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.

Merged and unmerged adapter forward passes agree

Example

Let $d = k = 2$ , $r = 1$ , $W_{0} = I_{2}$ , $c = 2$ , $A = (1, 2)$ , $B = (3, 4)^{T}$ , and $x = (1, 1)^{T}$ . The merged matrix is $W_{0} + c B A = (\begin{array}{cc} 7 & 12 \\ 8 & 17 \end{array})$ , and both the merged and unmerged forward passes return $(19, 25)^{T}$ .

Facts & Assumptions

Given: The specified matrices, scale, and input vector.

[F1]

Exact merging and gradients for a low-rank adapter: for every input $x$ , $W_{0} x + c B (A x) = (W_{0} + c B A) x$ .

Verification

1.1givenalgebra

Multiplication gives $B A = (\begin{array}{cc} 3 & 6 \\ 4 & 8 \end{array})$ , so $W_{0} + 2 B A = (\begin{array}{cc} 7 & 12 \\ 8 & 17 \end{array})$ and $(W_{0} + 2 B A) x = (19, 25)^{T}$ .

2.1givenF1step 1.1algebra∎

In the unmerged calculation, $W_{0} x = (1, 1)^{T}$ and $A x = 3$ , so $W_{0} x + 2 B (A x) = (1, 1)^{T} + 2 (3, 4)^{T} 3 = (19, 25)^{T}$ , in agreement with [F1].

Depends on

Exact merging and gradients for a low-rank adapter

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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.