Provedbalanced-conversionconversion-theoremattention-compute-lawlower-bound

Uniform proposal inside a block of m keys with proposal mass ρ contributes Σπ²/(ρ/m) ≤ m (Σπ)²/ρ

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

\htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\frac{\htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}^{2}}}{\htmlData{mx=6}{\frac{\htmlData{mx=4}{\rho}}{\htmlData{mx=5}{m}}}}}} \mathrel{\htmlData{mx=18}{\leq}} \htmlData{mx=17}{\frac{\htmlData{mx=15}{\htmlData{mx=9}{m} \cdot \htmlData{mx=14}{\left(\htmlData{mx=13}{\sum_{j} \htmlData{mx=12}{\htmlData{mx=10}{\pi}(\htmlData{mx=11}{j})}}\right)^{2}}}}{\htmlData{mx=16}{\rho}}}

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)
NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
∀For all: The statement after the comma holds for every j. (∀ j, 0 ≤ π j)
<Less than: 0 is strictly smaller than ρ. (0 < ρ)
∑Summation: Add up π(j)^2/(ρ/m) for every j. (∑ j, π j ^ 2 / (ρ / m))
RReal numbers: The real numbers: every point on the number line. (ℝ)
≤At most: 0 is less than or equal to π(j). (0 ≤ π j)
aᵇPower: π(j) raised to the power 2. (π j ^ 2)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
a/bDivision: ρ divided by m. In Lean, dividing by 0 gives 0. (ρ / m)
·Multiplication: m times (∑ j, π(j))^2. (m * (∑ j, π j) ^ 2)

theorem bc2-uniform-block-contribution {J : Type} [Fintype J] (π : J → ℝ) (ρ : ℝ) (m : ℕ) (hπ : ∀ j, 0 ≤ π j) (hρ : 0 < ρ) (hm : 0 < m) :
    ∑ j, π j ^ 2 / (ρ / m) ≤ m * (∑ j, π j) ^ 2 / ρ

With block mass ≤ n^{−3} and ρ ≥ (block mass)/K the truncated blocks contribute at most K m n^{−3} each, O(1) in total; closes the truncation step of the n^{O(ε²)} bound. 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: 19 lines of Lean

Uniform proposal inside a block of m keys with proposal mass ρ contributes Σπ²/(ρ/m) ≤ m (Σπ)²/ρ

How it was proved

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