Provedalphabaguettebaguette-zoptimizer

After global-norm clipping, every gradient coordinate is bounded by the clip norm

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{g : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=3}{i \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ , if $\htmlData{mx=5}{\sqrt{\htmlData{mx=4}{\sum_{i} \htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{g}(\htmlData{mx=1}{i})}^{2}}}}} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{c}$ , then

\htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{g}(\htmlData{mx=1}{i})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{c}

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)
√Square root: The non-negative number whose square is ∑ i, g(i)^2. (Real.sqrt (∑ i, g i ^ 2))
≤At most: √(∑ i, g(i)^2) is less than or equal to c. (Real.sqrt (∑ i, g i ^ 2) ≤ c)
|x|Absolute value: The size of g(i) without its sign. (|g i|)
RReal numbers: The real numbers: every point on the number line. (ℝ)
∑Summation: Add up g(i)^2 for every i. (∑ i, g i ^ 2)
aᵇPower: g(i) raised to the power 2. (g i ^ 2)

theorem bz-clip-coordinate-bound {n : ℕ} (g : Fin n → ℝ) (c : ℝ) (h : Real.sqrt (∑ i, g i ^ 2) ≤ c) (i : Fin n) :
    |g i| ≤ c

If ‖g‖₂ ≤ c then |gᵢ| ≤ c for every coordinate. Baguette clips at c = 1 (on its pre-scaled tree). Part of the plan 'Baguette-Z: closed-loop certified integer-state training'.

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: 15 lines of Lean

After global-norm clipping, every gradient coordinate is bounded by the clip norm

How it was proved

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