proofs
Provedproposal-principleattentionunifiedprinciple

Average-row target allocation: budgets b_i ∝ √V_i give average MSE exactly ε at total budget (Σ√V_i)²/(nε)

For all n∈N\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}, s:Fin⁡ n→R\htmlData{mx=5}{s : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}} and ε∈R\htmlData{mx=1}{\varepsilon \in \htmlData{mx=0}{\mathbb{R}}}, if 0<n0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}, ∀i, 0<s(i)\htmlData{mx=4}{\forall i,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{i})}} and 0<ε0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\varepsilon}, then

1n⋅∑is(i)2s(i)⋅(∑ks(k))n⋅ε=ε∧∑is(i)⋅(∑ks(k))n⋅ε=(∑ks(k))2n⋅ε\begin{aligned} & \htmlData{mx=22}{\htmlData{mx=3}{\frac{1}{\htmlData{mx=2}{\htmlData{mx=1}{n}}}} \cdot \htmlData{mx=21}{\sum_{i} \htmlData{mx=20}{\frac{\htmlData{mx=7}{\htmlData{mx=6}{\htmlData{mx=4}{s}(\htmlData{mx=5}{i})}^{2}}}{\htmlData{mx=19}{\frac{\htmlData{mx=15}{\htmlData{mx=10}{\htmlData{mx=8}{s}(\htmlData{mx=9}{i})} \cdot \left(\htmlData{mx=14}{\sum_{k} \htmlData{mx=13}{\htmlData{mx=11}{s}(\htmlData{mx=12}{k})}}\right)}}{\htmlData{mx=18}{\htmlData{mx=16}{n} \cdot \htmlData{mx=17}{\varepsilon}}}}}}}} \mathrel{\htmlData{mx=24}{=}} \htmlData{mx=23}{\varepsilon} \\ & \htmlData{mx=0}{\land} \htmlData{mx=37}{\sum_{i} \htmlData{mx=36}{\frac{\htmlData{mx=32}{\htmlData{mx=27}{\htmlData{mx=25}{s}(\htmlData{mx=26}{i})} \cdot \left(\htmlData{mx=31}{\sum_{k} \htmlData{mx=30}{\htmlData{mx=28}{s}(\htmlData{mx=29}{k})}}\right)}}{\htmlData{mx=35}{\htmlData{mx=33}{n} \cdot \htmlData{mx=34}{\varepsilon}}}}} \mathrel{\htmlData{mx=47}{=}} \htmlData{mx=46}{\frac{\htmlData{mx=42}{\left(\htmlData{mx=41}{\sum_{k} \htmlData{mx=40}{\htmlData{mx=38}{s}(\htmlData{mx=39}{k})}}\right)^{2}}}{\htmlData{mx=45}{\htmlData{mx=43}{n} \cdot \htmlData{mx=44}{\varepsilon}}}} \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 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)
<Less than
0 is strictly smaller than n. (0 < n)
∀For all
The statement after the comma holds for every i. (∀ i, 0 < s i)
=Equality
1/n · (∑ i, s(i)^2/(s(i) · (∑ k, s(k))/(n · ε))) and ε are the same. ((1 / (n : ℝ)) * ∑ i, s i ^ 2 / (s i * (…)
∧And
Both hold: 1/n · (∑ i, s(i)^2/(s(i) · (∑ k, s(k))/(n · ε))) = ε, and (∑ i, s(i) · (∑ k, s(k))/(n · ε)) = (∑ k, s(k))^2/(n · ε). ((1 / (n : ℝ)) * ∑ i, s i ^ 2 / (s i * (…)
a/bDivision
1 divided by n. In Lean, dividing by 0 gives 0. (1 / (n : ℝ))
(x : T)As real numbers
n read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((n : ℝ))
RReal numbers
The real numbers: every point on the number line. (ℝ)
aᵇPower
s(i) raised to the power 2. (s i ^ 2)
·Multiplication
s(i) times ∑ k, s(k). (s i * (∑ k, s k))
∑Summation
Add up s(k) for every k. (∑ k, s k)

Disproval Loop 3 repair: the Proposal Principle is target-relative. For an average-MSE target the optimal per-row sample budgets are proportional to √(1+χ²_i), and the cost is (Σ_i √(1+χ²_i))²/(nε) rather than Σ_i (1+χ²_i)/ε. 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
19 lines of Lean
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