Provedbalanced-conversionconversion-theoremattention-compute-lawlower-bound

Partial knowledge: with signs known on some levels, Σ_σ π_σ(b)² = Π_known 2 f_ℓ(b)² · Π_unknown (1/2 + 2ε²) at every leaf

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

\begin{aligned} & \htmlData{mx=40}{\sum_{\sigma : \htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{L}} \mathrel{\htmlData{mx=5}{\to}} \htmlData{mx=4}{\operatorname{Bool}}} \htmlData{mx=39}{\left(\htmlData{mx=38}{\prod_{\ell} \left(\htmlData{mx=37}{\begin{cases} \htmlData{mx=19}{\begin{cases} \htmlData{mx=8}{\htmlData{mx=6}{\frac{1}{2}} + \htmlData{mx=7}{\varepsilon}} & \text{if } \htmlData{mx=11}{\htmlData{mx=9}{\tau}(\htmlData{mx=10}{\ell})} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\htmlData{mx=12}{b}(\htmlData{mx=13}{\ell})} \\ \htmlData{mx=18}{\htmlData{mx=16}{\frac{1}{2}} - \htmlData{mx=17}{\varepsilon}} & \text{otherwise} \end{cases}} & \text{if } \htmlData{mx=22}{\htmlData{mx=20}{\mathit{known}}(\htmlData{mx=21}{\ell})} \\ \htmlData{mx=36}{\begin{cases} \htmlData{mx=25}{\htmlData{mx=23}{\frac{1}{2}} + \htmlData{mx=24}{\varepsilon}} & \text{if } \htmlData{mx=28}{\htmlData{mx=26}{\sigma}(\htmlData{mx=27}{\ell})} \mathrel{\htmlData{mx=32}{=}} \htmlData{mx=31}{\htmlData{mx=29}{b}(\htmlData{mx=30}{\ell})} \\ \htmlData{mx=35}{\htmlData{mx=33}{\frac{1}{2}} - \htmlData{mx=34}{\varepsilon}} & \text{otherwise} \end{cases}} & \text{otherwise} \end{cases}}\right)}\right)^{2}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=66}{\prod_{\ell} \left(\htmlData{mx=65}{\begin{cases} \htmlData{mx=56}{2 \cdot \htmlData{mx=55}{\left(\htmlData{mx=54}{\begin{cases} \htmlData{mx=43}{\htmlData{mx=41}{\frac{1}{2}} + \htmlData{mx=42}{\varepsilon}} & \text{if } \htmlData{mx=46}{\htmlData{mx=44}{\tau}(\htmlData{mx=45}{\ell})} \mathrel{\htmlData{mx=50}{=}} \htmlData{mx=49}{\htmlData{mx=47}{b}(\htmlData{mx=48}{\ell})} \\ \htmlData{mx=53}{\htmlData{mx=51}{\frac{1}{2}} - \htmlData{mx=52}{\varepsilon}} & \text{otherwise} \end{cases}}\right)^{2}}} & \text{if } \htmlData{mx=59}{\htmlData{mx=57}{\mathit{known}}(\htmlData{mx=58}{\ell})} \\ \htmlData{mx=64}{\htmlData{mx=60}{\frac{1}{2}} + \htmlData{mx=63}{2 \cdot \htmlData{mx=62}{\htmlData{mx=61}{\varepsilon}^{2}}}} & \text{otherwise} \end{cases}}\right)} \end{aligned}

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 τ(ℓ) = b(ℓ), else 1/2 − ε if known(ℓ), el… for every σ in Fin(L) → Bool. (∑ σ : Fin L → Bool, (∏ ℓ, (if known ℓ t…)
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)
aᵇPower: ∏ ℓ, 1/2 + ε if τ(ℓ) = b(ℓ), else 1/2 − ε if known(ℓ), els… raised to the power 2. ((∏ ℓ, (if known ℓ then (if τ ℓ = b ℓ th…)
∏Product: Multiply together 1/2 + ε if τ(ℓ) = b(ℓ), else 1/2 − ε if known(ℓ), else 1/2… for every ℓ. (∏ ℓ, (if known ℓ then (if τ ℓ = b ℓ the…)
{ a if c; b otherwiseCases (if … then … else): 1/2 + ε when τ(ℓ) = b(ℓ), and 1/2 − ε otherwise. (if τ ℓ = b ℓ then 1 / 2 + ε else 1 / 2 …)
=Equality: τ(ℓ) and b(ℓ) are the same. (τ 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)
·Multiplication: 2 times 1/2 + ε if τ(ℓ) = b(ℓ), else 1/2 − ε^2. (2 * (if τ ℓ = b ℓ then 1 / 2 + ε else 1…)

theorem bc2-partial-knowledge-second-moment (L : ℕ) (ε : ℝ) (known : Fin L → Bool) (τ : Fin L → Bool) (b : Fin L → Bool) :
    ∑ σ : Fin L → Bool, (∏ ℓ, (if known ℓ then (if τ ℓ = b ℓ then 1 / 2 + ε else 1 / 2 - ε) else (if σ ℓ = b ℓ then 1 / 2 + ε else 1 / 2 - ε))) ^ 2 = ∏ ℓ, (if known ℓ then 2 * (if τ ℓ = b ℓ then 1 / 2 + ε else 1 / 2 - ε) ^ 2 else 1 / 2 + 2 * ε ^ 2)

The family second moment when an algorithm knows the biases on a set of levels and nothing about the rest: the unknown levels contribute the constant factor 1/2 + 2ε² each. 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: 15 lines of Lean

Partial knowledge: with signs known on some levels, Σ_σ π_σ(b)² = Π_known 2 f_ℓ(b)² · Π_unknown (1/2 + 2ε²) at every leaf

How it was proved

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