Provedllmlarge-language-modelsformal-proofattention

Doubling the key doubles the logit (d=1)

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

\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{logit}}(1, \htmlData{mx=1}{q}, \htmlData{mx=6}{i \mapsto \htmlData{mx=5}{2\htmlData{mx=4}{\htmlData{mx=2}{k}(\htmlData{mx=3}{i})}}})} \mathrel{\htmlData{mx=13}{=}} \htmlData{mx=12}{2\htmlData{mx=11}{\htmlData{mx=8}{\operatorname{logit}}(1, \htmlData{mx=9}{q}, \htmlData{mx=10}{k})}}

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)
x : TVariables: q, k range over Fin 1 → ℝ: the statement must hold for every choice. ((q k : 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, i ↦ 2 · k(i)) and 2 · logit(1, q, k) are the same. (LLMAttention.logit 1 q (fun i => 2 * k …)
RReal numbers: The real numbers: every point on the number line. (ℝ)
x ↦ f(x)Function (λ-abstraction): The function sending i to 2 · k(i). Lean writes fun x => …; papers write x ↦ …. (fun i => 2 * k i)
·Multiplication: 2 times k(i). (2 * k i)

theorem attn-logit-double-key (q k : Fin 1 → ℝ) :
    LLMAttention.logit 1 q (fun i => 2 * k i) = 2 * LLMAttention.logit 1 q k

logit 1 q (2k) = 2 * logit 1 q k componentwise scaling in d=1.

How it was proved

Proved: 28 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: 16 lines of Lean

Doubling the key doubles the logit (d=1)

How it was proved

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