Provedbalanced-conversionconversion-theoremattention-compute-lawlower-bound

Biased product family: each π_σ(b) = Π_ℓ (1/2 ± ε) is a probability distribution on the 2^L leaves

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=1}{\varepsilon \in \htmlData{mx=0}{\mathbb{R}}}$ and $\htmlData{mx=5}{\sigma : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\operatorname{Bool}}}$ :

\htmlData{mx=20}{\sum_{b : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\operatorname{Bool}}} \htmlData{mx=19}{\prod_{\ell} \left(\htmlData{mx=18}{\begin{cases} \htmlData{mx=7}{\htmlData{mx=5}{\frac{1}{2}} + \htmlData{mx=6}{\varepsilon}} & \text{if } \htmlData{mx=10}{\htmlData{mx=8}{\sigma}(\htmlData{mx=9}{\ell})} \mathrel{\htmlData{mx=14}{=}} \htmlData{mx=13}{\htmlData{mx=11}{b}(\htmlData{mx=12}{\ell})} \\ \htmlData{mx=17}{\htmlData{mx=15}{\frac{1}{2}} - \htmlData{mx=16}{\varepsilon}} & \text{otherwise} \end{cases}}\right)}} \mathrel{\htmlData{mx=21}{=}} 1

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
RReal numbers: The real numbers: every point on the number line. (ℝ)
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 L. Lean writes f x; papers write f(x). (Fin L)
⟹Implies: If Fin(L), then Bool. (Between types, A → B is the type of functions from A to B.) (Fin L → Bool)
∑Summation: Add up ∏ ℓ, 1/2 + ε if σ(ℓ) = b(ℓ), else 1/2 − ε for every b in Fin(L) → Bool. (∑ b : Fin L → Bool, ∏ ℓ, (if σ ℓ = b ℓ …)
Bool: Bool: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Bool)
∏Product: Multiply together 1/2 + ε if σ(ℓ) = b(ℓ), else 1/2 − ε for every ℓ. (∏ ℓ, (if σ ℓ = b ℓ then 1 / 2 + ε else …)
{ a if c; b otherwiseCases (if … then … else): 1/2 + ε when σ(ℓ) = b(ℓ), and 1/2 − ε otherwise. (if σ ℓ = b ℓ then 1 / 2 + ε else 1 / 2 …)
=Equality: σ(ℓ) and b(ℓ) are the same. (σ ℓ = b ℓ)
a/bDivision: 1 divided by 2. In Lean, dividing by 0 gives 0. (1 / 2)
+Addition: 1/2 plus ε. (1 / 2 + ε)
-Subtraction: 1/2 minus ε. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 / 2 - ε)

theorem bc2-product-family-sums-to-one (L : ℕ) (ε : ℝ) (σ : Fin L → Bool) :
    ∑ b : Fin L → Bool, ∏ ℓ, (if σ ℓ = b ℓ then 1 / 2 + ε else 1 / 2 - ε) = 1

Members of the family used for the lower bound are genuine attention rows; every node's conditional mass is 1/2 ± ε, so an additive-ε oracle answering 1/2 everywhere is correct for all of them. 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: 13 lines of Lean

Biased product family: each π_σ(b) = Π_ℓ (1/2 ± ε) is a probability distribution on the 2^L leaves

How it was proved

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