ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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 numeric run of the multiplicative iteration in Lemma 2.9

Example

Take $c = 2^{- 4}, x = 2^{- 8}, b_{1} = 2, b_{2} = 3, b_{3} = 3.$ Then $b_{1} b_{2} = 6 = b_{2} + b_{3}$ , so the iteration lemma predicts an $x$ -sparse induced subgraph of size at least $x^{6} ∣ V (G) ∣ = 2^{- 48} ∣ V (G) ∣$ .

Facts & Assumptions

Given: The numerical parameters displayed above.

[L1]

Under the lemma's hypotheses, the output size is at least $x^{b_{1} b_{2}} ∣ V (G) ∣$ (Iterated sparse restriction reaches the target sparsity threshold).

Verification

technique · arithmetic unpacking

1.1

The equality $b_{1} b_{2} = 2 \cdot 3 = 6$ and $b_{2} + b_{3} = 3 + 3 = 6$ shows that the required inequality $b_{1} b_{2} \geq b_{2} + b_{3}$ is tight in this example.

givenalgebra

2.1

Applying [L1] gives a final induced subgraph of size at least $x^{b_{1} b_{2}} ∣ V (G) ∣ = (2^{- 8})^{6} ∣ V (G) ∣ = 2^{- 48} ∣ V (G) ∣$ . This makes the multiplicative exponent bookkeeping explicit in one concrete case.

step 1.1L1algebra∎

Depends on

Iterated sparse restriction reaches the target sparsity threshold

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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. VI. Bounded VC-dimension, Lemma 3.2 (standard reference, not scraped)