RemarkRemark: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

The scalar workspace result is not a hardware I/O bound

Remarks

The workspace theorem counts scalar storage and operations for the explicitly defined real-arithmetic procedure. It does not count reads and writes between GPU high-bandwidth memory and on-chip SRAM, and it gives no hardware speedup claim. The FlashAttention paper's $Θ (n^{2} d^{2} / M)$ HBM-access result assumes its stated memory model, block schedule, and range $d \leq M \leq n d$ ; its endpoint input-size argument is not a lower bound for every fixed memory size. Neither claim follows from the scalar workspace theorem here.

The maximum shift makes every exact-real exponent argument $s_{i j} - m_{i}$ nonpositive, but this does not specify rounded arithmetic or rule out underflow of very small finite-precision terms. Different floating-point accumulation orders can return different bit patterns. The base procedure includes no dropout; if an extension uses dropout, its fixed realized mask must be reused in the reverse pass because a different mask changes the differentiated function.

Remarks references

The scalar operation and workspace theorem proved for this algorithm

Dao et al., §§2.1–3.2 and Appendix B: the paper's hardware-memory model and recomputed backward pass

Depends on

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.

Sources

Tri Dao et al., FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness, §§2.1–3.2 and Appendix B (standard reference, not scraped)