Provedllmlarge-language-modelsformal-proofattention

A zero query scores every key at zero

For all $\htmlData{mx=1}{d \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{k : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ :

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{logit}}(\htmlData{mx=1}{d}, \htmlData{mx=2}{\cdot \mapsto 0}, \htmlData{mx=3}{k})} \mathrel{\htmlData{mx=5}{=}} 0

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 d. Lean writes f x; papers write f(x). (Fin d)
⟹Implies: If Fin(d), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin d → ℝ)
logit: LLMAttention.logit: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMAttention.logit)
=Equality: logit(d, _ ↦ 0, k) and 0 are the same. (LLMAttention.logit d (fun _ => 0) k = 0)
RReal numbers: The real numbers: every point on the number line. (ℝ)
x ↦ f(x)Function (λ-abstraction): The function sending _ to 0. Lean writes fun x => …; papers write x ↦ …. (fun _ => 0)

theorem attn-logit-zero-query (d : ℕ) (k : Fin d → ℝ) :
    LLMAttention.logit d (fun _ => 0) k = 0

logit d 0 k = 0: the zero query is orthogonal to everything.

How it was proved

Proved: 26 September 2026
By: Ajax, with the agent “ompmuse”
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: 14 lines of Lean

A zero query scores every key at zero

How it was proved

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