proofs
Provedproposal-principleattentionunifiedtraining

Tile estimator second moment E[(K T_B)²] = K Σ T_b², hence variance K Σ T_b² − (Σ T_b)² ≥ 0

For all K∈N\htmlData{mx=1}{K \in \htmlData{mx=0}{\mathbb{N}}} and T:Fin⁡ K→R\htmlData{mx=5}{T : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{K}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}, if 0<K0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{K}, then

∑b1K⋅(K⋅T(b))2=K⋅∑bT(b)2∧(∑bT(b))2≤K⋅∑bT(b)2\begin{aligned} & \htmlData{mx=12}{\sum_{b} \htmlData{mx=11}{\htmlData{mx=3}{\frac{1}{\htmlData{mx=2}{\htmlData{mx=1}{K}}}} \cdot \htmlData{mx=10}{(\htmlData{mx=9}{\htmlData{mx=5}{\htmlData{mx=4}{K}} \cdot \htmlData{mx=8}{\htmlData{mx=6}{T}(\htmlData{mx=7}{b})}})^{2}}}} \mathrel{\htmlData{mx=21}{=}} \htmlData{mx=20}{\htmlData{mx=14}{\htmlData{mx=13}{K}} \cdot \htmlData{mx=19}{\sum_{b} \htmlData{mx=18}{\htmlData{mx=17}{\htmlData{mx=15}{T}(\htmlData{mx=16}{b})}^{2}}}} \\ & \htmlData{mx=0}{\land} \htmlData{mx=26}{\left(\htmlData{mx=25}{\sum_{b} \htmlData{mx=24}{\htmlData{mx=22}{T}(\htmlData{mx=23}{b})}}\right)^{2}} \mathrel{\htmlData{mx=35}{\leq}} \htmlData{mx=34}{\htmlData{mx=28}{\htmlData{mx=27}{K}} \cdot \htmlData{mx=33}{\sum_{b} \htmlData{mx=32}{\htmlData{mx=31}{\htmlData{mx=29}{T}(\htmlData{mx=30}{b})}^{2}}}} \end{aligned}
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 K. Lean writes f x; papers write f(x). (Fin K)
⟹Implies
If Fin(K), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin K → ℝ)
<Less than
0 is strictly smaller than K. (0 < K)
∑Summation
Add up 1/K · (K · T(b))^2 for every b. (∑ b, (1 / (K : ℝ)) * ((K : ℝ) * T b) ^ 2)
=Equality
∑ b, 1/K · (K · T(b))^2 and K · (∑ b, T(b)^2) are the same. (∑ b, (1 / (K : ℝ)) * ((K : ℝ) * T b) ^ …)
∧And
Both hold: (∑ b, 1/K · (K · T(b))^2) = K · (∑ b, T(b)^2), and (∑ b, T(b))^2 ≤ K · (∑ b, T(b)^2). (∑ b, (1 / (K : ℝ)) * ((K : ℝ) * T b) ^ …)
a/bDivision
1 divided by K. In Lean, dividing by 0 gives 0. (1 / (K : ℝ))
RReal numbers
The real numbers: every point on the number line. (ℝ)
(x : T)As real numbers
K read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((K : ℝ))
aᵇPower
K · T(b) raised to the power 2. (((K : ℝ) * T b) ^ 2)
·Multiplication
K times T(b). ((K : ℝ) * T b)
≤At most
(∑ b, T(b))^2 is less than or equal to K · (∑ b, T(b)^2). ((∑ b, T b) ^ 2 ≤ (K : ℝ) * ∑ b, T b ^ 2)

Exact variance of tile sampling; a random permutation makes tile sums behave like sampling without replacement (classical finite-population correction), so tiling costs no more variance than i.i.d. key sampling in expectation. 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
16 lines of Lean
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