Provedalphabaguettenoether-checksumnorm

Checksum N1: RMSNorm's input cotangent satisfies ⟨dx, x⟩ = ε·r³·⟨v, x⟩

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

\begin{aligned} & \htmlData{mx=57}{\sum_{k} \htmlData{mx=56}{\htmlData{mx=52}{\htmlData{mx=11}{\frac{1}{\htmlData{mx=10}{\sqrt{\htmlData{mx=9}{\htmlData{mx=7}{\frac{\htmlData{mx=5}{\sum_{j} \htmlData{mx=4}{\htmlData{mx=3}{\htmlData{mx=1}{x}(\htmlData{mx=2}{j})}^{2}}}}{\htmlData{mx=6}{n}}} + \htmlData{mx=8}{\varepsilon}}}}}} \cdot \left(\htmlData{mx=51}{\htmlData{mx=14}{\htmlData{mx=12}{v}(\htmlData{mx=13}{k})} - \htmlData{mx=50}{\htmlData{mx=28}{\frac{\htmlData{mx=17}{\htmlData{mx=15}{x}(\htmlData{mx=16}{k})}}{\htmlData{mx=27}{\sqrt{\htmlData{mx=26}{\htmlData{mx=24}{\frac{\htmlData{mx=22}{\sum_{j} \htmlData{mx=21}{\htmlData{mx=20}{\htmlData{mx=18}{x}(\htmlData{mx=19}{j})}^{2}}}}{\htmlData{mx=23}{n}}} + \htmlData{mx=25}{\varepsilon}}}}}} \cdot \left(\htmlData{mx=49}{\frac{\htmlData{mx=47}{\sum_{i} \htmlData{mx=46}{\htmlData{mx=31}{\htmlData{mx=29}{v}(\htmlData{mx=30}{i})} \cdot \left(\htmlData{mx=45}{\frac{\htmlData{mx=34}{\htmlData{mx=32}{x}(\htmlData{mx=33}{i})}}{\htmlData{mx=44}{\sqrt{\htmlData{mx=43}{\htmlData{mx=41}{\frac{\htmlData{mx=39}{\sum_{j} \htmlData{mx=38}{\htmlData{mx=37}{\htmlData{mx=35}{x}(\htmlData{mx=36}{j})}^{2}}}}{\htmlData{mx=40}{n}}} + \htmlData{mx=42}{\varepsilon}}}}}}\right)}}}{\htmlData{mx=48}{n}}}\right)}}\right)} \cdot \htmlData{mx=55}{\htmlData{mx=53}{x}(\htmlData{mx=54}{k})}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=80}{\htmlData{mx=71}{\htmlData{mx=58}{\varepsilon} \cdot \htmlData{mx=70}{\left(\htmlData{mx=69}{\frac{1}{\htmlData{mx=68}{\sqrt{\htmlData{mx=67}{\htmlData{mx=65}{\frac{\htmlData{mx=63}{\sum_{j} \htmlData{mx=62}{\htmlData{mx=61}{\htmlData{mx=59}{x}(\htmlData{mx=60}{j})}^{2}}}}{\htmlData{mx=64}{n}}} + \htmlData{mx=66}{\varepsilon}}}}}}\right)^{3}}} \cdot \htmlData{mx=79}{\sum_{k} \htmlData{mx=78}{\htmlData{mx=74}{\htmlData{mx=72}{v}(\htmlData{mx=73}{k})} \cdot \htmlData{mx=77}{\htmlData{mx=75}{x}(\htmlData{mx=76}{k})}}}} \end{aligned}

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)
x : TVariables: x, v range over Fin n → ℝ: the statement must hold for every choice. ((x v : Fin n → ℝ))
=Equality: ∑ k, 1/√((∑ j, x(j)^2)/n + ε) · (v(k) − x(k)/√((∑ j, x(j)^… and ε · (1/√((∑ j, x(j)^2)/n + ε))^3 · (∑ k, v(k) · x(k)) are the same. (∑ k, (1 / Real.sqrt ((∑ j, x j ^ 2) / n…)
√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)
-Subtraction: v(k) minus x(k)/√((∑ j, x(j)^2)/n + ε) · ((∑ i, v(i) · (x(i)/√((∑ j, …. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (v k - x k / Real.sqrt ((∑ j, x j ^ 2) /…)
·Multiplication: v(i) times x(i)/√((∑ j, x(j)^2)/n + ε). (v i * (x i / Real.sqrt ((∑ j, x j ^ 2) …)

theorem nc-rmsnorm-scale-identity {n : ℕ} (hn : 0 < n) (ε : ℝ) (hε : 0 < ε) (x v : Fin n → ℝ) :
    ∑ k, (1 / Real.sqrt ((∑ j, x j ^ 2) / n + ε) * (v k - x k / Real.sqrt ((∑ j, x j ^ 2) / n + ε) * ((∑ i, v i * (x i / Real.sqrt ((∑ j, x j ^ 2) / n + ε))) / n))) * x k = ε * (1 / Real.sqrt ((∑ j, x j ^ 2) / n + ε)) ^ 3 * ∑ k, v k * x k

For Alpha's RMSNorm pullback dx = r·(v − z·mean(v⊙z)), z = r·x, r = (mean(x²)+ε)^(-1/2): ⟨dx, x⟩ = ε·r³·⟨v, x⟩ exactly. With ε = 0 the norm is scale-invariant and ⟨dx, x⟩ = 0. Baguette's ε = 1e-5 breaks the symmetry by a known, computable amount, so this per-row identity checks the normalization pullback kernel. Part of 'Noether checksums for Baguette's backward kernels'.

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: 31 lines of Lean

Checksum N1: RMSNorm's input cotangent satisfies ⟨dx, x⟩ = ε·r³·⟨v, x⟩

How it was proved

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