Provedproposal-principleattentionunifiedtraining

Summing scattered row contributions over rows reproduces the column sum Σ_i π_ij g_i

For all $\htmlData{mx=1}{m, n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=9}{\pi : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{m}} \mathrel{\htmlData{mx=8}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{n}} \mathrel{\htmlData{mx=7}{\to}} \htmlData{mx=6}{\mathbb{R}}}$ , $\htmlData{mx=5}{g : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{m}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=3}{j \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ :

\begin{aligned} & \htmlData{mx=14}{\sum_{i} \htmlData{mx=13}{\sum_{j'} \htmlData{mx=12}{\htmlData{mx=4}{\htmlData{mx=1}{\pi}(\htmlData{mx=2}{i}, \htmlData{mx=3}{j'})} \cdot \left(\htmlData{mx=11}{\begin{cases} \htmlData{mx=7}{\htmlData{mx=5}{g}(\htmlData{mx=6}{i})} & \text{if } \htmlData{mx=8}{j'} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{j} \\ 0 & \text{otherwise} \end{cases}}\right)}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=23}{\sum_{i} \htmlData{mx=22}{\htmlData{mx=18}{\htmlData{mx=15}{\pi}(\htmlData{mx=16}{i}, \htmlData{mx=17}{j})} \cdot \htmlData{mx=21}{\htmlData{mx=19}{g}(\htmlData{mx=20}{i})}}} \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)
x ∈ TVariables: m, n range over natural numbers: the statement must hold for every choice. ((m n : 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 m. Lean writes f x; papers write f(x). (Fin m)
∑Summation: Add up π(i, j') · g(i) if j' = j, else 0 for every j'. (∑ j', π i j' * (if j' = j then g i else…)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
·Multiplication: π(i, j') times g(i) if j' = j, else 0. (π i j' * (if j' = j then g i else 0))
RReal numbers: The real numbers: every point on the number line. (ℝ)
{ a if c; b otherwiseCases (if … then … else): g(i) when j' = j, and 0 otherwise. (if j' = j then g i else 0)
=Equality: j' and j are the same. (j' = j)

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

Completes the training-step coverage of the Proposal Principle: the full forward + backward attention step is estimated unbiasedly from row proposals alone. 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

Summing scattered row contributions over rows reproduces the column sum Σ_i π_ij g_i

How it was proved

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