Provedunbiased-attentionrandomized-algorithmssoftmaxhardness

Whole-instance reduction: NO (all x ≤ s) ⇒ every row ≤ 1/4; YES (some x = 1) ⇒ some row ≥ 1/2

For all $\htmlData{mx=1}{m, n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=9}{x : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{m}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{n}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ and $\htmlData{mx=1}{s \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}$ and $\htmlData{mx=0}{s} \mathrel{\htmlData{mx=1}{<}} 1$ , then

\begin{aligned} & \left(\right. (\htmlData{mx=8}{\forall i, k,\ \htmlData{mx=5}{\htmlData{mx=2}{x}(\htmlData{mx=3}{i}, \htmlData{mx=4}{k})} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{s}}) \\ & \htmlData{mx=1}{\implies} \htmlData{mx=9}{\forall} i, \\ & \quad \htmlData{mx=45}{\frac{\htmlData{mx=23}{\sum_{k} \htmlData{mx=22}{e^{\htmlData{mx=21}{\htmlData{mx=16}{\frac{\htmlData{mx=13}{\log (\htmlData{mx=12}{4\htmlData{mx=11}{n}})}}{\htmlData{mx=15}{1 - \htmlData{mx=14}{s}}}} \cdot \htmlData{mx=20}{\htmlData{mx=17}{x}(\htmlData{mx=18}{i}, \htmlData{mx=19}{k})}}}}}}{\htmlData{mx=44}{\htmlData{mx=30}{e^{\htmlData{mx=29}{\frac{\htmlData{mx=26}{\log (\htmlData{mx=25}{4\htmlData{mx=24}{n}})}}{\htmlData{mx=28}{1 - \htmlData{mx=27}{s}}}}}} + \htmlData{mx=43}{\sum_{k} \htmlData{mx=42}{e^{\htmlData{mx=41}{\htmlData{mx=36}{\frac{\htmlData{mx=33}{\log (\htmlData{mx=32}{4\htmlData{mx=31}{n}})}}{\htmlData{mx=35}{1 - \htmlData{mx=34}{s}}}} \cdot \htmlData{mx=40}{\htmlData{mx=37}{x}(\htmlData{mx=38}{i}, \htmlData{mx=39}{k})}}}}}}}} \\ & \quad \quad \htmlData{mx=10}{\leq} \htmlData{mx=46}{\frac{1}{4}} \left.\right) \\ & \htmlData{mx=0}{\land} \left(\right. (\htmlData{mx=53}{\exists i, k,\ \htmlData{mx=51}{\htmlData{mx=48}{x}(\htmlData{mx=49}{i}, \htmlData{mx=50}{k})} \mathrel{\htmlData{mx=52}{=}} 1}) \\ & \htmlData{mx=47}{\implies} \htmlData{mx=54}{\exists} i, \\ & \quad \htmlData{mx=56}{\frac{1}{2}} \\ & \quad \quad \htmlData{mx=55}{\leq} \htmlData{mx=91}{\frac{\htmlData{mx=69}{\sum_{k} \htmlData{mx=68}{e^{\htmlData{mx=67}{\htmlData{mx=62}{\frac{\htmlData{mx=59}{\log (\htmlData{mx=58}{4\htmlData{mx=57}{n}})}}{\htmlData{mx=61}{1 - \htmlData{mx=60}{s}}}} \cdot \htmlData{mx=66}{\htmlData{mx=63}{x}(\htmlData{mx=64}{i}, \htmlData{mx=65}{k})}}}}}}{\htmlData{mx=90}{\htmlData{mx=76}{e^{\htmlData{mx=75}{\frac{\htmlData{mx=72}{\log (\htmlData{mx=71}{4\htmlData{mx=70}{n}})}}{\htmlData{mx=74}{1 - \htmlData{mx=73}{s}}}}}} + \htmlData{mx=89}{\sum_{k} \htmlData{mx=88}{e^{\htmlData{mx=87}{\htmlData{mx=82}{\frac{\htmlData{mx=79}{\log (\htmlData{mx=78}{4\htmlData{mx=77}{n}})}}{\htmlData{mx=81}{1 - \htmlData{mx=80}{s}}}} \cdot \htmlData{mx=86}{\htmlData{mx=83}{x}(\htmlData{mx=84}{i}, \htmlData{mx=85}{k})}}}}}}}} \left.\right) \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables: m, n range over natural numbers: the statement must hold for every choice. ((m n : 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)
<Less than: 0 is strictly smaller than n. (0 < n)
∧And: Both hold: (∀ i, k, x(i, k) ≤ s) → (∀ i, (∑ k, exp(log(4 · n)/(1 − s)…, and (∃ i, k, x(i, k) = 1) → (∃ i, 1/2 ≤ (∑ k, exp(log(4 · n)/(…. (((∀ i k, x i k ≤ s) → ∀ i, (∑ k, Real.e…)
∀For all: The statement after the comma holds for every i, k. (∀ i k, x i k ≤ s)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
≤At most: x(i, k) is less than or equal to s. (x i k ≤ s)
RReal numbers: The real numbers: every point on the number line. (ℝ)
∑Summation: Add up exp(log(4 · n)/(1 − s) · x(i, k)) for every k. (∑ k, Real.exp (Real.log (4 * n) / (1 - …)
eˣExponential function: e raised to the power log(4 · n)/(1 − s) · x(i, k), where e ≈ 2.718. (Real.exp (Real.log (4 * n) / (1 - s) * …)
logNatural logarithm: The logarithm of 4 · n to base e: the power e must be raised to to give 4 · n. (Real.log (4 * n))
a/bDivision: log(4 · n) divided by 1 − s. In Lean, dividing by 0 gives 0. (Real.log (4 * n) / (1 - s))
·Multiplication: 4 times n. (4 * n)
-Subtraction: 1 minus s. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 - s)
+Addition: exp(log(4 · n)/(1 − s)) plus ∑ k, exp(log(4 · n)/(1 − s) · x(i, k)). (Real.exp (Real.log (4 * n) / (1 - s)) +…)
∃There exists: At least one i, k makes the statement after the comma true. (∃ i k, x i k = 1)
=Equality: x(i, k) and 1 are the same. (x i k = 1)

theorem ua-reduction-all-rows {m n : ℕ} (x : Fin m → Fin n → ℝ) (s : ℝ) (hn : 0 < n) (hs : s < 1) :
    ((∀ i k, x i k ≤ s) → ∀ i, (∑ k, Real.exp (Real.log (4 * n) / (1 - s) * x i k)) / (Real.exp (Real.log (4 * n) / (1 - s)) + ∑ k, Real.exp (Real.log (4 * n) / (1 - s) * x i k)) ≤ 1 / 4) ∧ ((∃ i k, x i k = 1) → ∃ i, 1 / 2 ≤ (∑ k, Real.exp (Real.log (4 * n) / (1 - s) * x i k)) / (Real.exp (Real.log (4 * n) / (1 - s)) + ∑ k, Real.exp (Real.log (4 * n) / (1 - s) * x i k)))

For an instance with rows i and keys k, normalized inner products x i k, baseline at β = log(4n)/(1−s): the YES/NO promise transfers to the attention outputs of all rows at once. 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: 45 lines of Lean

Whole-instance reduction: NO (all x ≤ s) ⇒ every row ≤ 1/4; YES (some x = 1) ⇒ some row ≥ 1/2

How it was proved

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