Provedalphabaguettealgorithm-ideab6

Skipped SwiGLU units contribute at most Σ εᵤ·|upᵤ|·|wᵤ| to any output coordinate

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{s, \mathit{up}, w, \varepsilon : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $\htmlData{mx=8}{\forall u,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{u})} \right\rvert} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{\htmlData{mx=4}{\varepsilon}(\htmlData{mx=5}{u})}}$ , then

\htmlData{mx=12}{\left\lvert \htmlData{mx=11}{\sum_{u} \htmlData{mx=10}{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{u})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{\mathit{up}}(\htmlData{mx=4}{u})}} \cdot \htmlData{mx=9}{\htmlData{mx=7}{w}(\htmlData{mx=8}{u})}}} \right\rvert} \mathrel{\htmlData{mx=27}{\leq}} \htmlData{mx=26}{\sum_{u} \htmlData{mx=25}{\htmlData{mx=20}{\htmlData{mx=15}{\htmlData{mx=13}{\varepsilon}(\htmlData{mx=14}{u})} \cdot \htmlData{mx=19}{\left\lvert \htmlData{mx=18}{\htmlData{mx=16}{\mathit{up}}(\htmlData{mx=17}{u})} \right\rvert}} \cdot \htmlData{mx=24}{\left\lvert \htmlData{mx=23}{\htmlData{mx=21}{w}(\htmlData{mx=22}{u})} \right\rvert}}}

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 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: s, up, w, ε range over Fin n → ℝ: the statement must hold for every choice. ((s up w ε : Fin n → ℝ))
∀For all: The statement after the comma holds for every u. (∀ u, |s u| ≤ ε u)
∑Summation: Add up s(u) · up(u) · w(u) for every u. (∑ u, s u * up u * w u)
|x|Absolute value: The size of s(u) without its sign. (|s u|)
≤At most: |s(u)| is less than or equal to ε(u). (|s u| ≤ ε u)
·Multiplication: s(u) times up(u). (s u * up u)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem gate-skip-contribution-bound {n : ℕ} (s up w ε : Fin n → ℝ) (hs : ∀ u, |s u| ≤ ε u) :
    |∑ u, s u * up u * w u| ≤ ∑ u, ε u * |up u| * |w u|

Idea B6. If each skipped unit's gate activation satisfies |sᵤ| ≤ εᵤ (e.g. from silu-negative-tail-bound), the total contribution of the skipped units to an output coordinate, Σ sᵤ·upᵤ·wᵤ (wᵤ the down-projection entry), is bounded by Σ εᵤ·|upᵤ|·|wᵤ|.

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: 14 lines of Lean

Skipped SwiGLU units contribute at most Σ εᵤ·|upᵤ|·|wᵤ| to any output coordinate

How it was proved

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