Provedproposal-principleattentionunifiedprinciple

The average target never costs more than the worst-row target: (Σ√V)²/n ≤ Σ V

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

\htmlData{mx=6}{\frac{\htmlData{mx=4}{\left(\htmlData{mx=3}{\sum_{i} \htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{i})}}\right)^{2}}}{\htmlData{mx=5}{n}}} \mathrel{\htmlData{mx=12}{\leq}} \htmlData{mx=11}{\sum_{i} \htmlData{mx=10}{\htmlData{mx=9}{\htmlData{mx=7}{s}(\htmlData{mx=8}{i})}^{2}}}

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)
aᵇPower: ∑ i, s(i) raised to the power 2. ((∑ i, s i) ^ 2)
a/bDivision: (∑ i, s(i))^2 divided by n. In Lean, dividing by 0 gives 0. ((∑ i, s i) ^ 2 / n)
≤At most: (∑ i, s(i))^2/n is less than or equal to ∑ i, s(i)^2. ((∑ i, s i) ^ 2 / n ≤ ∑ i, s i ^ 2)
∑Summation: Add up s(i) for every i. (∑ i, s i)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem pp-average-target-cheaper {n : ℕ} (s : Fin n → ℝ) (hn : 0 < n) :
    (∑ i, s i) ^ 2 / n ≤ ∑ i, s i ^ 2

Cauchy–Schwarz. Aggregate targets live on the (row, key) pair index set, where row subsampling is itself a proposal — so the round-15 refutation fits inside the target-relative Principle. 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

The average target never costs more than the worst-row target: (Σ√V)²/n ≤ Σ V

How it was proved

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