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

Union bound on a finite probability space: N events of mass ≤ q each have union of mass ≤ N q

For all $\htmlData{mx=1}{\Omega \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=1}{N \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=3}{w : \htmlData{mx=0}{\Omega} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , $\htmlData{mx=7}{E : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{N}} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=3}{\Omega} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\operatorname{Prop}}}$ and $\htmlData{mx=1}{q \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall \omega,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{\omega})}}$ and $\htmlData{mx=11}{\forall i,\ \htmlData{mx=8}{\sum_{\omega \in \htmlData{mx=4}{\htmlData{mx=0}{\operatorname{filter}}(\htmlData{mx=3}{\htmlData{mx=1}{E}(\htmlData{mx=2}{i})})}} \htmlData{mx=7}{\htmlData{mx=5}{w}(\htmlData{mx=6}{\omega})}} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{q}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\Omega}}$ and $\htmlData{mx=5}{\forall i,\ \htmlData{mx=4}{\htmlData{mx=0}{\operatorname{DecidablePred}}\,\htmlData{mx=3}{\htmlData{mx=1}{E}(\htmlData{mx=2}{i})}}}$ ), then

\htmlData{mx=11}{\sum_{\omega \in \htmlData{mx=7}{\htmlData{mx=0}{\operatorname{filter}}(\htmlData{mx=6}{\omega \mapsto \htmlData{mx=5}{\exists i,\ \htmlData{mx=4}{\htmlData{mx=1}{E}(\htmlData{mx=2}{i}, \htmlData{mx=3}{\omega})}}})}} \htmlData{mx=10}{\htmlData{mx=8}{w}(\htmlData{mx=9}{\omega})}} \mathrel{\htmlData{mx=15}{\leq}} \htmlData{mx=14}{\htmlData{mx=12}{N} \cdot \htmlData{mx=13}{q}}

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)
NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
⟹Implies: If Ω, then ℝ. (Between types, A → B is the type of functions from A to B.) (Ω → ℝ)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
∀For all: The statement after the comma holds for every i. (∀ i, DecidablePred (E i))
RReal numbers: The real numbers: every point on the number line. (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)
≤At most: 0 is less than or equal to w(ω). (0 ≤ w ω)
∑Summation: Add up w(ω) for each ω in filter(E(i)). (∑ ω ∈ Finset.univ.filter (E i), w 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)
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)
x ↦ f(x)Function (λ-abstraction): The function sending ω to ∃ i, E(i, ω). Lean writes fun x => …; papers write x ↦ …. (fun ω => ∃ i, E i ω)
∃There exists: At least one i makes the statement after the comma true. (∃ i, E i ω)
·Multiplication: N times q. (N * q)

theorem rc-union-bound {Ω : Type} [Fintype Ω] {N : ℕ} (w : Ω → ℝ) (E : Fin N → Ω → Prop) [∀ i, DecidablePred (E i)] (q : ℝ) (hw : ∀ ω, 0 ≤ w ω) (hE : ∀ i, ∑ ω ∈ Finset.univ.filter (E i), w ω ≤ q) :
    ∑ ω ∈ Finset.univ.filter (fun ω => ∃ i, E i ω), w ω ≤ N * q

Used over the 2n nodes of the descent tree: if each boosted node estimate fails with probability ≤ q, all nodes are simultaneously accurate except with probability ≤ 2n q. 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: 29 lines of Lean

Union bound on a finite probability space: N events of mass ≤ q each have union of mass ≤ N q

How it was proved

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