proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxhardness

Soundness amplification: 0 ≤ x ≤ s ⇒ 0 ≤ x^k ≤ s^k

For all x,s∈R\htmlData{mx=1}{x, s \in \htmlData{mx=0}{\mathbb{R}}} and k∈N\htmlData{mx=1}{k \in \htmlData{mx=0}{\mathbb{N}}}, if 0≤x0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{x} and x≤s\htmlData{mx=0}{x} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{s}, then

0≤xk∧xk≤sk0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{x}^{\htmlData{mx=1}{k}}} \mathbin{\htmlData{mx=11}{\land}} \htmlData{mx=6}{\htmlData{mx=4}{x}^{\htmlData{mx=5}{k}}} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{\htmlData{mx=7}{s}^{\htmlData{mx=8}{k}}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
x, s range over real numbers: the statement must hold for every choice. ((x s : ℝ))
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
≤At most
0 is less than or equal to x. (0 ≤ x)
∧And
Both hold: 0 ≤ x^k, and x^k ≤ s^k. (0 ≤ x ^ k ∧ x ^ k ≤ s ^ k)
aᵇPower
x raised to the power k. (x ^ k)

The normalized NO-case inner product after k-fold tensoring. 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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