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.

The epsilon^(5d) substitution in Claim 4.5.1 and Lemma 4.5

Example

Take

ϵ : = \frac{1}{4}, c_{4}^{'} : = 4, d : = 232 = 58 c_{4}^{'}, x : = ϵ^{5 d} .

Then the four exponent comparisons in the final property-(*) reduction become explicit.

Facts & Assumptions

Given: The displayed values of $ϵ$ , $d$ , $c_{4}^{'}$ , and $x$ .

[A1]

Since $ϵ = \frac{1}{4} < \frac{1}{2}$ , the powers of $ϵ$ decrease as their exponents increase.

Verification

technique · direct arithmetic

1.1

The defining substitution gives $x = ϵ^{5 d} = 2^{- 10 d} < 2^{- d} .$ Together with $d = 58 c_{4}^{'}$ , this verifies the two parameter inequalities required before applying the Rödl-initialized theorem; its separate graph-order hypothesis must also be checked in any application.

A1algebra

1.2

For the restricted-set branch, $x^{23 c_{4}^{'}} ϵ^{d} = ϵ^{115 c_{4}^{'} d + d} \geq ϵ^{116 c_{4}^{'} d},$ because $115 c_{4}^{'} + 1 \leq 116 c_{4}^{'}$ when $c_{4}^{'} = 4$ .

A1algebra

1.3

For the clique-or-stable-set branch, $x^{31 c_{4}^{'}} ϵ^{d} = ϵ^{155 c_{4}^{'} d + d} \geq ϵ^{156 c_{4}^{'} d},$ because $155 c_{4}^{'} + 1 \leq 156 c_{4}^{'}$ .

A1algebra

2.1

If $k \geq x^{- 1}$ , then $ϵ^{d} = x^{1 / 5} \geq k^{- 1 / 5},$ so $\frac{ϵ^{d}}{k^{d}} \geq \frac{1}{k^{2 d}} .$ This is exactly the comparison used to turn the complete-or-anticomplete blockade branch into the generalized-nice blockade outcome.

step 1.1A1algebra

3.1

These computations are the concrete numerical version of the four exponent transfers behind the local blockade claim and the final proof of generalized niceness.

step 1.2step 1.3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Claim 4.5.1 and Lemma 4.5 (standard reference, not scraped)