Provedalphabaguettebaguette-zrange

Attention output never exceeds the largest value magnitude

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{w, v : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{M \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ and $\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{v}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{M}}$ , then

\htmlData{mx=8}{\left\lvert \htmlData{mx=7}{\sum_{j} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{v}(\htmlData{mx=4}{j})}}} \right\rvert} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{M}

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 n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
x : TVariables: w, v range over Fin n → ℝ: the statement must hold for every choice. ((w v : Fin n → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 ≤ w j)
∑Summation: Add up w(j) for every j. (∑ j, w j)
=Equality: ∑ j, w(j) and 1 are the same. (∑ j, w j = 1)
≤At most: 0 is less than or equal to w(j). (0 ≤ w j)
|x|Absolute value: The size of v(j) without its sign. (|v j|)
·Multiplication: w(j) times v(j). (w j * v j)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem bz-convex-combination-bound {n : ℕ} (w v : Fin n → ℝ) (M : ℝ) (hw : ∀ j, 0 ≤ w j) (hsum : ∑ j, w j = 1) (hv : ∀ j, |v j| ≤ M) :
    |∑ j, w j * v j| ≤ M

With nonnegative weights summing to 1 (softmax) and |vⱼ| ≤ M, |Σ wⱼvⱼ| ≤ M. Attention cannot amplify its value range. Part of the plan 'Baguette-Z: closed-loop certified integer-state training'.

How it was proved

Proved: 3 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: 16 lines of Lean

Attention output never exceeds the largest value magnitude

How it was proved

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