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 loglog bound

Example

For $ℓ (x) = \log_{2} (1 / x)$ on $(0, 1 / 2)$ , the function is subreciprocal. At $x = 1 / 16$ we have $ℓ (x) = 4$ and $\log_{2} ℓ (x) = 2$ , and the loglog density fraction with symbolic constant $C > 0$ is $2^{- 8 C}$ .

Facts & Assumptions

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

[F1]

From Qid logarithmic and constant divisibility: Every nonempty finite graph $H$ is $ℓ$ -divisive for each of $ℓ (x) = \log_{2} (1 / x)$ and $ℓ (x) = 2$ . Both functions are subreciprocal on $(0, 1 / 2)$ .

[F2]

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

Verification

1.1F1given

For $0 < x < 1 / 2$ , reciprocation and the increasing logarithm show $ℓ (x) > 1$ and make $ℓ$ nonincreasing. The bound $\log_{2} y \leq y$ at $y = 1 / x > 2$ implicit in the subreciprocity assertion [F1] gives $ℓ (x) \leq 1 / x$ . Thus all subreciprocal conditions hold, including positivity of $\log_{2} ℓ (x)$ .

2.1F2step 1.1algebra∎

At $x = 1 / 16$ , $1 / x = 16 = 2^{4}$ , so $ℓ (x) = 4 = 2^{2}$ and $\log_{2} ℓ (x) = 2$ . Substituting in [F2] with $C_{H} = C$ gives exponent $C \cdot 4^{2} / 2 = 8 C$ and fraction $2^{- 8 C}$ . The parameter $C$ remains symbolic.

Source notes

Proof/convention locator: Bucic, Nguyen, Scott and Seymour, Induced subgraph density I, 5.1 and 5.2.

Depends on

Used by

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Dependency tree · two levels

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Sources

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