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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 Two U(2) Fixed Points at Charge One

Example

Assume full AC. Let $T = (S^{1})^{2} \times T_{G}$ , with $T_{G} = (S^{1})^{2}$ acting diagonally on $W = C^{2}$ . Denote its two representation characters on $W$ by $e_{1}^{- 1}, e_{2}^{- 1}$ , and let $e_{1}, e_{2}$ be their dual characters, so $W = e_{1}^{- 1} + e_{2}^{- 1}$ . For a framing element $h \in T_{G} \subset U (W)$ , 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 $a_{α} = c_{1}^{T} (e_{α})$ , and work over $K = Q (ϵ_{1}, ϵ_{2}, a_{1}, a_{2})$ with algebraically independent generators. Let $M_{1, 2}^{ζ}$ be the quotient of $V = C$ , $W = C^{2}$ data $(B_{1}, B_{2}, I, J)$ satisfying $[B_{1}, B_{2}] + I J = 0$ and $span {B_{1}^{m_{1}} B_{2}^{m_{2}} I (W) : m_{1}, m_{2} \geq 0} = V$ , modulo $G L (V)$ , in the positive stability chamber, and let $q$ be the formal variable in the fixed-point series.

The two fixed points are $({(0, 0)}, \emptyset)$ and $(\emptyset, {(0, 0)})$ . Their tangent characters are respectively $t_{1} + t_{2} + \frac{e_{2}}{e_{1}} + t_{1} t_{2} \frac{e_{1}}{e_{2}} and t_{1} + t_{2} + \frac{e_{1}}{e_{2}} + t_{1} t_{2} \frac{e_{2}}{e_{1}} .$ For each isolated fixed point, define its tangent Euler product as the product, with multiplicity, of the additive equivariant weights in its tangent representation. The products are $E_{1} = ϵ_{1} ϵ_{2} (a_{2} - a_{1}) (ϵ_{1} + ϵ_{2} + a_{1} - a_{2}), E_{2} = ϵ_{1} ϵ_{2} (a_{1} - a_{2}) (ϵ_{1} + ϵ_{2} + a_{2} - a_{1}) .$ The theorem defines $Z_{2}^{f p} (q) = \sum_{k \geq 0} q^{k} \sum_{∣ \vec{Y} ∣ = k} E_{\vec{Y}}^{- 1}$ using these tangent Euler products. At total size one, its coefficient is $E_{1}^{- 1} + E_{2}^{- 1}$ .

The diagram places the one box in either framing color. Each square is a one-dimensional weight space of $V$ , and the $Y_{α}$ labels identify the framing color.

Example references

Fixed-point classification and tangent character

Facts & Assumptions

Given: Full AC, $N = 2$ , $k = 1$ , $V = C$ , $W = C^{2}$ , $ζ > 0$ , the displayed action and algebraically independent formal torus weights.

[A1]

The Axiom of Choice: full AC is assumed and used in the theorem supplier’s stable-quotient and fixed-scheme arguments that yield the fixed-point and tangent results; no logical necessity or optimality of this axiom is asserted; this example’s finite character calculation uses no additional choices.

[F1]

ADHM linear data, stability, costability and FI levels: for $N = 2$ , 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, fixed points are indexed by two-tuples of partitions of total size one, and their tangent character is $V^{\lor} V (t_{1} + t_{2} - 1 - t_{1} t_{2}) + V^{\lor} W^{\lor} + t_{1} t_{2} W V$ . The degree-wise formal coefficient sums the inverse tangent Euler products.

Verification

1.1givenA1F2algebra

A two-tuple of partitions with total size one places its single box in exactly one of the two colors, so the only possibilities are $({(0, 0)}, \emptyset)$ and $(\emptyset, {(0, 0)})$ . Their quotient representatives have $V$ -characters $e_{1}$ and $e_{2}$ , respectively.

2.1givenA1F1F2step 1.1algebra

For the first point, the ADHM convention [F1] gives $W = e_{1}^{- 1} + e_{2}^{- 1}$ and $W^{\lor} = e_{1} + e_{2}$ , while $V = e_{1}$ . Substitution in [F2] gives $t_{1} + t_{2} + \frac{e_{2}}{e_{1}} + t_{1} t_{2} \frac{e_{1}}{e_{2}} .$ The corresponding additive weights are $ϵ_{1}, ϵ_{2}, a_{2} - a_{1}, ϵ_{1} + ϵ_{2} + a_{1} - a_{2}$ , so their product is $E_{1}$ . Interchanging the two colors gives the second displayed character and $E_{2}$ .

3.1givenA1F2step 2.1algebra∎

Each fixed-point contribution is the reciprocal of its Euler product, so the degree-one coefficient is $E_{1}^{- 1} + E_{2}^{- 1}$ . The TikZ drawing only records the two one-box colorings and is not used in this calculation.

The Two U(2) Fixed Points at Charge One

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels