proofs
Provedalphabaguettebaguette-zoptimizer

If m² ≤ C²v then the Adam ratio |m/(√v + ε)| ≤ C

For all m,v,ε,C∈R\htmlData{mx=1}{m, v, \varepsilon, C \in \htmlData{mx=0}{\mathbb{R}}}, if 0<ε0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\varepsilon}, 0≤C0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{C} and m2≤C2⋅v\htmlData{mx=1}{\htmlData{mx=0}{m}^{2}} \mathrel{\htmlData{mx=6}{\leq}} \htmlData{mx=5}{\htmlData{mx=3}{\htmlData{mx=2}{C}^{2}} \cdot \htmlData{mx=4}{v}}, then

∣mv+ε∣≤C\htmlData{mx=6}{\left\lvert \htmlData{mx=5}{\frac{\htmlData{mx=0}{m}}{\htmlData{mx=4}{\htmlData{mx=2}{\sqrt{\htmlData{mx=1}{v}}} + \htmlData{mx=3}{\varepsilon}}}} \right\rvert} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{C}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
m, v, ε, C range over real numbers: the statement must hold for every choice. ((m v ε C : ℝ))
<Less than
0 is strictly smaller than ε. (0 < ε)
≤At most
0 is less than or equal to C. (0 ≤ C)
aᵇPower
m raised to the power 2. (m ^ 2)
|x|Absolute value
The size of m/(√(v) + ε) without its sign. (|m / (Real.sqrt v + ε)|)
a/bDivision
m divided by √(v) + ε. In Lean, dividing by 0 gives 0. (m / (Real.sqrt v + ε))
√Square root
The non-negative number whose square is v. (Real.sqrt v)
+Addition
√(v) plus ε. (Real.sqrt v + ε)
·Multiplication
C^2 times v. (C ^ 2 * v)

The per-coordinate Adam direction is bounded by C regardless of gradient scale; ε only makes it smaller. 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
18 lines of Lean
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