proofs
Provedalphabaguettebaguette-zrange

With non-expanding layers, quantization error grows at most linearly: e(L) ≤ L·δ

For all e:N→R\htmlData{mx=3}{e : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}} and δ∈R\htmlData{mx=1}{\delta \in \htmlData{mx=0}{\mathbb{R}}}, if e(0)=0\htmlData{mx=1}{\htmlData{mx=0}{e}(0)} \mathrel{\htmlData{mx=2}{=}} 0 and ∀l, e(l+1)≤e(l)+δ\htmlData{mx=10}{\forall l,\ \htmlData{mx=3}{\htmlData{mx=0}{e}(\htmlData{mx=2}{\htmlData{mx=1}{l} + 1})} \mathrel{\htmlData{mx=9}{\leq}} \htmlData{mx=8}{\htmlData{mx=6}{\htmlData{mx=4}{e}(\htmlData{mx=5}{l})} + \htmlData{mx=7}{\delta}}}, then

∀L∈N, e(L)≤L⋅δ\htmlData{mx=8}{\forall L \in \htmlData{mx=7}{\mathbb{N}},\ \htmlData{mx=2}{\htmlData{mx=0}{e}(\htmlData{mx=1}{L})} \mathrel{\htmlData{mx=6}{\leq}} \htmlData{mx=5}{\htmlData{mx=3}{L} \cdot \htmlData{mx=4}{\delta}}}
What the notation means · or point at any symbol
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
⟹Implies
If ℕ, then ℝ. (Between types, A → B is the type of functions from A to B.) (ℕ → ℝ)
f(x)Function application
e applied to 0. Lean writes f x; papers write f(x). (e 0)
=Equality
e(0) and 0 are the same. (e 0 = 0)
∀For all
The statement after the comma holds for every l. (∀ l, e (l + 1) ≤ e l + δ)
RReal numbers
The real numbers: every point on the number line. (ℝ)
≤At most
e(l + 1) is less than or equal to e(l) + δ. (e (l + 1) ≤ e l + δ)
+Addition
l plus 1. (l + 1)
·Multiplication
L times δ. (L * δ)

The λ = 0 case of the Grönwall bound. 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
17 lines of Lean
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