Provedllmlarge-language-modelsformal-proofattention

A zero key scores zero in one dimension

For all $\htmlData{mx=4}{q : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Fin}}\,1} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ :

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

What the notation means · or point at any symbol

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 1. Lean writes f x; papers write f(x). (Fin 1)
⟹Implies: If Fin(1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 1 → 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(1, q, _ ↦ 0) and 0 are the same. (LLMAttention.logit 1 q (fun _ => 0) = 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-d1-zero-key (q : Fin 1 → ℝ) :
    LLMAttention.logit 1 q (fun _ => 0) = 0

For d = 1 and k = 0: the logit is 0.

How it was proved

Proved: 27 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: 15 lines of Lean