Provedbalanced-conversionconversion-theoremattention-compute-lawlower-bound

Descent recursion: Σ_c (p_c²/q_c) X(c) ≤ (Σ_c p_c²/q_c) · max X(c)

For all $\htmlData{mx=1}{C \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{p, q, X : \htmlData{mx=0}{C} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{M \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall c,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{q}(\htmlData{mx=1}{c})}}$ , $\htmlData{mx=5}{\forall c,\ \htmlData{mx=2}{\htmlData{mx=0}{X}(\htmlData{mx=1}{c})} \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{M}}$ , $\htmlData{mx=4}{\forall c,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{X}(\htmlData{mx=1}{c})}}$ and $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{M}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{C}}$ ), then

\htmlData{mx=12}{\sum_{c} \htmlData{mx=11}{\htmlData{mx=7}{\frac{\htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{p}(\htmlData{mx=1}{c})}^{2}}}{\htmlData{mx=6}{\htmlData{mx=4}{q}(\htmlData{mx=5}{c})}}} \cdot \htmlData{mx=10}{\htmlData{mx=8}{X}(\htmlData{mx=9}{c})}}} \mathrel{\htmlData{mx=24}{\leq}} \htmlData{mx=23}{\left(\htmlData{mx=21}{\sum_{c} \htmlData{mx=20}{\frac{\htmlData{mx=16}{\htmlData{mx=15}{\htmlData{mx=13}{p}(\htmlData{mx=14}{c})}^{2}}}{\htmlData{mx=19}{\htmlData{mx=17}{q}(\htmlData{mx=18}{c})}}}}\right) \cdot \htmlData{mx=22}{M}}

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 C. Lean writes f x; papers write f(x). (Fintype C)
⟹Implies: If C, then ℝ. (Between types, A → B is the type of functions from A to B.) (C → ℝ)
x : TVariables: p, q, X range over C → ℝ: the statement must hold for every choice. ((p q X : C → ℝ))
∀For all: The statement after the comma holds for every c. (∀ c, 0 < q c)
∑Summation: Add up p(c)^2/q(c) · X(c) for every c. (∑ c, p c ^ 2 / q c * X c)
RReal numbers: The real numbers: every point on the number line. (ℝ)
<Less than: 0 is strictly smaller than q(c). (0 < q c)
≤At most: X(c) is less than or equal to M. (X c ≤ M)
aᵇPower: p(c) raised to the power 2. (p c ^ 2)
a/bDivision: p(c)^2 divided by q(c). In Lean, dividing by 0 gives 0. (p c ^ 2 / q c)
·Multiplication: p(c)^2/q(c) times X(c). (p c ^ 2 / q c * X c)

theorem bc2-recursion-step {C : Type} [Fintype C] (p q X : C → ℝ) (M : ℝ) (hq : ∀ c, 0 < q c) (hX : ∀ c, X c ≤ M) (hXnn : ∀ c, 0 ≤ X c) (hM : 0 ≤ M) :
    ∑ c, p c ^ 2 / q c * X c ≤ (∑ c, p c ^ 2 / q c) * M

The chi-square of a tree proposal satisfies X(v) = Σ_c (p_c²/q_c) X(c); bounding the children by M gives X(v) ≤ (node factor) · M. 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: 11 lines of Lean

Descent recursion: Σ_c (p_c²/q_c) X(c) ≤ (Σ_c p_c²/q_c) · max X(c)

How it was proved

More proved statements