ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Integral geometric layers for fourteen ordered blocks

Example

For $ℓ = 4$ and $t = 14$ , the integral cutoffs are $m_{1} = 2, m_{2} = 4, m_{3} = 8, m_{4} = 14.$ Thus the four layers have respectively $2, 2, 4, 6$ blocks. The final layer is truncated at the integral endpoint $14$ , rather than referring to a nonintegral block index.

Facts & Assumptions

Given: A decreasing ordered partition of $14$ blocks and $ℓ = 4$ .

[F1]

The cutoff $m_{r}$ is the largest integer at most both $t$ and $ℓ^{r / 2}$ , and layers are successive cutoff differences (Integral geometric layers of a decreasing block partition).

[F2]

These cutoffs produce nonempty layers covering all blocks (Integral geometric layers exist, cover the partition, and retain the required cutoff bounds).

Verification

technique · direct calculation

1.1

The bounds $4^{1 / 2}, 4^{2 / 2}, 4^{3 / 2}, 4^{4 / 2}$ are $2, 4, 8, 16$ ; intersecting their allowed integer indices with $[14]$ gives the stated cutoffs $2, 4, 8, 14$ by [F1].

F1algebra

2.1

Successive differences are $2$ , $4 - 2 = 2$ , $8 - 4 = 4$ , and $14 - 8 = 6$ . Their sum is $14$ , agreeing with the coverage conclusion in [F2].

F2step 1.1algebra

3.1

In particular, the last cutoff is the integer $14$ , so no expression such as a fifteenth or nonintegrally numbered block has been used.

F1step 1.1∎

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

Huang, Ju, and Zhou, Erdős–Hajnal beyond the five-vertex path, geometric layers in Lemma 5.1 (standard reference, not scraped)