proofs
Provedalphabaguettebaguette-zp2norm

Scaling the energy by 4ʲ shifts the exponent by exactly j

For all S,j∈N\htmlData{mx=1}{S, j \in \htmlData{mx=0}{\mathbb{N}}}, if S≠0\htmlData{mx=0}{S} \mathrel{\htmlData{mx=1}{\neq}} 0, then

log⁡(2,4j⋅S)2=log⁡(2,S)2+j\htmlData{mx=6}{\frac{\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{log}}(2, \htmlData{mx=4}{\htmlData{mx=2}{4^{\htmlData{mx=1}{j}}} \cdot \htmlData{mx=3}{S}})}}{2}} \mathrel{\htmlData{mx=13}{=}} \htmlData{mx=12}{\htmlData{mx=10}{\frac{\htmlData{mx=9}{\htmlData{mx=7}{\operatorname{log}}(2, \htmlData{mx=8}{S})}}{2}} + \htmlData{mx=11}{j}}
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, j range over natural numbers: the statement must hold for every choice. ((S j : ℕ))
≠Not equal
S and 0 are different. (S ≠ 0)
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, 4^j · S. Lean writes f x; papers write f(x). (Nat.log 2 (4 ^ j * S))
a/bDivision
log(2, 4^j · S) divided by 2. In Lean, dividing by 0 gives 0. (Nat.log 2 (4 ^ j * S) / 2)
=Equality
log(2, 4^j · S)/2 and log(2, S)/2 + j are the same. (Nat.log 2 (4 ^ j * S) / 2 = Nat.log 2 S…)
aᵇPower
4 raised to the power j. (4 ^ j)
·Multiplication
4^j times S. (4 ^ j * S)
+Addition
log(2, S)/2 plus j. (Nat.log 2 S / 2 + j)

k(4ʲ·S) = k(S) + j. Because the exponent comes from the bit length, it commutes exactly with power-of-two rescaling. 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
23 lines of Lean
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