Provedproposal-principleattentionunifiedtraining

Backward scatter: one exact sample J ~ π_i scattering g_i into dV_J is unbiased: E[1{J=j} g_i] = π_ij g_i

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

\htmlData{mx=9}{\sum_{j'} \htmlData{mx=8}{\htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j'})} \cdot \left(\htmlData{mx=7}{\begin{cases} \htmlData{mx=3}{g} & \text{if } \htmlData{mx=4}{j'} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{j} \\ 0 & \text{otherwise} \end{cases}}\right)}} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\htmlData{mx=12}{\htmlData{mx=10}{\pi}(\htmlData{mx=11}{j})} \cdot \htmlData{mx=13}{g}}

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

theorem pp-backward-scatter-unbiased {n : ℕ} (π : Fin n → ℝ) (g : ℝ) (j : Fin n) :
    ∑ j', π j' * (if j' = j then g else 0) = π j * g

The value gradient dV_j = Σ_i π_ij g_i is a column sum, yet needs no column proposals: the forward row samples, scattered, estimate it unbiasedly. One proposal pass serves forward, dQ, dK and dV. 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: 10 lines of Lean

Backward scatter: one exact sample J ~ π_i scattering g_i into dV_J is unbiased: E[1{J=j} g_i] = π_ij g_i

How it was proved

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