LemmaStatement: Literature-sourcedProof: AI-generatedprecheck 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.

Refining the largest layout block forces local blockade length at least $ϵ^{- 1}$

Statement

In the setting of the previous lemma, let $A$ be the largest block of the maximal layout. If $A$ is refined by a pure or $x$ -sparse polynomial blockade, then that local blockade has length at least $ϵ^{- 1}$ .

Facts & Assumptions

Given: A maximal layout, its largest block $A$ , and a pure or $x$ -sparse polynomial blockade inside $A$ .

[L1]

The cited source claim proves that substituting a local blockade of length below $ϵ^{- 1}$ into the largest layout block preserves the three defining layout bounds while strictly increasing the number of blocks.

Proof

technique · contradiction by substitution

1.1

Suppose the local blockade inside $A$ had length $k < ϵ^{- 1}$ . By [L1], substituting its pattern for the layout vertex corresponding to $A$ produces another admissible layout with strictly more blocks.

L1assume-contra

2.1

This contradicts the maximal choice of the original layout. Hence the local blockade has length at least $ϵ^{- 1}$ .

step 1.1discharge-contradiction∎

Depends on

Used by

Local pure or x-sparse blockades yield a nice blockade Theorem

Dependency tree · two levels

15 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 6.1.2 (standard reference, not scraped)

Refining the largest layout block forces local blockade length at least ϵ−1ϵ−1