Provedproposal-principleattentionunifiedcrux

Bounded ratio ⇒ bounded χ²: π_j ≤ M r_j for all j gives Σ π²/r ≤ M

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{\pi, r : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{M \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ and $\htmlData{mx=9}{\forall j,\ \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\htmlData{mx=3}{M} \cdot \htmlData{mx=6}{\htmlData{mx=4}{r}(\htmlData{mx=5}{j})}}}$ , 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=10}{\leq}} \htmlData{mx=9}{M}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
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)
f(x)Function application: Fin applied to n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
x : TVariables: π, r range over Fin n → ℝ: the statement must hold for every choice. ((π r : Fin n → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 < r 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)
<Less than: 0 is strictly smaller than r(j). (0 < r j)
≤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)
RReal numbers: The real numbers: every point on the number line. (ℝ)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
·Multiplication: M times r(j). (M * r j)

theorem cv-ratio-to-chi-square {n : ℕ} (π r : Fin n → ℝ) (M : ℝ) (hr : ∀ j, 0 < r j) (hπ : ∀ j, 0 ≤ π j) (hsum : ∑ j, π j = 1) (hM : ∀ j, π j ≤ M * r j) :
    ∑ j, π j ^ 2 / r j ≤ M

So the induced proposal has 1 + χ² ≤ (1 + 2ε)^{log₂ n}; SNIS + MLMC then gives unbiased attention without the normaliser. Part of 'The Proposal Principle: attention complexity is proposal complexity'.

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: 14 lines of Lean

Bounded ratio ⇒ bounded χ²: π_j ≤ M r_j for all j gives Σ π²/r ≤ M

How it was proved

More proved statements