Provedproposal-principleattentionunifiedtraining

Key-gradient scatter: E over J ~ π_i of 1{J=j}·u_J·c = π_ij u_j c

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{\pi, u : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=3}{j \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ :

\htmlData{mx=13}{\sum_{j'} \htmlData{mx=12}{\htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j'})} \cdot \left(\htmlData{mx=11}{\begin{cases} \htmlData{mx=7}{\htmlData{mx=5}{\htmlData{mx=3}{u}(\htmlData{mx=4}{j'})} \cdot \htmlData{mx=6}{c}} & \text{if } \htmlData{mx=8}{j'} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{j} \\ 0 & \text{otherwise} \end{cases}}\right)}} \mathrel{\htmlData{mx=23}{=}} \htmlData{mx=22}{\htmlData{mx=20}{\htmlData{mx=16}{\htmlData{mx=14}{\pi}(\htmlData{mx=15}{j})} \cdot \htmlData{mx=19}{\htmlData{mx=17}{u}(\htmlData{mx=18}{j})}} \cdot \htmlData{mx=21}{c}}

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, u range over Fin n → ℝ: the statement must hold for every choice. ((π u : Fin n → ℝ))
∑Summation: Add up π(j') · u(j') · c if j' = j, else 0 for every j'. (∑ j', π j' * (if j' = j then u j' * c e…)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
RReal numbers: The real numbers: every point on the number line. (ℝ)
{ a if c; b otherwiseCases (if … then … else): u(j') · c when j' = j, and 0 otherwise. (if j' = j then u j' * c else 0)
=Equality: j' and j are the same. (j' = j)
·Multiplication: u(j') times c. (u j' * c)

theorem pp-key-gradient-scatter {n : ℕ} (π u : Fin n → ℝ) (c : ℝ) (j : Fin n) :
    ∑ j', π j' * (if j' = j then u j' * c else 0) = π j * u j * c

With c = q_i and u_j = g·(v_j − ô_i) using an independent unbiased ô_i, the scattered dK_j is 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: 11 lines of Lean

Key-gradient scatter: E over J ~ π_i of 1{J=j}·u_J·c = π_ij u_j c

How it was proved

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