proofs
Provedalphabaguettebaguette-zp2norm

The integer bit length brackets S: 2^log₂S ≤ S < 2^(log₂S+1)

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

2log⁡(2,S)≤S∧S<2log⁡(2,S)+1\htmlData{mx=3}{2^{\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{log}}(2, \htmlData{mx=1}{S})}}} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{S} \mathbin{\htmlData{mx=13}{\land}} \htmlData{mx=6}{S} \mathrel{\htmlData{mx=12}{<}} \htmlData{mx=11}{2^{\htmlData{mx=10}{\htmlData{mx=9}{\htmlData{mx=7}{\operatorname{log}}(2, \htmlData{mx=8}{S})} + 1}}}
What the notation means · or point at any symbol
NNatural numbers
The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
≠Not equal
S and 0 are different. (S ≠ 0)
aᵇPower
2 raised to the power log(2, S). (2 ^ Nat.log 2 S)
≤At most
2^log(2, S) is less than or equal to S. (2 ^ Nat.log 2 S ≤ S)
∧And
Both hold: 2^log(2, S) ≤ S, and S < 2^(log(2, S) + 1). (2 ^ Nat.log 2 S ≤ S ∧ S < 2 ^ (Nat.log …)
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)
<Less than
S is strictly smaller than 2^(log(2, S) + 1). (S < 2 ^ (Nat.log 2 S + 1))
+Addition
log(2, S) plus 1. (Nat.log 2 S + 1)

Nat.log 2 S is a count-leading-zeros instruction. The normalization exponent is computed exactly with integer operations, with no square root or division. 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
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