ExampleConstruction: AI-adaptedVerification: 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.

The Fano plane has incidence matrix $N$ with $N N^{T} = 2 I + J$

Example

Order the Fano lines as $L_{0} = {0, 1, 3}, L_{1} = {1, 2, 4}, L_{2} = {2, 3, 5}, L_{3} = {3, 4, 6}, L_{4} = {4, 5, 0}, L_{5} = {5, 6, 1}, L_{6} = {6, 0, 2},$ and order the points as $0, 1, \dots, 6$ . Then the point-block incidence matrix is $N = (\begin{array}{ccccccc} 1 & 0 & 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 \end{array}) .$

Facts & Assumptions

Given: The Fano plane from The seven translates of ${0, 1, 3}$ form the Fano plane.

[L1]

Every two distinct points of the Fano plane lie on exactly one line (The seven translates of ${0, 1, 3}$ form the Fano plane).

Verification

technique · direct

1.1

Each row of $N$ has three ones, because each point lies on three of the seven displayed lines. Hence every diagonal entry of $N N^{T}$ is $3$ .

givenalgebra

1.2

Distinct rows of $N$ have dot product $1$ , because [L1] says that any two distinct points lie on exactly one common line. Hence every off-diagonal entry of $N N^{T}$ is $1$ .

L1algebra

2.1

Therefore $N N^{T}$ has diagonal entries $3$ and off-diagonal entries $1$ , so $N N^{T} = 2 I_{7} + J_{7}$ . In particular this example realizes equality in Fisher's inequality, since the plane has $7$ points and $7$ lines.

step 1.1step 1.2algebra∎

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

L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §4.1 (standard reference, not scraped)

The Fano plane has incidence matrix NN with NNT=2I+JNNT=2I+J