proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxeasy-side

Easy side in exponent form: e^{α log n} = n^α

For all n,α∈R\htmlData{mx=1}{n, \alpha \in \htmlData{mx=0}{\mathbb{R}}}, if 0<n0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}, then

eα⋅log⁡n=nα\htmlData{mx=4}{e^{\htmlData{mx=3}{\htmlData{mx=0}{\alpha} \cdot \htmlData{mx=2}{\log \htmlData{mx=1}{n}}}}} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{\htmlData{mx=5}{n}^{\htmlData{mx=6}{\alpha}}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
n, α range over real numbers: the statement must hold for every choice. ((n α : ℝ))
<Less than
0 is strictly smaller than n. (0 < n)
eˣExponential function
e raised to the power α · log(n), where e ≈ 2.718. (Real.exp (α * Real.log n))
=Equality
exp(α · log(n)) and n^α are the same. (Real.exp (α * Real.log n) = n ^ α)
·Multiplication
α times log(n). (α * Real.log n)
logNatural logarithm
The logarithm of n to base e: the power e must be raised to to give n. (Real.log n)
aᵇPower
n raised to the power α. (n ^ α)

With R = α ln n the uniform-SNIS sample count 16F²e^R/ε is 16F²n^α/ε per row: total work n^{1+α}, subquadratic exactly when α < 1. Matches the hard side's α > 1: the transition is at score range ln n. Part of 'Unbiased randomized attention: cost x variance thresholds', round 3 (sharp threshold at score range ln n).

How it was proved

Proved
4 October 2026
By
Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
Checked by
Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms
Classical.choice, Quot.sound, propext
Proof
10 lines of Lean
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