Provedalphabaguettebaguette-zp2norm

The normalization exponent is monotone in the energy

For all $\htmlData{mx=1}{S, S' \in \htmlData{mx=0}{\mathbb{N}}}$ , if $\htmlData{mx=0}{S} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{S'}$ , then

\htmlData{mx=3}{\frac{\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{log}}(2, \htmlData{mx=1}{S})}}{2}} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\frac{\htmlData{mx=6}{\htmlData{mx=4}{\operatorname{log}}(2, \htmlData{mx=5}{S'})}}{2}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables: S, S' range over natural numbers: the statement must hold for every choice. ((S S' : ℕ))
≤At most: S is less than or equal to S'. (S ≤ S')
log: Nat.log: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Nat.log)
f(x)Function application: Nat.log applied to 2, S. Lean writes f x; papers write f(x). (Nat.log 2 S)
a/bDivision: log(2, S) divided by 2. In Lean, dividing by 0 gives 0. (Nat.log 2 S / 2)

theorem bz-exponent-monotone (S S' : ℕ) (h : S ≤ S') :
    Nat.log 2 S / 2 ≤ Nat.log 2 S' / 2

S ≤ S' implies k(S) ≤ k(S'): larger rows are never scaled up relative to smaller ones. 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: 12 lines of Lean

The normalization exponent is monotone in the energy

How it was proved

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