Provedbalanced-conversionconversion-theoremattention-compute-lawlower-bound

Partial-knowledge lower bound: knowing the biases on all but u levels, every single proposal has family-average Σπ²/r ≥ (1+4ε²)^u

For all $\htmlData{mx=1}{L \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=1}{\varepsilon \in \htmlData{mx=0}{\mathbb{R}}}$ , $\htmlData{mx=5}{\mathit{known}, \tau : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\operatorname{Bool}}}$ and $\htmlData{mx=7}{r : (\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{L}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\operatorname{Bool}}) \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall b,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{b})}}$ and $\htmlData{mx=3}{\sum_{b} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{b})}} \mathrel{\htmlData{mx=4}{=}} 1$ , then

\begin{aligned} & \htmlData{mx=22}{(\htmlData{mx=4}{1 + \htmlData{mx=3}{4 \cdot \htmlData{mx=2}{\htmlData{mx=1}{\varepsilon}^{2}}}})^{\htmlData{mx=21}{\lvert \htmlData{mx=20}{\htmlData{mx=13}{\operatorname{filter}}(\htmlData{mx=19}{\ell \mapsto \htmlData{mx=16}{\htmlData{mx=14}{\mathit{known}}(\htmlData{mx=15}{\ell})} \mathrel{\htmlData{mx=18}{=}} \htmlData{mx=17}{\operatorname{false}}})} \rvert}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=76}{\htmlData{mx=25}{\frac{1}{\htmlData{mx=24}{2^{\htmlData{mx=23}{L}}}}} \cdot \htmlData{mx=75}{\sum_{\sigma : \htmlData{mx=28}{\htmlData{mx=26}{\operatorname{Fin}}\,\htmlData{mx=27}{L}} \mathrel{\htmlData{mx=30}{\to}} \htmlData{mx=29}{\operatorname{Bool}}} \htmlData{mx=74}{\sum_{b : \htmlData{mx=33}{\htmlData{mx=31}{\operatorname{Fin}}\,\htmlData{mx=32}{L}} \mathrel{\htmlData{mx=35}{\to}} \htmlData{mx=34}{\operatorname{Bool}}} \htmlData{mx=73}{\frac{\htmlData{mx=69}{\left(\htmlData{mx=68}{\prod_{\ell} \left(\htmlData{mx=67}{\begin{cases} \htmlData{mx=49}{\begin{cases} \htmlData{mx=38}{\htmlData{mx=36}{\frac{1}{2}} + \htmlData{mx=37}{\varepsilon}} & \text{if } \htmlData{mx=41}{\htmlData{mx=39}{\tau}(\htmlData{mx=40}{\ell})} \mathrel{\htmlData{mx=45}{=}} \htmlData{mx=44}{\htmlData{mx=42}{b}(\htmlData{mx=43}{\ell})} \\ \htmlData{mx=48}{\htmlData{mx=46}{\frac{1}{2}} - \htmlData{mx=47}{\varepsilon}} & \text{otherwise} \end{cases}} & \text{if } \htmlData{mx=52}{\htmlData{mx=50}{\mathit{known}}(\htmlData{mx=51}{\ell})} \\ \htmlData{mx=66}{\begin{cases} \htmlData{mx=55}{\htmlData{mx=53}{\frac{1}{2}} + \htmlData{mx=54}{\varepsilon}} & \text{if } \htmlData{mx=58}{\htmlData{mx=56}{\sigma}(\htmlData{mx=57}{\ell})} \mathrel{\htmlData{mx=62}{=}} \htmlData{mx=61}{\htmlData{mx=59}{b}(\htmlData{mx=60}{\ell})} \\ \htmlData{mx=65}{\htmlData{mx=63}{\frac{1}{2}} - \htmlData{mx=64}{\varepsilon}} & \text{otherwise} \end{cases}} & \text{otherwise} \end{cases}}\right)}\right)^{2}}}{\htmlData{mx=72}{\htmlData{mx=70}{r}(\htmlData{mx=71}{b})}}}}}} \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)
x : TVariables: known, τ range over Fin L → Bool: the statement must hold for every choice. ((known τ : Fin L → Bool))
∀For all: The statement after the comma holds for every b. (∀ b, 0 < r b)
∑Summation: Add up r(b) for every b. (∑ b, r b)
=Equality: ∑ b, r(b) and 1 are the same. (∑ b, r b = 1)
≤At most: (1 + 4 · ε^2)^card(filter(ℓ ↦ known(ℓ) = false)) is less than or equal to 1/2^L · (∑ σ, ∑ b, (∏ ℓ, 1/2 + ε if τ(ℓ) = b(ℓ), else 1/2 …. ((1 + 4 * ε ^ 2) ^ (Finset.univ.filter (…)
+Addition: 1 plus 4 · ε^2. (1 + 4 * ε ^ 2)
<Less than: 0 is strictly smaller than r(b). (0 < r b)
·Multiplication: 4 times ε^2. (4 * ε ^ 2)
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: ε raised to the power 2. (ε ^ 2)
|s|Cardinality: The number of elements of filter(ℓ ↦ known(ℓ) = false). ((Finset.univ.filter (fun ℓ => known ℓ =…)
filter: Finset.univ.filter: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ.filter)
x ↦ f(x)Function (λ-abstraction): The function sending ℓ to known(ℓ) = false. Lean writes fun x => …; papers write x ↦ …. (fun ℓ => known ℓ = false)
false: false: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (false)
a/bDivision: 1 divided by 2^L. In Lean, dividing by 0 gives 0. (1 / 2 ^ L)
∏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 …)
-Subtraction: 1/2 minus ε. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (1 / 2 - ε)

theorem bc2-partial-knowledge-lower-bound (L : ℕ) (ε : ℝ) (known τ : Fin L → Bool) (r : (Fin L → Bool) → ℝ) (hr : ∀ b, 0 < r b) (hrs : ∑ b, r b = 1) :
    (1 + 4 * ε ^ 2) ^ (Finset.univ.filter (fun ℓ => known ℓ = false)).card ≤ (1 / 2 ^ L) * ∑ σ : Fin L → Bool, ∑ b : 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 / r b

Per-level formal version of the general-query candidate: an algorithm that has learned the biases of some levels still pays (1+4ε²) per unknown level. The candidate that polylog additive-ε queries leave L − O(log log n) node signs unknown would make the n^{Θ(ε²)} deterministic residual tight in general; that step is recorded as open. 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: 79 lines of Lean

Partial-knowledge lower bound: knowing the biases on all but u levels, every single proposal has family-average Σπ²/r ≥ (1+4ε²)^u

How it was proved

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