Provedalphabaguettebaguette-zv2

Re-scoped: Baguette's existing FP32 master weights never leave [−730, 730], for any schedule

For all $\htmlData{mx=3}{\theta, u, \eta : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{K \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=0}{K} \mathrel{\htmlData{mx=1}{\leq}} 73$ , $\htmlData{mx=4}{\forall t,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\eta}(\htmlData{mx=1}{t})}}$ , $\htmlData{mx=6}{\forall t,\ \htmlData{mx=4}{\htmlData{mx=2}{\htmlData{mx=0}{\eta}(\htmlData{mx=1}{t})} \cdot \left(\htmlData{mx=3}{\frac{1}{10}}\right)} \mathrel{\htmlData{mx=5}{\leq}} 1}$ , $\htmlData{mx=6}{\forall t,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{u}(\htmlData{mx=1}{t})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{K}}$ , $\htmlData{mx=2}{\left\lvert \htmlData{mx=1}{\htmlData{mx=0}{\theta}(0)} \right\rvert} \mathrel{\htmlData{mx=3}{\leq}} 1$ and $\htmlData{mx=23}{\forall t,\ \htmlData{mx=3}{\htmlData{mx=0}{\theta}(\htmlData{mx=2}{\htmlData{mx=1}{t} + 1})} \mathrel{\htmlData{mx=22}{=}} \htmlData{mx=21}{\htmlData{mx=13}{\left(\htmlData{mx=9}{1 - \htmlData{mx=8}{\htmlData{mx=6}{\htmlData{mx=4}{\eta}(\htmlData{mx=5}{t})} \cdot \left(\htmlData{mx=7}{\frac{1}{10}}\right)}}\right) \cdot \htmlData{mx=12}{\htmlData{mx=10}{\theta}(\htmlData{mx=11}{t})}} - \htmlData{mx=20}{\htmlData{mx=16}{\htmlData{mx=14}{\eta}(\htmlData{mx=15}{t})} \cdot \htmlData{mx=19}{\htmlData{mx=17}{u}(\htmlData{mx=18}{t})}}}}$ , then

\htmlData{mx=9}{\forall t,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{\theta}(\htmlData{mx=1}{t})} \right\rvert} \mathrel{\htmlData{mx=4}{\leq}} 730 \mathbin{\htmlData{mx=8}{\land}} \htmlData{mx=5}{730} \mathrel{\htmlData{mx=7}{<}} \htmlData{mx=6}{2^{127}}}

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)
x : TVariables: θ, u, η range over ℕ → ℝ: the statement must hold for every choice. ((θ u η : ℕ → ℝ))
≤At most: K is less than or equal to 73. (K ≤ 73)
∀For all: The statement after the comma holds for every t. (∀ t, 0 ≤ η t)
RReal numbers: The real numbers: every point on the number line. (ℝ)
·Multiplication: η(t) times 1/10. (η t * (1 / 10))
|x|Absolute value: The size of u(t) without its sign. (|u t|)
=Equality: θ(t + 1) and (1 − η(t) · (1/10)) · θ(t) − η(t) · u(t) are the same. (θ (t + 1) = (1 - η t * (1 / 10)) * θ t …)
∧And: Both hold: |θ(t)| ≤ 730, and 730 < 2^127. (|θ t| ≤ 730 ∧ (730 : ℝ) < 2 ^ 127)
+Addition: t plus 1. (t + 1)
f(x)Function application: η applied to t. Lean writes f x; papers write f(x). (η t)
a/bDivision: 1 divided by 10. In Lean, dividing by 0 gives 0. (1 / 10)
-Subtraction: 1 minus η(t) · (1/10). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 - η t * (1 / 10))
(x : T)As real numbers: 730 read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((730 : ℝ))
<Less than: 730 is strictly smaller than 2^127. ((730 : ℝ) < 2 ^ 127)
aᵇPower: 2 raised to the power 127. (2 ^ 127)

theorem bz2-fp32-state-in-range (θ u η : ℕ → ℝ) (K : ℝ) (hK : K ≤ 73) (hη : ∀ t, 0 ≤ η t) (hηl : ∀ t, η t * (1 / 10) ≤ 1) (hu : ∀ t, |u t| ≤ K) (hθ0 : |θ 0| ≤ 1) (hθ : ∀ t, θ (t + 1) = (1 - η t * (1 / 10)) * θ t - η t * u t) :
    ∀ t, |θ t| ≤ 730 ∧ (730 : ℝ) < 2 ^ 127

Answers M1 honestly. With clipping and the Adam ratio the per-step direction is ≤ K ≤ 73. With Baguette's decay λ = 0.1 and |θ₀| ≤ 1, every FP32 master weight stays within 730 for the whole run, under any learning-rate schedule. That is far inside FP32's range (> 2¹²⁷). Integer state is NOT needed for overflow safety. Part of 'Baguette-Z v2' (revision after the hostile review).

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: 35 lines of Lean

Re-scoped: Baguette's existing FP32 master weights never leave [−730, 730], for any schedule

How it was proved

More proved statements