ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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 symmetric local-lemma condition for a sparse uniform hypergraph

Example

For any natural $q$ , form a $5$ -uniform hypergraph from $q$ disjoint blocks, each block consisting of two edges that meet in one vertex and are otherwise disjoint. Every edge meets exactly one other edge. The Local Lemma proves the hypergraph two-colourable for every $q$ , while the first-moment edge-count criterion applies only when $2 q < 16$ .

Facts & Assumptions

Given: The hypergraph construction in the Example.

[L1]

Fewer than $2^{k - 1}$ edges is sufficient for first-moment two-colourability (Every $k$ -uniform hypergraph with fewer than $2^{k - 1}$ edges is $2$ -colourable).

[L2]

The Local-Lemma criterion is $e (d + 1) \leq 2^{k - 1}$ when every edge meets at most $d$ other edges (A $k$ -uniform hypergraph is $2$ -colourable when every edge meets at most $d$ other edges and $e (d + 1) \leq 2^{k - 1}$ ).

[L3]

$e = \sum_{j \geq 0} 1 / j!$ , and at $x = 1, N = 2$ the exponential tail from $j = 3$ onward is at most $1 / 3$ (The real exponential function and the number $e$ by a power series, A geometric bound for tails of the exponential series).

Verification

technique · direct

1.1

Every edge has five vertices and meets only its block partner, so $k = 5$ and $d = 1$ .

given

2.1

By [L3], $e \leq 1 + 1 + 1 / 2 + 1 / 3 = 17 / 6 < 3$ , so $2 e < 6 < 16$ . Hence [L2] gives a proper two-colouring for every $q$ , including $q = 0$ .

step 1.1L2L3algebra

3.1

The hypergraph has $2 q$ edges, so [L1] applies only when $2 q < 2^{4} = 16$ . For $q \geq 8$ , the Local Lemma still applies while this first-moment criterion does not.

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 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

J. Matousek and J. Vondrak, The Probabilistic Method, Sections 5.1-5.2 (standard reference, not scraped)