Provedproposal-principleattentionunifiedtraining

Permuted-tile sampling is unbiased: pick one of K key tiles uniformly and scale its sum by K

For all $\htmlData{mx=1}{n, K \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=7}{\mathit{tile} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{K}}}$ and $\htmlData{mx=5}{x : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{K}$ , then

\htmlData{mx=19}{\sum_{b} \htmlData{mx=18}{\htmlData{mx=2}{\frac{1}{\htmlData{mx=1}{\htmlData{mx=0}{K}}}} \cdot \left(\htmlData{mx=17}{\htmlData{mx=4}{\htmlData{mx=3}{K}} \cdot \htmlData{mx=16}{\sum_{j \in \htmlData{mx=12}{\htmlData{mx=5}{\operatorname{filter}}(\htmlData{mx=11}{j \mapsto \htmlData{mx=8}{\htmlData{mx=6}{\mathit{tile}}(\htmlData{mx=7}{j})} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{b}})}} \htmlData{mx=15}{\htmlData{mx=13}{x}(\htmlData{mx=14}{j})}}}\right)}} \mathrel{\htmlData{mx=24}{=}} \htmlData{mx=23}{\sum_{j} \htmlData{mx=22}{\htmlData{mx=20}{x}(\htmlData{mx=21}{j})}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables: n, K range over natural numbers: the statement must hold for every choice. ((n K : 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 Fin(K). (Between types, A → B is the type of functions from A to B.) (Fin n → Fin K)
<Less than: 0 is strictly smaller than K. (0 < K)
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))
·Multiplication: K times ∑ j, x(j). ((K : ℝ) * ∑ j ∈ Finset.univ.filter (fun…)
∑Summation: Add up x(j) for each j in filter(j ↦ tile(j) = b). (∑ j ∈ Finset.univ.filter (fun j => tile…)
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)
x ↦ f(x)Function (λ-abstraction): The function sending j to tile(j) = b. Lean writes fun x => …; papers write x ↦ …. (fun j => tile j = b)
=Equality: tile(j) and b are the same. (tile j = b)

theorem pp-tile-sampling-unbiased {n K : ℕ} (tile : Fin n → Fin K) (x : Fin n → ℝ) (hK : 0 < K) :
    ∑ b, (1 / (K : ℝ)) * ((K : ℝ) * ∑ j ∈ Finset.univ.filter (fun j => tile j = b), x j) = ∑ j, x j

Energy-aware proposal for GPUs (Disproval Loop 2): permute keys once, sample contiguous 64-key tiles. Coalesced reads make the random-access penalty γ ≈ 1 while the estimator stays unbiased. 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

Permuted-tile sampling is unbiased: pick one of K key tiles uniformly and scale its sum by K

How it was proved

More proved statements