proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxdebiasing

Composed easy side: level MSE ≤ K/2^l and V_l ≤ 2(MSE_{l+1}+MSE_l) give c_l V_l ≤ 6K at cost c_l = 2^{l+1}

For all K∈R\htmlData{mx=1}{K \in \htmlData{mx=0}{\mathbb{R}}}, l∈N\htmlData{mx=1}{l \in \htmlData{mx=0}{\mathbb{N}}} and M0,M1,V∈R\htmlData{mx=1}{M_{0}, M_{1}, V \in \htmlData{mx=0}{\mathbb{R}}}, if M0≤K2l\htmlData{mx=0}{M_{0}} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{\frac{\htmlData{mx=1}{K}}{\htmlData{mx=3}{2^{\htmlData{mx=2}{l}}}}}, M1≤K2l+1\htmlData{mx=0}{M_{1}} \mathrel{\htmlData{mx=6}{\leq}} \htmlData{mx=5}{\frac{\htmlData{mx=1}{K}}{\htmlData{mx=4}{2^{\htmlData{mx=3}{\htmlData{mx=2}{l} + 1}}}}} and V≤2(M1+M0)\htmlData{mx=0}{V} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{2(\htmlData{mx=3}{\htmlData{mx=1}{M_{1}} + \htmlData{mx=2}{M_{0}}})}, then

2l+1⋅V≤6K\htmlData{mx=4}{\htmlData{mx=2}{2^{\htmlData{mx=1}{\htmlData{mx=0}{l} + 1}}} \cdot \htmlData{mx=3}{V}} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{6\htmlData{mx=5}{K}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables
M0, M1, V range over real numbers: the statement must hold for every choice. ((M0 M1 V : ℝ))
≤At most
M0 is less than or equal to K/2^l. (M0 ≤ K / 2 ^ l)
·Multiplication
2 times M1 + M0. (2 * (M1 + M0))
a/bDivision
K divided by 2^l. In Lean, dividing by 0 gives 0. (K / 2 ^ l)
aᵇPower
2 raised to the power l. (2 ^ l)
+Addition
l plus 1. (l + 1)

Feeds the inverse-cost allocation lemma with A = 6K, K = 16F²(1+χ²): unbiased attention with cost × variance ≤ 6 L² K. Part of 'Unbiased randomized attention: cost x variance thresholds', round 9 (true derivative, whole-instance reduction, compositions).

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
Sign in to see the proofThe proof, its signed report and its history.

More proved statements