LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 minimal sparsity parameter drops below the target

Statement

Let $x \in (0, \frac{1}{2})$ . In the minimal-threshold setup of the source, the least sparsity parameter $y$ for which a linear-sized $y$ -sparse induced subgraph exists satisfies $y < x$ .

Facts & Assumptions

Given: A parameter $x \in (0, \frac{1}{2})$ and a counterexample graph $G$ in which $y$ is minimal with the stated property.

[L1]

Claim 7.3.1 of the cited source proves exactly that the minimal parameter in its fully quantified threshold setup satisfies $y < x$ .

Proof

technique · direct translation

1.1

The cited source claim applies the exact sparsity, size, and exponent bounds to the minimal witness and excludes its blockade outcome by the enclosing counterexample hypothesis.

L1given

2.1

The remaining outcome contradicts minimality unless $y < x$ , which proves the statement.

step 1.1L1∎

Depends on

Used by

P₅-free graphs yield a polynomial restricted set or a complete or anticomplete blockade Theorem

Dependency tree · two levels

10 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

Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. VII. The five-vertex path, Claim 7.3.1 (standard reference, not scraped)