Provedllmlarge-language-modelsformal-proofsoftmax

Exact two-to-one odds at the log-2 boundary

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

\htmlData{mx=1}{\htmlData{mx=0}{w}(0)} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{\htmlData{mx=3}{\htmlData{mx=2}{w}(1)} + \htmlData{mx=4}{\log 2}} \mathrel{\htmlData{mx=12}{\implies}} \htmlData{mx=9}{\htmlData{mx=7}{\operatorname{sm}}(\htmlData{mx=8}{w}, 0)} \mathrel{\htmlData{mx=11}{=}} \htmlData{mx=10}{\frac{2}{3}}

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 2. Lean writes f x; papers write f(x). (Fin 2)
⟹Implies: If Fin(2), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 2 → ℝ)
=Equality: w(0) and w(1) + log(2) are the same. (w 0 = w 1 + Real.log 2)
+Addition: w(1) plus log(2). (w 1 + Real.log 2)
RReal numbers: The real numbers: every point on the number line. (ℝ)
logNatural logarithm: The logarithm of 2 to base e: the power e must be raised to to give 2. (Real.log 2)
sm: LLMSoftmax.sm: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMSoftmax.sm)
a/bDivision: 2 divided by 3. In Lean, dividing by 0 gives 0. (2 / 3)

theorem smx-two-boundary-dominance (w : Fin 2 → ℝ) :
    w 0 = w 1 + Real.log 2 → LLMSoftmax.sm w 0 = 2 / 3

For Fin 2, w 0 = w 1 + log 2 gives exactly sm w 0 = 2/3.

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: 17 lines of Lean