proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxhardness

Sharp gadget range: x ∈ [0,1] ⇒ scores β·x ∈ [0, β]

For all x,β∈R\htmlData{mx=1}{x, \beta \in \htmlData{mx=0}{\mathbb{R}}}, if 0≤β0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\beta}, 0≤x0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{x} and x≤1\htmlData{mx=0}{x} \mathrel{\htmlData{mx=1}{\leq}} 1, then

0≤β⋅x∧β⋅x≤β0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\beta} \cdot \htmlData{mx=1}{x}} \mathbin{\htmlData{mx=9}{\land}} \htmlData{mx=6}{\htmlData{mx=4}{\beta} \cdot \htmlData{mx=5}{x}} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\beta}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
x, β range over real numbers: the statement must hold for every choice. ((x β : ℝ))
≤At most
0 is less than or equal to β. (0 ≤ β)
∧And
Both hold: 0 ≤ β · x, and β · x ≤ β. (0 ≤ β * x ∧ β * x ≤ β)
·Multiplication
β times x. (β * x)

With the baseline at β, every score of the gadget row lies in [0, β], so the row's score range is ≤ β. 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
11 lines of Lean
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