Provedllmlarge-language-modelsformal-proofdecoding

Constant logits make every selection attain the constant

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=6}{w : \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=2}{\htmlData{mx=1}{n} + 1}} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\mathbb{R}}}$ and $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ :

(\htmlData{mx=5}{\forall j,\ \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{j})} \mathrel{\htmlData{mx=4}{=}} \htmlData{mx=3}{c}}) \mathrel{\htmlData{mx=13}{\implies}} \htmlData{mx=10}{\htmlData{mx=6}{w}(\htmlData{mx=9}{\htmlData{mx=7}{\operatorname{argmax}}(\htmlData{mx=8}{w})})} \mathrel{\htmlData{mx=12}{=}} \htmlData{mx=11}{c}

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 + 1. Lean writes f x; papers write f(x). (Fin (n + 1))
⟹Implies: If Fin(n + 1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin (n + 1) → ℝ)
∀For all: The statement after the comma holds for every j. (∀ j, w j = c)
+Addition: n plus 1. (n + 1)
=Equality: w(j) and c are the same. (w j = c)
RReal numbers: The real numbers: every point on the number line. (ℝ)
argmax: LLMDecode.argmax: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMDecode.argmax)

theorem dec-argmax-const (n : ℕ) (w : Fin (n + 1) → ℝ) (c : ℝ) :
    (∀ j, w j = c) → w (LLMDecode.argmax w) = c

If all logits equal c then the argmax's logit is c.

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: 13 lines of Lean

Constant logits make every selection attain the constant

How it was proved

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