

A fixed smooth compactly supported background-field calculation inside the perturbative local U(1)_V threshold model; it is illustrative and makes no empirical or global-anomaly claim.

Example

In the trivial-bundle $H = S U (3) \times U (1)_{V}$ threshold model on natural-unit Minkowski space $M = R^{1, 3}$ with $η = diag (- 1, 1, 1, 1)$ , orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ , and zero $S U (3)$ background, use a smooth real compactly supported $U (1)_{V}$ potential $B$ and parameter $α$ , with $G = d B$ and $B \mapsto B + d α$ , $θ \mapsto θ - α$ . Choose a smooth compactly supported cutoff $χ$ that equals $1$ on a neighborhood of $supp α$ , with $α \geq 0$ . Let $λ > 0$ have mass dimension two and set $B = λ χ (x) (x^{0} d x^{1} + x^{2} d x^{3})$ , normalizing $α$ so that $λ^{2} \int_{M} α d^{4} x = 1$ . For the charge-one positive-chirality field, neutral mass partner, and $M_{H} > 0$ of the stated formal model, this evaluates the exact local identity from the preceding theorem. It makes no assertion about a finite-momentum vertex or the Fourier support of the chosen compactly supported backgrounds. The consistent heavy-sector and Wess–Zumino variations are equal, nonzero, and each equal $+ 1 / (12 π^{2})$ .

Example references

Conditional local threshold-matching theorem and anomaly normalization

Facts & Assumptions

Given: The theorem's charge, transformation, and consistent-anomaly convention, together with the smooth compactly supported cutoff and normalized nonnegative test parameter.

[F1]

Heavy-chiral threshold Wess–Zumino anomaly match: the heavy-sector consistent variation and Wess–Zumino variation both equal $+ \frac{1}{24 π^{2}} \int_{M} α G \land G$ .

[A1]

The support of $α$ lies where $χ = 1$ , $α \geq 0$ , and $λ^{2} \int_{M} α d^{4} x = 1$ .

Verification

Given: [F1], [A1], and the orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ .

1.1givenA1constructalgebra

On a neighborhood of $supp α$ , $B = λ (x^{0} d x^{1} + x^{2} d x^{3})$ , so $G = d B = λ (d x^{0} \land d x^{1} + d x^{2} \land d x^{3})$ . The cutoff makes $B$ globally smooth and compactly supported.

2.1step 1.1algebra

The two summands of $G$ have degree two, so they commute under the wedge product; each self-wedge is zero. Hence $G \land G = 2 λ^{2} d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ on $supp α$ .

3.1F1A1step 2.1algebra∎

Integrating against $α$ and using [A1] gives $\int_{M} α G \land G = 2 λ^{2} \int_{M} α d^{4} x = 2$ . Therefore both variations in [F1] equal $+ 2 / (24 π^{2}) = + 1 / (12 π^{2})$ .

Depends on

Heavy-chiral threshold Wess–Zumino anomaly match