Provedalphabaguettermsnormdeferred-normalization

Deferred RMSNorm: the pre-scale inner product ⟨dy, u⟩ is ⟨dy, r·u⟩ / r

For all $\htmlData{mx=1}{m \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{\mathit{dy}, u : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{m}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{r \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=0}{r} \mathrel{\htmlData{mx=1}{\neq}} 0$ , then

\htmlData{mx=7}{\sum_{i} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{\mathit{dy}}(\htmlData{mx=1}{i})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{u}(\htmlData{mx=4}{i})}}} \mathrel{\htmlData{mx=20}{=}} \htmlData{mx=19}{\frac{\htmlData{mx=17}{\sum_{i} \htmlData{mx=16}{\htmlData{mx=10}{\htmlData{mx=8}{\mathit{dy}}(\htmlData{mx=9}{i})} \cdot (\htmlData{mx=15}{\htmlData{mx=11}{r} \cdot \htmlData{mx=14}{\htmlData{mx=12}{u}(\htmlData{mx=13}{i})}})}}}{\htmlData{mx=18}{r}}}

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 m. Lean writes f x; papers write f(x). (Fin m)
⟹Implies: If Fin(m), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin m → ℝ)
x : TVariables: dy, u range over Fin m → ℝ: the statement must hold for every choice. ((dy u : Fin m → ℝ))
≠Not equal: r and 0 are different. (r ≠ 0)
∑Summation: Add up dy(i) · u(i) for every i. (∑ i, dy i * u i)
=Equality: ∑ i, dy(i) · u(i) and (∑ i, dy(i) · (r · u(i)))/r are the same. (∑ i, dy i * u i = (∑ i, dy i * (r * u i…)
·Multiplication: dy(i) times u(i). (dy i * u i)
RReal numbers: The real numbers: every point on the number line. (ℝ)
a/bDivision: ∑ i, dy(i) · (r · u(i)) divided by r. In Lean, dividing by 0 gives 0. ((∑ i, dy i * (r * u i)) / r)

theorem drn-row-scalar {m : ℕ} (dy u : Fin m → ℝ) (r : ℝ) (hr : r ≠ 0) :
    ∑ i, dy i * u i = (∑ i, dy i * (r * u i)) / r

The deferred backward needs ⟨dy, u⟩ for the unscaled projection output u. It is recoverable from the saved, scaled output y = r·u (r ≠ 0) as ⟨dy, y⟩/r, so u never has to be stored. Part of the idea 'Deferred RMS normalization for Baguette'.

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: 12 lines of Lean

Deferred RMSNorm: the pre-scale inner product ⟨dy, u⟩ is ⟨dy, r·u⟩ / r

How it was proved

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