ExampleConstruction: AI-generatedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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.

$W_{1, 2} (4)$ written out, and its rank computed

Example

Ordering the rows by the points $0, 1, 2, 3$ and the columns by the pairs $01, 02, 03, 12, 13, 23$ , one has

$W_{1, 2} (4) = (\begin{array}{cccccc} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{array}) .$

Facts & Assumptions

Given: the matrix above.

[L1]

The point-inclusion matrix has rank $4$ ( ${rank}_{Q} W_{1, k} (n) = n$ for $1 \leq k \leq n - 1$ ).

Verification

technique · direct

1.1

Row operations reduce the matrix to a row-echelon form with four pivot rows, so its rank is $4$ .

given

2.1

This agrees with [L1].

L1step 1.1

3.1

Direct multiplication also gives $W_{0, 1} (4) W_{1, 2} (4) = 2 W_{0, 2} (4),$ since every pair contains exactly two points.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §7.1 (standard reference, not scraped)

W1,2(4)W1,2​(4) written out, and its rank computed