ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

A numerical square-root rescaling identity

Example

As a standalone numerical illustration, take

c_{4} : = 4, d : = 232 = 58 c_{4}, ℓ : = 256, k : = \sqrt{ℓ} = 16,

and assume $∣ F ∣ \geq ∣ G ∣$ .

Facts & Assumptions

Given: The numerical choices in the Example.

[A1]

The sample values satisfy the numerical relation $d = 58 c_{4}$ . They are not asserted to be the existential constants supplied by the source lemma.

Verification

technique · direct arithmetic

1.1

Since $k = \sqrt{ℓ}$ , one has $ℓ^{29 c_{4}} = (k^{2})^{29 c_{4}} = k^{58 c_{4}} = k^{d} .$

A1algebra

2.1

Therefore $\frac{∣ F ∣}{ℓ^{29 c_{4}}} = \frac{∣ F ∣}{k^{d}} \geq \frac{∣ G ∣}{k^{d}},$ using the assumption $∣ F ∣ \geq ∣ G ∣$ .

step 1.1algebra

3.1

This standalone calculation isolates the square-root renormalization: after replacing $ℓ$ by $k = \sqrt{ℓ}$ , the width bound takes exactly the target form $∣ G ∣ / k^{d}$ when the numerical relation $d = 58 c_{4}$ holds.

step 2.1∎

Depends on

Rödl initialization removes the constant-scale restriction in the property (*) four-outcome theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Lemma 4.4 (standard reference, not scraped)