proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxhardness

Range constant: s ≤ 1 − 1/α ⇒ log(4n)/(1−s) ≤ α·log(4n)

For all L,s,α∈R\htmlData{mx=1}{L, s, \alpha \in \htmlData{mx=0}{\mathbb{R}}}, if 0≤L0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{L}, 1<α1 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\alpha} and s≤1−1α\htmlData{mx=0}{s} \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{1 - \htmlData{mx=2}{\frac{1}{\htmlData{mx=1}{\alpha}}}}, then

L1−s≤α⋅L\htmlData{mx=3}{\frac{\htmlData{mx=0}{L}}{\htmlData{mx=2}{1 - \htmlData{mx=1}{s}}}} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{\htmlData{mx=4}{\alpha} \cdot \htmlData{mx=5}{L}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
L, s, α range over real numbers: the statement must hold for every choice. ((L s α : ℝ))
≤At most
0 is less than or equal to L. (0 ≤ L)
<Less than
1 is strictly smaller than α. (1 < α)
-Subtraction
1 minus 1/α. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 - 1 / α)
a/bDivision
1 divided by α. In Lean, dividing by 0 gives 0. (1 / α)
·Multiplication
α times L. (α * L)

For every α > 1 a constant tensor power k (2^{-k} ≤ 1 − 1/α) puts the hard instances at score range ≤ α ln(4n). 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
17 lines of Lean
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