Proveddrift-lawdecodeinnovationreview-response

Innovation estimator core: for reference row r with mean m, acceptance a ∈ [a_min, 1] and β = (1+a_min)/2, E_r[(a−β)(v−m)] = p(o−m) with p = E_r[a], and |(a−β)(v−m)| ≤ F(1−a_min)

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{r, v, a : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{m, p, o, F, \mathit{amin}, \beta \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $\htmlData{mx=0}{m} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\htmlData{mx=3}{\htmlData{mx=1}{r}(\htmlData{mx=2}{j})} \cdot \htmlData{mx=6}{\htmlData{mx=4}{v}(\htmlData{mx=5}{j})}}}$ , $\htmlData{mx=0}{p} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\htmlData{mx=3}{\htmlData{mx=1}{r}(\htmlData{mx=2}{j})} \cdot \htmlData{mx=6}{\htmlData{mx=4}{a}(\htmlData{mx=5}{j})}}}$ , $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{p}$ , $\htmlData{mx=0}{o} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\sum_{j} \htmlData{mx=13}{\htmlData{mx=9}{\frac{\htmlData{mx=7}{\htmlData{mx=3}{\htmlData{mx=1}{r}(\htmlData{mx=2}{j})} \cdot \htmlData{mx=6}{\htmlData{mx=4}{a}(\htmlData{mx=5}{j})}}}{\htmlData{mx=8}{p}}} \cdot \htmlData{mx=12}{\htmlData{mx=10}{v}(\htmlData{mx=11}{j})}}}$ , $\htmlData{mx=0}{\beta} \mathrel{\htmlData{mx=4}{=}} \htmlData{mx=3}{\frac{\htmlData{mx=2}{1 + \htmlData{mx=1}{\mathit{amin}}}}{2}}$ , $\htmlData{mx=10}{\forall j,\ \htmlData{mx=0}{\mathit{amin}} \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=1}{a}(\htmlData{mx=2}{j})} \mathbin{\htmlData{mx=9}{\land}} \htmlData{mx=7}{\htmlData{mx=5}{a}(\htmlData{mx=6}{j})} \mathrel{\htmlData{mx=8}{\leq}} 1}$ , $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\mathit{amin}}$ and $\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{v}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{F}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

\begin{aligned} & \htmlData{mx=16}{\sum_{j} \htmlData{mx=15}{\htmlData{mx=3}{\htmlData{mx=1}{r}(\htmlData{mx=2}{j})} \cdot (\htmlData{mx=14}{(\htmlData{mx=8}{\htmlData{mx=6}{\htmlData{mx=4}{a}(\htmlData{mx=5}{j})} - \htmlData{mx=7}{\beta}}) \cdot (\htmlData{mx=13}{\htmlData{mx=11}{\htmlData{mx=9}{v}(\htmlData{mx=10}{j})} - \htmlData{mx=12}{m}})})}} \mathrel{\htmlData{mx=22}{=}} \htmlData{mx=21}{\htmlData{mx=17}{p} \cdot (\htmlData{mx=20}{\htmlData{mx=18}{o} - \htmlData{mx=19}{m}})} \\ & \htmlData{mx=0}{\land} \htmlData{mx=40}{\forall j,\ \htmlData{mx=34}{\left\lvert \htmlData{mx=33}{(\htmlData{mx=27}{\htmlData{mx=25}{\htmlData{mx=23}{a}(\htmlData{mx=24}{j})} - \htmlData{mx=26}{\beta}}) \cdot (\htmlData{mx=32}{\htmlData{mx=30}{\htmlData{mx=28}{v}(\htmlData{mx=29}{j})} - \htmlData{mx=31}{m}})} \right\rvert} \mathrel{\htmlData{mx=39}{\leq}} \htmlData{mx=38}{\htmlData{mx=35}{F} \cdot (\htmlData{mx=37}{1 - \htmlData{mx=36}{\mathit{amin}}})}} \end{aligned}

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)
x : TVariables: r, v, a range over J → ℝ: the statement must hold for every choice. ((r v a : J → ℝ))
x ∈ TVariables: m, p, o, F, amin, β range over real numbers: the statement must hold for every choice. ((m p o F amin β : ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 ≤ r j)
∑Summation: Add up r(j) for every j. (∑ j, r j)
=Equality: ∑ j, r(j) and 1 are the same. (∑ j, r j = 1)
<Less than: 0 is strictly smaller than p. (0 < p)
RReal numbers: The real numbers: every point on the number line. (ℝ)
≤At most: 0 is less than or equal to r(j). (0 ≤ r j)
+Addition: 1 plus amin. (1 + amin)
∧And: Both hold: amin ≤ a(j), and a(j) ≤ 1. (amin ≤ a j ∧ a j ≤ 1)
|x|Absolute value: The size of v(j) without its sign. (|v j|)
·Multiplication: r(j) times v(j). (r j * v j)
a/bDivision: r(j) · a(j) divided by p. In Lean, dividing by 0 gives 0. (r j * a j / p)
-Subtraction: a(j) minus β. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (a j - β)

theorem innov-core-identity {J : Type} [Fintype J] (r v a : J → ℝ) (m p o F amin β : ℝ) (hr : ∀ j, 0 ≤ r j) (hrs : ∑ j, r j = 1) (hm : m = ∑ j, r j * v j) (hp : p = ∑ j, r j * a j) (hp0 : 0 < p) (ho : o = ∑ j, (r j * a j / p) * v j) (hβ : β = (1 + amin) / 2) (ha : ∀ j, amin ≤ a j ∧ a j ≤ 1) (hamin : 0 ≤ amin) (hv : ∀ j, |v j| ≤ F) :
    ∑ j, r j * ((a j - β) * (v j - m)) = p * (o - m) ∧ ∀ j, |(a j - β) * (v j - m)| ≤ F * (1 - amin)

The two finite identities behind the stopped-rejection innovation estimator proposed by an external reviewer (2026-10-05): ô = m + Σ_{i≤T} (a_{J_i} − β)(v_{J_i} − m), summed over every trial up to the first acceptance. With π_j = r_j a_j / p the current row, the per-trial reward has mean p(o − m), so (paper step: Wald's identity with the geometric stopping time) E[ô] = o exactly, with no normaliser and no terminal level; and each reward is bounded by F(1 − a_min), which with E[T²] ≤ 2e^{2Δ} gives Var(ô) ≤ 2F²(e^Δ − 1)² = O(F²Δ²) as the drift oscillation Δ → 0.

How it was proved

Proved: 5 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: 43 lines of Lean

Innovation estimator core: for reference row r with mean m, acceptance a ∈ [a_min, 1] and β = (1+a_min)/2, E_r[(a−β)(v−m)] = p(o−m) with p = E_r[a], and |(a−β)(v−m)| ≤ F(1−a_min)

How it was proved

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