ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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 zero-charge framed monad

Example

For $k = 0$ and any $N \geq 1$ , the ADHM vector space is zero, all four maps are the unique zero-dimensional maps, and both quotient presentations are a point. The monad is $0 \to W \otimes O_{P^{2}} \to 0$ ; its middle cohomology is the framed trivial bundle $W \otimes O_{P^{2}}$ and its Hilbert polynomial is $N (\binom{m + 2}{2})$ .

Example references

ADHM linear data, stability, costability and FI levels

Framed P² monads and stable ADHM data

Facts & Assumptions

Given: $N \geq 1$ , $k = 0$ , and the local ADHM definitions.

[F1]

For $V = 0$ , every endomorphism and map to or from $V$ is unique; stability and costability are vacuous, and $G L (0)$ and $U (0)$ are trivial. ADHM linear data, stability, costability and FI levels

[F2]

The ADHM-to-monad construction uses source $V O (- 1)$ , middle $(V \oplus V \oplus W) O$ , target $V O (1)$ , and the displayed Hilbert polynomial. Framed P² monads and stable ADHM data

Proof

Given: The zero vector space $V = C^{0}$ .

1.1F1algebra

There is exactly one linear map between any of the zero spaces, so $B_{1}, B_{2}, I, J$ are all zero maps. The complex and real equations are equalities between zero endomorphisms. The span of the empty family is $0 = V$ , and no nonzero subspace of $V$ exists; hence stability and costability hold. The gauge groups have one element, so each quotient consists of one point.

2.1F2step 1.1algebra∎

Substituting $V = 0$ into the monad leaves $0 \to W O \to 0$ . Its kernel at the right is $W O$ and the image at the left is zero, so the middle cohomology is canonically $W O$ . Restriction to $L_{\infty}$ is the identity framing on the prescribed basis of $W$ . Since $χ (O (m)) = (\binom{m + 2}{2})$ , its Hilbert polynomial is $N (\binom{m + 2}{2})$ , the case $k = 0$ of [F2]. No Euler-block or empty-matrix convention changes these calculations.

The zero-charge framed monad

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels