Provedunbiased-attentionrandomized-algorithmssoftmaxhardness

Finite Chebyshev: mass of {|error| ≥ t} ≤ MSE / t²

For all $\htmlData{mx=1}{m \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{w, e : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{m}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{t \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall i,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{i})}}$ and $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{t}$ , then

\htmlData{mx=12}{\sum_{i \in \htmlData{mx=8}{\htmlData{mx=0}{\operatorname{filter}}(\htmlData{mx=7}{i \mapsto \htmlData{mx=1}{t} \mathrel{\htmlData{mx=6}{\leq}} \htmlData{mx=5}{\left\lvert \htmlData{mx=4}{\htmlData{mx=2}{e}(\htmlData{mx=3}{i})} \right\rvert}})}} \htmlData{mx=11}{\htmlData{mx=9}{w}(\htmlData{mx=10}{i})}} \mathrel{\htmlData{mx=25}{\leq}} \htmlData{mx=24}{\frac{\htmlData{mx=21}{\sum_{i} \htmlData{mx=20}{\htmlData{mx=15}{\htmlData{mx=13}{w}(\htmlData{mx=14}{i})} \cdot \htmlData{mx=19}{\htmlData{mx=18}{\htmlData{mx=16}{e}(\htmlData{mx=17}{i})}^{2}}}}}{\htmlData{mx=23}{\htmlData{mx=22}{t}^{2}}}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to m. Lean writes f x; papers write f(x). (Fin m)
⟹Implies: If Fin(m), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin m → ℝ)
x : TVariables: w, e range over Fin m → ℝ: the statement must hold for every choice. ((w e : Fin m → ℝ))
∀For all: The statement after the comma holds for every i. (∀ i, 0 ≤ w i)
<Less than: 0 is strictly smaller than t. (0 < t)
∑Summation: Add up w(i) for each i in filter(i ↦ t ≤ |e(i)|). (∑ i ∈ Finset.univ.filter (fun i => t ≤ …)
≤At most: 0 is less than or equal to w(i). (0 ≤ w i)
filter: Finset.univ.filter: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ.filter)
RReal numbers: The real numbers: every point on the number line. (ℝ)
x ↦ f(x)Function (λ-abstraction): The function sending i to t ≤ |e(i)|. Lean writes fun x => …; papers write x ↦ …. (fun i => t ≤ |e i|)
|x|Absolute value: The size of e(i) without its sign. (|e i|)
a/bDivision: ∑ i, w(i) · e(i)^2 divided by t^2. In Lean, dividing by 0 gives 0. ((∑ i, w i * e i ^ 2) / t ^ 2)
·Multiplication: w(i) times e(i)^2. (w i * e i ^ 2)
aᵇPower: e(i) raised to the power 2. (e i ^ 2)

theorem ua-finite-chebyshev {m : ℕ} (w e : Fin m → ℝ) (hw : ∀ i, 0 ≤ w i) (t : ℝ) (ht : 0 < t) :
    ∑ i ∈ Finset.univ.filter (fun i => t ≤ |e i|), w i ≤ (∑ i, w i * e i ^ 2) / t ^ 2

Over a finite outcome space with weights w ≥ 0, Σ_{|e_i| ≥ t} w_i ≤ (Σ w_i e_i²)/t². With MSE ≤ 1/256 and t = 1/8, failure probability ≤ 1/4, which feeds the gap decision. Part of 'Unbiased randomized attention: cost x variance thresholds' (open theory problem).

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: 20 lines of Lean

Finite Chebyshev: mass of {|error| ≥ t} ≤ MSE / t²

How it was proved

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