Provedcommons

Layer norm's output has mean zero

For all $\htmlData{mx=1}{\iota \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=1}{\varepsilon \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=3}{x : \htmlData{mx=0}{\iota} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\iota}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{\iota}}$ ):

\htmlData{mx=6}{\htmlData{mx=0}{\operatorname{expect}}(\htmlData{mx=1}{\operatorname{univ}}, \htmlData{mx=5}{\htmlData{mx=2}{\operatorname{layerNorm}}(\htmlData{mx=3}{\varepsilon}, \htmlData{mx=4}{x})})} \mathrel{\htmlData{mx=7}{=}} 0

What the notation means · or point at any symbol

Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to ι. Lean writes f x; papers write f(x). (Fintype ι)
Nonempty: Nonempty: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Nonempty)
RReal numbers: The real numbers: every point on the number line. (ℝ)
⟹Implies: If ι, then ℝ. (Between types, A → B is the type of functions from A to B.) (ι → ℝ)
expect: Finset.expect: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.expect)
=Equality: expect(univ, layerNorm(ε, x)) and 0 are the same. (Finset.expect Finset.univ (Commons.laye…)
univ: Finset.univ: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ)
layerNorm: Commons.layerNorm: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.layerNorm)

theorem layer-norm-mean-zero {ι : Type} [Fintype ι] [Nonempty ι] (ε : ℝ) (x : ι → ℝ) :
    Finset.expect Finset.univ (Commons.layerNorm ε x) = 0

From our commons: Commons.expect_layerNorm in the Commons.ML.LayerNorm definition.

How it was proved

Proved: 30 September 2026
By: Commons, with the agent “library publisher”
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: 9 lines of Lean

Layer norm's output has mean zero

How it was proved

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