Provedalphabaguettebaguette-zquantizer

Round-half-up ⌊x + 1/2⌋ is within 1/2 of x

For all $\htmlData{mx=1}{x \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=6}{\left\lvert \htmlData{mx=5}{\htmlData{mx=0}{x} - \htmlData{mx=4}{\left\lfloor \htmlData{mx=3}{\htmlData{mx=1}{x} + \htmlData{mx=2}{\frac{1}{2}}} \right\rfloor}} \right\rvert} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\frac{1}{2}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
|x|Absolute value: The size of x − ⌊x + 1/2⌋ without its sign. (|x - ⌊x + 1 / 2⌋|)
≤At most: |x − ⌊x + 1/2⌋| is less than or equal to 1/2. (|x - ⌊x + 1 / 2⌋| ≤ 1 / 2)
-Subtraction: x minus ⌊x + 1/2⌋. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (x - ⌊x + 1 / 2⌋)
⌊x⌋Floor: The largest integer not above x + 1/2. (⌊x + 1 / 2⌋)
+Addition: x plus 1/2. (x + 1 / 2)
a/bDivision: 1 divided by 2. In Lean, dividing by 0 gives 0. (1 / 2)

theorem bz-nearest-rounding-error (x : ℝ) :
    |x - ⌊x + 1 / 2⌋| ≤ 1 / 2

The nearest-rounding variant of the quantizer halves the worst-case error; used for sublayer increments. 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: 15 lines of Lean

Round-half-up ⌊x + 1/2⌋ is within 1/2 of x

How it was proved

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