Provedvariance-share-lawtraining-energystochastic-attentionlaw

Variance-Share Law, attainment: at b*² σ² = A c₀ the energy equals (√(σ²c₀) + √A)²

For all $\htmlData{mx=1}{\sigma_{2}, A, c_{0}, b \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\sigma_{2}}$ , $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{A}$ , $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{c_{0}}$ , $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{b}$ and $\htmlData{mx=3}{\htmlData{mx=1}{\htmlData{mx=0}{b}^{2}} \cdot \htmlData{mx=2}{\sigma_{2}}} \mathrel{\htmlData{mx=7}{=}} \htmlData{mx=6}{\htmlData{mx=4}{A} \cdot \htmlData{mx=5}{c_{0}}}$ , then

\htmlData{mx=8}{\left(\htmlData{mx=4}{\htmlData{mx=0}{\sigma_{2}} + \htmlData{mx=3}{\frac{\htmlData{mx=1}{A}}{\htmlData{mx=2}{b}}}}\right) \cdot (\htmlData{mx=7}{\htmlData{mx=5}{c_{0}} + \htmlData{mx=6}{b}})} \mathrel{\htmlData{mx=17}{=}} \htmlData{mx=16}{\left(\htmlData{mx=15}{\htmlData{mx=12}{\sqrt{\htmlData{mx=11}{\htmlData{mx=9}{\sigma_{2}} \cdot \htmlData{mx=10}{c_{0}}}}} + \htmlData{mx=14}{\sqrt{\htmlData{mx=13}{A}}}}\right)^{2}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: σ2, A, c0, b range over real numbers: the statement must hold for every choice. ((σ2 A c0 b : ℝ))
<Less than: 0 is strictly smaller than σ2. (0 < σ2)
≤At most: 0 is less than or equal to A. (0 ≤ A)
aᵇPower: b raised to the power 2. (b ^ 2)
·Multiplication: b^2 times σ2. (b ^ 2 * σ2)
=Equality: b^2 · σ2 and A · c0 are the same. (b ^ 2 * σ2 = A * c0)
+Addition: σ2 plus A/b. (σ2 + A / b)
a/bDivision: A divided by b. In Lean, dividing by 0 gives 0. (A / b)
√Square root: The non-negative number whose square is σ2 · c0. (Real.sqrt (σ2 * c0))

theorem vs-energy-optimal (σ2 A c0 b : ℝ) (hσ : 0 < σ2) (hA : 0 ≤ A) (hc : 0 < c0) (hb : 0 < b) (hopt : b ^ 2 * σ2 = A * c0) :
    (σ2 + A / b) * (c0 + b) = (Real.sqrt (σ2 * c0) + Real.sqrt A) ^ 2

The optimal sampling budget is b* = √(A c₀/σ²): it grows with the attention variance constant and the non-attention cost, and shrinks with the intrinsic noise. Part of 'Variance-Share Law: compute-optimal sampling budgets for stochastic attention in training'.

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: 22 lines of Lean

Variance-Share Law, attainment: at b*² σ² = A c₀ the energy equals (√(σ²c₀) + √A)²

How it was proved

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