ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

A positive rate matrix has NMFs beyond scale and permutation

Example

Let $X$ be the $2 \times 2$ all-ones count matrix and set $r = 2$ . For $0 < α < 1$ , define

$W_{α} = (\begin{array}{cc} α & 1 - α \\ α & 1 - α \end{array}), H = (\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}) .$

Every pair $(W_{α}, H)$ is strictly positive and gives rate matrix $W_{α} H = X$ . The factorizations for $α = 1 / 2$ and $α = 1 / 3$ are not related by positive component rescaling or permutation.

Facts & Assumptions

Given: The count matrix and factors defined in the example statement.

[F1]

Poisson nonnegative matrix-factorization model: the Poisson rate matrix is the product $W H$ .

[F2]

Positive component rescaling and permutation preserve an NMF rate matrix: the elementary component ambiguities rescale one factor column and its corresponding factor row inversely, or permute corresponding components.

Verification

1.1F1givenalgebra

Each row of $W_{α}$ sums to $1$ , while each entry of $H$ is $1$ , so $(W_{α} H)_{i j} = α + (1 - α) = 1$ for every cell. This verifies the common positive rate matrix.

2.1F2step 1.1algebra∎

Under the transformations in [F2], each rank-one component matrix $W_{\cdot k} H_{k \cdot}$ is unchanged except for reordering. At $α = 1 / 2$ the multiset of component matrices is ${\frac{1}{2} X, \frac{1}{2} X}$ , whereas at $α = 1 / 3$ it is ${\frac{1}{3} X, \frac{2}{3} X}$ . These multisets differ, so the factorizations are not related by scale or permutation.

Depends on

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