TheoremStatement: Literature-sourcedProof: 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.

For a $2$ -design, $N N^{T} = (r - λ) I + λ J$

Statement

Let $(P, B)$ be a $2$ - $(v, k, λ)$ design, let $N$ be its point-block incidence matrix, let $r$ be the common number of blocks through a point, let $I_{v}$ be the $v \times v$ identity matrix, and let $J_{v}$ be the $v \times v$ all-ones matrix. Then $N N^{T} = (r - λ) I_{v} + λ J_{v} .$

Facts & Assumptions

Given: A $2$ - $(v, k, λ)$ design $(P, B)$ and its incidence matrix $N$ .

[L1]

Every point lies in exactly $r$ blocks (A $2$ -design satisfies $b k = v r$ and $r (k - 1) = λ (v - 1)$ ).

[L2]

Every two distinct points lie together in exactly $λ$ blocks (A $2$ - $(v, k, λ)$ design).

Proof

technique · direct

1.1

The $(p, p)$ entry of $N N^{T}$ counts the blocks containing $p$ , so every diagonal entry is $r$ by [L1].

L1algebra

1.2

If $p \neq q$ , then the $(p, q)$ entry of $N N^{T}$ counts the blocks containing both $p$ and $q$ , so every off-diagonal entry is $λ$ by [L2].

L2algebra

2.1

The matrix on the right has diagonal entries $(r - λ) + λ = r$ and off-diagonal entries $λ$ , so steps 1.1 and 1.2 identify it with $N N^{T}$ .

step 1.1step 1.2algebra∎

Depends on

Used by

Dependency tree · two levels

4 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

Noam D. Elkies, Math 155 notes: Jan. 29 (standard reference, not scraped)

For a 22-design, NNT=(r−λ)I+λJNNT=(r−λ)I+λJ