ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Two merge orders give the same attention summary

Example

Take $n = 2$ , $d = v = 1$ , both rows of $Q$ equal to $[1]$ , $K_{1} = [0]$ , $K_{2} = [\ln 2]$ , $V_{1} = [0]$ , $V_{2} = [3]$ , scale $c = 1$ , block size $b = 1$ , and $A_{1} = A_{2} = {1, 2}$ . For either query, merging the summaries of blocks ${1}$ then ${2}$ , or ${2}$ then ${1}$ , gives $(m, ℓ, z) = (\ln 2, 3 / 2, 3)$ and output $2$ .

Example references

Block summaries and the merge operation

Summary invariant and order-independent exact output

Facts & Assumptions

Given: The two query rows, two key/value rows, and allowed sets in the example statement.

[F1]

Stable summary of an attention key block: a singleton block summary uses its score maximum, shifted exponential sum, and shifted value sum; a merge rescales both blocks to their common maximum.

[F2]

Tiled attention and its recomputed reverse pass are exact: the completed summary divided by its normalizer is the rowwise masked attention output.

Verification

1.1F1givenalgebra

The two scores are $s_{1} = 0$ and $s_{2} = \ln 2$ . Thus $Σ ({1}) = (0, 1, 0)$ and $Σ ({2}) = (\ln 2, 1, 3)$ .

2.1F1F2step 1.1algebra∎

Merging in either order gives maximum $\ln 2$ , normalizer $e^{- \ln 2} + 1 = 3 / 2$ , and weighted sum $3$ . The output is $3 / (3 / 2) = 2$ in either order.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Tri Dao et al., FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness, §3.1 and Appendix C (standard reference, not scraped)
Maxim Milakov and Natalia Gimelshein, Online normalizer calculation for softmax, §3.1 (standard reference, not scraped)