DefinitionDefinition: 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.

Tight exponential tail for a positive loss derivative

A function $f : R \to R$ has a tight exponential tail if there are positive constants $c, a, μ_{+}, μ_{-}, u_{+}, u_{-} > 0$ such that

$f (u) \leq c (1 + e^{- μ_{+} u}) e^{- a u} (u > u_{+}),$

and

$f (u) \geq c (1 - e^{- μ_{-} u}) e^{- a u} (u > u_{-}) .$

For a differentiable strictly decreasing loss $ℓ$ , the source applies this definition to $f = - ℓ^{'}$ , the negative loss derivative.

Used by

Nothing. This result depends on no other item in the library.