ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingjudge 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.

The One-Box Fixed Point for U(1)

Example

Assume full AC. Let $V = W = C$ , let $ζ > 0$ , and let $T = (S^{1})^{2} \times S^{1}$ have rotation characters $t_{1}, t_{2}$ and a framing-circle factor acting on $W$ in its defining representation. Write $h \in U (W) = S^{1}$ for a framing element; its character on $W$ is $e_{1}^{- 1}$ , and $e_{1}$ is the dual character. The action on ADHM data is $(t_{1}, t_{2}, h) \cdot (B_{1}, B_{2}, I, J) = (t_{1} B_{1}, t_{2} B_{2}, I h^{- 1}, t_{1} t_{2} h J) .$ Put $ϵ_{i} = c_{1}^{T} (t_{i})$ and $a_{1} = c_{1}^{T} (e_{1})$ , and work over $K = Q (ϵ_{1}, ϵ_{2}, a_{1})$ with algebraically independent generators. Let $M_{1, 1}^{ζ}$ be the quotient by $G L (V)$ of data satisfying $[B_{1}, B_{2}] + I J = 0$ and $span {B_{1}^{m_{1}} B_{2}^{m_{2}} I (W) : m_{1}, m_{2} \geq 0} = V$ , in the positive stability chamber, and let $q$ be the formal variable in the theorem's fixed-point series.

The datum $B_{1} = B_{2} = J = 0$ , $I = 1$ is stable and satisfies the complex moment-map equation. The full-AC fixed-point theorem classifies the fixed points by one-color partitions; at total size one the unique tuple is $Y_{1} = {(0, 0)}$ . Its tangent character is $t_{1} + t_{2}$ , with additive tangent weights $ϵ_{1}, ϵ_{2}$ . Define its tangent Euler product as the product of those additive weights, so $E_{Y_{1}} = ϵ_{1} ϵ_{2}$ . The theorem's formal series $Z_{1}^{f p} (q)$ has degree- $k$ coefficient $\sum_{∣ Y_{1} ∣ = k} E_{Y_{1}}^{- 1}$ ; hence the degree-one contribution is $(ϵ_{1} ϵ_{2})^{- 1}$ .

Example references

Fixed-point classification and tangent character

Facts & Assumptions

Given: Full AC, $N = k = 1$ , $V = W = C$ , $ζ > 0$ , the stated ADHM equation, stability condition and exact $T$ -action, and algebraically independent formal weights.

[A1]

The Axiom of Choice: full AC is required by the theorem supplier’s stable-quotient and fixed-scheme arguments that yield the fixed-point and tangent results; this example’s finite character calculation uses no additional choices.

[F1]

ADHM linear data, stability, costability and FI levels: for $N = 1$ , the raw ADHM maps, equation, word stability, and dual-framing torus action have the conventions displayed in the Example.

[F2]

ADHM Fixed Points and the Equivariant Instanton Count: under full AC, the stable quotient is smooth, the fixed points are indexed by one-tuples of partitions, their tangent characters are computed from the ADHM deformation complex, and the formal series sums inverse tangent Euler products.

Verification

1.1givenF1algebra

The chosen datum has $I J = 0$ , so the complex moment-map equation holds. Since $I = 1$ maps $W$ onto $V$ , its image already spans $V$ , so the datum is stable. The $G L (V) = C^{\times}$ action removes the nonzero scalar choice of $I$ ; the quotient coordinates are $B_{1}, B_{2}$ .

2.1givenA1F2step 1.1algebra

There is exactly one partition of $k = 1$ for $N = 1$ , namely $Y_{1} = {(0, 0)}$ , so [F2] gives the unique fixed point. Here $V_{\vec{Y}} = e_{1}$ and $W = e_{1}^{- 1}$ , whence $V^{\lor} V = 1$ , $V^{\lor} W^{\lor} = 1$ , and $t_{1} t_{2} W V = t_{1} t_{2}$ . Substitution in the tangent character gives $t_{1} + t_{2}$ ; the two additive weights are $ϵ_{1}, ϵ_{2}$ , so the Euler class is $ϵ_{1} ϵ_{2}$ .

3.1givenA1F2step 2.1algebra∎

The degree-one coefficient is the sum over the single fixed point, so it equals $(ϵ_{1} ϵ_{2})^{- 1}$ . This is a formal equivariant fixed-point contribution, not an asserted continuum path-integral value.

The One-Box Fixed Point for U(1)

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels