ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

Checking the subreciprocal condition for the quadratic log bound

Example

For $ℓ (x) = 2$ on $(0, 1 / 2)$ , the subreciprocal inequalities are $1 < 2 < 1 / x$ and $\log_{2} ℓ (x) = 1$ . At $x = 1 / 16$ the classical density fraction with symbolic constant $C > 0$ is $2^{- 16 C}$ .

Facts & Assumptions

Given: $ℓ (x) = 2$ , $0 < x < 1 / 2$ , and $C > 0$ symbolic.

[F1]

From Subreciprocal function and ell divisibility: A function $ℓ : (0, 1 / 2) \to (0, \infty)$ is subreciprocal when it is nonincreasing and satisfies $1 < ℓ (x) \leq 1 / x$ throughout its domain.

[F2]

From Fox sudakov quantitative induced density bound: $δ = 2^{- C_{H} (\log_{2} (1 / x))^{2}}$

Verification

1.1F1given

The constant function is positive and nonincreasing. For $0 < x < 1 / 2$ , reciprocation gives $1 / x > 2 > 1$ , so it meets [F1]. Also $2^{1} = 2$ , hence $\log_{2} ℓ (x) = 1$ .

2.1F2step 1.1algebra∎

The value $1 / 16$ lies in that interval, and $\log_{2} (1 / (1 / 16)) = \log_{2} 16 = 4$ . Substitution into [F2] with $C_{H} = C$ yields exponent $C \cdot 4^{2} = 16 C$ and fraction $2^{- 16 C}$ . Here $C$ is a symbolic admissible constant, not a numerically certified constant for arbitrary $H$ .

Source notes

Proof/convention locator: Bucic, Nguyen, Scott and Seymour, Induced subgraph density I, Section 5, ell=2.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Bucic, Nguyen, Scott and Seymour, Induced subgraph density I (standard reference, not scraped)