Provedbalanced-conversionconversion-theoremattention-compute-lawlower-bound

Weighted Cauchy–Schwarz for proposals: for a ≥ 0 and a probability proposal r, (Σa)² ≤ Σ a²/r

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ and $\htmlData{mx=3}{a, r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ and $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

\htmlData{mx=4}{\left(\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{a}(\htmlData{mx=1}{j})}}\right)^{2}} \mathrel{\htmlData{mx=14}{\leq}} \htmlData{mx=13}{\sum_{j} \htmlData{mx=12}{\frac{\htmlData{mx=8}{\htmlData{mx=7}{\htmlData{mx=5}{a}(\htmlData{mx=6}{j})}^{2}}}{\htmlData{mx=11}{\htmlData{mx=9}{r}(\htmlData{mx=10}{j})}}}}

What the notation means · or point at any symbol

Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to J. Lean writes f x; papers write f(x). (Fintype J)
⟹Implies: If J, then ℝ. (Between types, A → B is the type of functions from A to B.) (J → ℝ)
x : TVariables: a, r range over J → ℝ: the statement must hold for every choice. ((a r : J → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 < r j)
∑Summation: Add up r(j) for every j. (∑ j, r j)
=Equality: ∑ j, r(j) and 1 are the same. (∑ j, r j = 1)
aᵇPower: ∑ j, a(j) raised to the power 2. ((∑ j, a j) ^ 2)
≤At most: (∑ j, a(j))^2 is less than or equal to ∑ j, a(j)^2/r(j). ((∑ j, a j) ^ 2 ≤ ∑ j, a j ^ 2 / r j)
RReal numbers: The real numbers: every point on the number line. (ℝ)
<Less than: 0 is strictly smaller than r(j). (0 < r j)
a/bDivision: a(j)^2 divided by r(j). In Lean, dividing by 0 gives 0. (a j ^ 2 / r j)

theorem bc2-weighted-cauchy-schwarz {J : Type} [Fintype J] (a r : J → ℝ) (hr : ∀ j, 0 < r j) (hrs : ∑ j, r j = 1) :
    (∑ j, a j) ^ 2 ≤ ∑ j, a j ^ 2 / r j

The one inequality behind every proposal lower bound in the programme: a proposal cannot be good for a family of rows whose root-sum-of-squares exceeds 1. Part of Balanced Conversion: the Conversion Theorem's two-way descent factor is exactly 1 + (p−q)²/(q(1−q)), so additive oracle error is second order at mass-balanced nodes; a mass-balanced descent converts any deterministic additive-ε oracle with loss n^{O(ε²)}, and the biased product family shows this is tight. Quantifies the last residual class of the Attention Compute Law.

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: 20 lines of Lean

Weighted Cauchy–Schwarz for proposals: for a ≥ 0 and a probability proposal r, (Σa)² ≤ Σ a²/r

How it was proved

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