Provedalphabaguettermsnorm

RMSNorm with ε > 0: the normalized row's squared norm is strictly below the width

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=1}{\varepsilon \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=5}{x : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}$ and $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\varepsilon}$ , then

\htmlData{mx=15}{\sum_{i} \htmlData{mx=14}{\left(\htmlData{mx=13}{\frac{\htmlData{mx=2}{\htmlData{mx=0}{x}(\htmlData{mx=1}{i})}}{\htmlData{mx=12}{\sqrt{\htmlData{mx=11}{\htmlData{mx=9}{\frac{\htmlData{mx=7}{\sum_{j} \htmlData{mx=6}{\htmlData{mx=5}{\htmlData{mx=3}{x}(\htmlData{mx=4}{j})}^{2}}}}{\htmlData{mx=8}{n}}} + \htmlData{mx=10}{\varepsilon}}}}}}\right)^{2}}} \mathrel{\htmlData{mx=17}{<}} \htmlData{mx=16}{n}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
<Less than: 0 is strictly smaller than n. (0 < n)
RReal numbers: The real numbers: every point on the number line. (ℝ)
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)
√Square root: The non-negative number whose square is (∑ j, x(j)^2)/n + ε. (Real.sqrt ((∑ j, x j ^ 2) / n + ε))
a/bDivision: ∑ j, x(j)^2 divided by n. In Lean, dividing by 0 gives 0. ((∑ j, x j ^ 2) / n)
+Addition: (∑ j, x(j)^2)/n plus ε. ((∑ j, x j ^ 2) / n + ε)
∑Summation: Add up x(j)^2 for every j. (∑ j, x j ^ 2)
aᵇPower: x(j) raised to the power 2. (x j ^ 2)

theorem rmsnorm-output-energy-lt-width (n : ℕ) (hn : 0 < n) (ε : ℝ) (hε : 0 < ε) (x : Fin n → ℝ) :
    ∑ i, (x i / Real.sqrt ((∑ j, x j ^ 2) / n + ε)) ^ 2 < n

Alpha's mean-square normalization (Component.MeanSquareNormalization.normalizeMeanSquareWith) computes z = x · (Σx²/n + ε)^(-1/2), with no centering and no bias. Baguette uses ε = 1e-5 and n = 576 (baguetteRmsEpsilonWord, Alpha commit e3330947, systems/baguette/src/Baguette/Model.alpha). For any input, ‖z‖² = n·m/(m+ε) < n, where m = Σx²/n. So every normalized coordinate is bounded by √n before the gain, whatever the residual stream's magnitude.

How it was proved

Proved: 3 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
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

RMSNorm with ε > 0: the normalized row's squared norm is strictly below the width

How it was proved

More proved statements