proofs
Provedbatch-sample-lawstochastic-attentionbatch-sizelaw

The optimal budget b* = √(C/(B + B_noise)) strictly decreases with the batch size B

For all C,Bn,B1,B2∈R\htmlData{mx=1}{C, \mathit{Bn}, B_{1}, B_{2} \in \htmlData{mx=0}{\mathbb{R}}}, if 0<C0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{C}, 0<B1+Bn0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{B_{1}} + \htmlData{mx=1}{\mathit{Bn}}} and B1<B2\htmlData{mx=0}{B_{1}} \mathrel{\htmlData{mx=2}{<}} \htmlData{mx=1}{B_{2}}, then

CB2+Bn<CB1+Bn\htmlData{mx=5}{\sqrt{\htmlData{mx=4}{\frac{\htmlData{mx=0}{C}}{\htmlData{mx=3}{\htmlData{mx=1}{B_{2}} + \htmlData{mx=2}{\mathit{Bn}}}}}}} \mathrel{\htmlData{mx=12}{<}} \htmlData{mx=11}{\sqrt{\htmlData{mx=10}{\frac{\htmlData{mx=6}{C}}{\htmlData{mx=9}{\htmlData{mx=7}{B_{1}} + \htmlData{mx=8}{\mathit{Bn}}}}}}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
C, Bn, B1, B2 range over real numbers: the statement must hold for every choice. ((C Bn B1 B2 : ℝ))
<Less than
0 is strictly smaller than C. (0 < C)
√Square root
The non-negative number whose square is C/(B2 + Bn). (Real.sqrt (C / (B2 + Bn)))
+Addition
B1 plus Bn. (B1 + Bn)
a/bDivision
C divided by B2 + Bn. In Lean, dividing by 0 gives 0. (C / (B2 + Bn))

The Batch–Sample Law: large-batch training tolerates noisier attention; quadrupling the batch (B ≫ B_noise) halves the optimal attention-sample budget. Part of 'Batch–Sample Law: the optimal attention-sample budget shrinks as the batch grows'.

How it was proved

Proved
4 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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