ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (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.

Vectorlike charges cancel cubic and linear sums

Example

Let $g = R t$ be the one-dimensional Abelian real Lie algebra and take $V_{+} = V_{-} = C$ . For $q \in R$ , define complex-linear representations by $ρ_{\pm} (z t) = \pm q z {id}_{C}$ for $z \in C$ , and attach the left-handed bookkeeping label to both. This opposite-charge pair is called vectorlike at the representation-data level here. It does not construct spinor fields or a charge-conjugation map; an interpretation as a field pair requires a separately specified field model and convention. The coefficient sums satisfy $q^{3} + (- q)^{3} = 0$ and $q + (- q) = 0$ . With the no-half anticommutator convention ${A, B} = A B + B A$ , the cubic coefficient is $d = 2 (q^{3} + (- q)^{3}) = 0$ , and the separate linear representation coefficient is $ℓ = q + (- q) = 0$ . This verifies cancellation of these two algebraic coefficients in that interface only; it does not identify them with a regulated anomaly, prove an all-orders theorem, or establish a nonperturbative gauge theory or cancellation of global anomalies.

Example references

Four-dimensional chiral anomaly coefficient interface

Facts & Assumptions

Given: The two one-dimensional representations $ρ_{\pm}$ of $g = R t$ and their left-handed bookkeeping labels from the Example.

[A1]

The representation matrices for $t$ are the scalars $T_{+} = q$ and $T_{-} = - q$ , and both species are included once in the finite coefficient sum.

[F1]

Four-dimensional chiral anomaly coefficient interface: with ${A, B} = A B + B A$ , $d^{a b c} = \sum_{i} {tr}_{V_{i}} (T_{R_{i}}^{a} {T_{R_{i}}^{b}, T_{R_{i}}^{c}})$ ; the separate linear representation coefficient is $ℓ^{a} = \sum_{i} {tr}_{V_{i}} (T_{R_{i}}^{a})$ . These coefficient definitions do not construct a field model or assert an anomaly/descent identification.

Verification

Given: [A1] and [F1].

1.1givenA1F1algebra

In one dimension the trace of a scalar is that scalar, and ${q, q} = 2 q^{2}$ , ${- q, - q} = 2 q^{2}$ . Therefore the two contributions to the cubic coefficient sum to $q (2 q^{2}) + (- q) (2 q^{2}) = 2 (q^{3} + (- q)^{3}) = 0$ .

2.1givenA1F1step 1.1algebra∎

The linear representation coefficient is $q + (- q) = 0$ . Thus both the cubic coefficient and the separately defined linear coefficient vanish by direct arithmetic. This calculation asserts no descent identification, loop theorem, global-anomaly conclusion, or nonperturbative existence claim.

Depends on

Four-dimensional chiral anomaly coefficient interface