Provedllmlarge-language-modelsformal-proofsoftmax

Scaling logits with temperature rescaling leaves sampling unchanged

For all $\htmlData{mx=1}{T, c \in \htmlData{mx=0}{\mathbb{R}}}$ , $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{w : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=3}{i \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ :

\htmlData{mx=0}{T} \mathrel{\htmlData{mx=1}{\neq}} 0 \mathrel{\htmlData{mx=23}{\implies}} \htmlData{mx=2}{c} \mathrel{\htmlData{mx=3}{\neq}} 0 \mathrel{\htmlData{mx=22}{\implies}} \htmlData{mx=15}{\htmlData{mx=4}{\operatorname{smT}}(\htmlData{mx=7}{\htmlData{mx=5}{c} \cdot \htmlData{mx=6}{T}}, \htmlData{mx=13}{j \mapsto \htmlData{mx=12}{\htmlData{mx=8}{c} \cdot \htmlData{mx=11}{\htmlData{mx=9}{w}(\htmlData{mx=10}{j})}}}, \htmlData{mx=14}{i})} \mathrel{\htmlData{mx=21}{=}} \htmlData{mx=20}{\htmlData{mx=16}{\operatorname{smT}}(\htmlData{mx=17}{T}, \htmlData{mx=18}{w}, \htmlData{mx=19}{i})}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: T, c range over real numbers: the statement must hold for every choice. ((T c : ℝ))
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)
≠Not equal: T and 0 are different. (T ≠ 0)
smT: LLMSoftmax.smT: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMSoftmax.smT)
=Equality: smT(c · T, j ↦ c · w(j), i) and smT(T, w, i) are the same. (LLMSoftmax.smT (c * T) (fun j => c * w …)
·Multiplication: c times T. (c * T)
x ↦ f(x)Function (λ-abstraction): The function sending j to c · w(j). Lean writes fun x => …; papers write x ↦ …. (fun j => c * w j)

theorem smx-temp-scale (T c : ℝ) (n : ℕ) (w : Fin n → ℝ) (i : Fin n) :
    T ≠ 0 → c ≠ 0 → LLMSoftmax.smT (c * T) (fun j => c * w j) i = LLMSoftmax.smT T w i

smT (c*T) (c·w) = smT T w for c ≠ 0, T ≠ 0: logit scale can be absorbed into temperature.

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: 18 lines of Lean

Scaling logits with temperature rescaling leaves sampling unchanged

How it was proved

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