Provedvariance-share-lawtraining-energystochastic-attentionlaw

At the optimum, attention's variance share equals sampling's cost share: (A/b)/σ² = b/c₀

For all $\htmlData{mx=1}{\sigma_{2}, A, c_{0}, b \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\sigma_{2}}$ , $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{c_{0}}$ , $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{b}$ 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=4}{\frac{\htmlData{mx=2}{\frac{\htmlData{mx=0}{A}}{\htmlData{mx=1}{b}}}}{\htmlData{mx=3}{\sigma_{2}}}} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{\frac{\htmlData{mx=5}{b}}{\htmlData{mx=6}{c_{0}}}}

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)
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)
a/bDivision: A divided by b. In Lean, dividing by 0 gives 0. (A / b)

theorem vs-variance-share-equals-cost-share (σ2 A c0 b : ℝ) (hσ : 0 < σ2) (hc : 0 < c0) (hb : 0 < b) (hopt : b ^ 2 * σ2 = A * c0) :
    (A / b) / σ2 = b / c0

A tuning rule for stochastic-attention training: increase samples until the fraction of step cost spent on attention sampling equals the ratio of attention-induced to intrinsic gradient variance. The mathematics is the standard AM–GM / MLMC allocation; the contribution is the framing for attention budgets and energy-to-target. 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: 11 lines of Lean

At the optimum, attention's variance share equals sampling's cost share: (A/b*)/σ² = b*/c₀

How it was proved

More proved statements

At the optimum, attention's variance share equals sampling's cost share: (A/b)/σ² = b/c₀