proofs
Provedunbiased-attentionrandomized-algorithmssoftmaxgradient-hardness

From mass to output: S ≤ E/8 ⇒ S/(E + S) ≤ 1/8

For all S,E∈R\htmlData{mx=1}{S, E \in \htmlData{mx=0}{\mathbb{R}}}, if 0<E0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{E}, 0≤S0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{S} and S≤E8\htmlData{mx=0}{S} \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\frac{\htmlData{mx=1}{E}}{8}}, then

SE+S≤18\htmlData{mx=4}{\frac{\htmlData{mx=0}{S}}{\htmlData{mx=3}{\htmlData{mx=1}{E} + \htmlData{mx=2}{S}}}} \mathrel{\htmlData{mx=6}{\leq}} \htmlData{mx=5}{\frac{1}{8}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
S, E range over real numbers: the statement must hold for every choice. ((S E : ℝ))
<Less than
0 is strictly smaller than E. (0 < E)
≤At most
0 is less than or equal to S. (0 ≤ S)
a/bDivision
E divided by 8. In Lean, dividing by 0 gives 0. (E / 8)
+Addition
E plus S. (E + S)

Arithmetic link from the far-mass bound to the NO-row output bound. Part of 'Unbiased randomized attention: cost x variance thresholds', round 7 (gradient hardness via isolation).

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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