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 rank-one adapter cannot realize a full-rank two-dimensional update

Example

Let $d = k = 2$ , $r = 1$ , $c = 1$ , and $W_{0} = 0$ . The target update $Δ = I_{2}$ cannot be represented as $B A$ with $B \in R^{2 \times 1}$ and $A \in R^{1 \times 2}$ .

Facts & Assumptions

Given: The target matrix $I_{2}$ and the stated rank-one factor shapes.

[F1]

Rank capacity and factorization ambiguity of a low-rank adapter: every product $c B A$ with the stated factor shapes has rank at most $r$ .

Verification

1.1givenalgebra

The identity matrix $I_{2}$ maps $R^{2}$ onto $R^{2}$ , so $rank (I_{2}) = 2$ .

2.1givenF1step 1.1algebra∎

For every $B \in R^{2 \times 1}$ and $A \in R^{1 \times 2}$ , $im (B A) \subseteq im (B)$ and $\dim im (B) \leq 1$ ; hence $rank (B A) \leq 1$ by [F1]. This contradicts step 1.1 if $B A = I_{2}$ , so no such factors represent the target.

Depends on

Rank capacity and factorization ambiguity of a low-rank adapter

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.