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

Worst-case chi-square: if the proposal is ≥ f everywhere, Σπ²/r ≤ 1/f

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{\pi, r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{f \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{f}$ , $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $\htmlData{mx=4}{\forall j,\ \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})} \mathrel{\htmlData{mx=3}{\leq}} 1}$ and $\htmlData{mx=5}{\forall j,\ \htmlData{mx=0}{f} \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=1}{r}(\htmlData{mx=2}{j})}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

\htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\frac{\htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}^{2}}}{\htmlData{mx=6}{\htmlData{mx=4}{r}(\htmlData{mx=5}{j})}}}} \mathrel{\htmlData{mx=11}{\leq}} \htmlData{mx=10}{\frac{1}{\htmlData{mx=9}{f}}}

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 J. Lean writes f x; papers write f(x). (Fintype J)
⟹Implies: If J, then ℝ. (Between types, A → B is the type of functions from A to B.) (J → ℝ)
x : TVariables: π, r range over J → ℝ: the statement must hold for every choice. ((π r : J → ℝ))
<Less than: 0 is strictly smaller than f. (0 < f)
∀For all: The statement after the comma holds for every j. (∀ j, 0 ≤ π j)
∑Summation: Add up π(j) for every j. (∑ j, π j)
=Equality: ∑ j, π(j) and 1 are the same. (∑ j, π j = 1)
RReal numbers: The real numbers: every point on the number line. (ℝ)
≤At most: 0 is less than or equal to π(j). (0 ≤ π j)
aᵇPower: π(j) raised to the power 2. (π j ^ 2)
a/bDivision: π(j)^2 divided by r(j). In Lean, dividing by 0 gives 0. (π j ^ 2 / r j)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)

theorem rc-chi-square-worst-case {J : Type} [Fintype J] (π r : J → ℝ) (f : ℝ) (hf : 0 < f) (hπ : ∀ j, 0 ≤ π j) (hπs : ∑ j, π j = 1) (hπ1 : ∀ j, π j ≤ 1) (hr : ∀ j, f ≤ r j) :
    ∑ j, π j ^ 2 / r j ≤ 1 / f

The chi-square cost factor of any proposal floored by f is at most 1/f; with f = (ε/(1+2ε))^L this is the bad-event 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: 17 lines of Lean

Worst-case chi-square: if the proposal is ≥ f everywhere, Σπ²/r ≤ 1/f

How it was proved

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