Provedalphabaguettebaguette-zrange

A projection's output is bounded by its row ℓ₁ norm times the input range

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{W, z : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{Z \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{z}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{Z}}$ , then

\htmlData{mx=8}{\left\lvert \htmlData{mx=7}{\sum_{j} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{W}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{z}(\htmlData{mx=4}{j})}}} \right\rvert} \mathrel{\htmlData{mx=16}{\leq}} \htmlData{mx=15}{\left(\htmlData{mx=13}{\sum_{j} \htmlData{mx=12}{\left\lvert \htmlData{mx=11}{\htmlData{mx=9}{W}(\htmlData{mx=10}{j})} \right\rvert}}\right) \cdot \htmlData{mx=14}{Z}}

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

theorem bz-row-sum-bound {n : ℕ} (W z : Fin n → ℝ) (Z : ℝ) (hz : ∀ j, |z j| ≤ Z) :
    |∑ j, W j * z j| ≤ (∑ j, |W j|) * Z

|Σⱼ Wⱼ zⱼ| ≤ (Σⱼ|Wⱼ|)·Z when |zⱼ| ≤ Z. Combined with |z| < 2 after power-of-two normalization, every projection output is bounded by twice a row norm that the optimizer can compute. 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: 16 lines of Lean

A projection's output is bounded by its row ℓ₁ norm times the input range

How it was proved

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