Provedrandomized-conversionconversion-theoremattention-compute-lawmedian-of-means

Good/bad split of an expectation: X ≤ G on a good event and X ≤ W off it gives E[X] ≤ G + δ W when P(bad) ≤ δ

For all $\htmlData{mx=1}{\Omega \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{w, X : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , $\htmlData{mx=3}{\mathit{good} : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\operatorname{Prop}}}$ and $\htmlData{mx=1}{G, W, \delta \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall \omega,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})}}$ , $\htmlData{mx=3}{\sum_{\omega} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{G}$ , $\htmlData{mx=9}{\forall \omega,\ \htmlData{mx=2}{\htmlData{mx=0}{\mathit{good}}(\htmlData{mx=1}{\omega})} \mathrel{\htmlData{mx=8}{\implies}} \htmlData{mx=5}{\htmlData{mx=3}{X}(\htmlData{mx=4}{\omega})} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{G}}$ , $\htmlData{mx=10}{\forall \omega,\ \htmlData{mx=3}{\lnot \htmlData{mx=2}{\htmlData{mx=0}{\mathit{good}}(\htmlData{mx=1}{\omega})}} \mathrel{\htmlData{mx=9}{\implies}} \htmlData{mx=6}{\htmlData{mx=4}{X}(\htmlData{mx=5}{\omega})} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{W}}$ , $\htmlData{mx=10}{\sum_{\omega \in \htmlData{mx=6}{\htmlData{mx=0}{\operatorname{filter}}(\htmlData{mx=5}{\omega \mapsto \htmlData{mx=4}{\lnot \htmlData{mx=3}{\htmlData{mx=1}{\mathit{good}}(\htmlData{mx=2}{\omega})}}})}} \htmlData{mx=9}{\htmlData{mx=7}{w}(\htmlData{mx=8}{\omega})}} \mathrel{\htmlData{mx=12}{\leq}} \htmlData{mx=11}{\delta}$ and $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{W}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\Omega}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{DecidablePred}}\,\htmlData{mx=1}{\mathit{good}}}$ ), then

\htmlData{mx=7}{\sum_{\omega} \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})} \cdot \htmlData{mx=5}{\htmlData{mx=3}{X}(\htmlData{mx=4}{\omega})}}} \mathrel{\htmlData{mx=13}{\leq}} \htmlData{mx=12}{\htmlData{mx=8}{G} + \htmlData{mx=11}{\htmlData{mx=9}{\delta} \cdot \htmlData{mx=10}{W}}}

What the notation means · or point at any symbol

Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to Ω. Lean writes f x; papers write f(x). (Fintype Ω)
⟹Implies: If Ω, then ℝ. (Between types, A → B is the type of functions from A to B.) (Ω → ℝ)
x : TVariables: w, X range over Ω → ℝ: the statement must hold for every choice. ((w X : Ω → ℝ))
DecidablePred: DecidablePred: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (DecidablePred)
x ∈ TVariables: G, W, δ range over real numbers: the statement must hold for every choice. ((G W δ : ℝ))
∀For all: The statement after the comma holds for every ω. (∀ ω, 0 ≤ w ω)
∑Summation: Add up w(ω) for every ω. (∑ ω, w ω)
=Equality: ∑ ω, w(ω) and 1 are the same. (∑ ω, w ω = 1)
RReal numbers: The real numbers: every point on the number line. (ℝ)
Prop: Prop: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Prop)
≤At most: 0 is less than or equal to w(ω). (0 ≤ w ω)
¬Not: It is not the case that good(ω). (¬ good ω)
·Multiplication: w(ω) times X(ω). (w ω * X ω)
filter: Finset.univ.filter: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ.filter)
+Addition: G plus δ · W. (G + δ * W)
x ↦ f(x)Function (λ-abstraction): The function sending ω to ¬good(ω). Lean writes fun x => …; papers write x ↦ …. (fun ω => ¬ good ω)

theorem rc-expected-chi-square-split {Ω : Type} [Fintype Ω] (w X : Ω → ℝ) (good : Ω → Prop) [DecidablePred good] (G W δ : ℝ) (hw : ∀ ω, 0 ≤ w ω) (hsum : ∑ ω, w ω = 1) (hG : 0 ≤ G) (hgood : ∀ ω, good ω → X ω ≤ G) (hbad : ∀ ω, ¬ good ω → X ω ≤ W) (hδ : ∑ ω ∈ Finset.univ.filter (fun ω => ¬ good ω), w ω ≤ δ) (hW : 0 ≤ W) :
    ∑ ω, w ω * X ω ≤ G + δ * W

The elementary split that converts a high-probability bound on a randomized oracle into an expected-cost bound. Part of the Randomized Conversion Theorem: the Conversion Theorem (counting ⇒ sampling descent) extended to every randomized bounded-variance attention algorithm via median-of-means boosting, a union bound over the 2n descent nodes, and ε-smoothing of each node; closes the residual scalability crux of the Attention Compute Law.

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: 20 lines of Lean

Good/bad split of an expectation: X ≤ G on a good event and X ≤ W off it gives E[X] ≤ G + δ W when P(bad) ≤ δ

How it was proved

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