Provedcertified-residual-attentiondriftexponential-tiltsharp-bound

Sharp exponential-tilt second-moment bound from score oscillation

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{w, \delta : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{l, u \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $\htmlData{mx=0}{l} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{u}$ and $\htmlData{mx=11}{\forall j,\ \htmlData{mx=0}{l} \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=1}{\delta}(\htmlData{mx=2}{j})} \mathbin{\htmlData{mx=10}{\land}} \htmlData{mx=7}{\htmlData{mx=5}{\delta}(\htmlData{mx=6}{j})} \mathrel{\htmlData{mx=9}{\leq}} \htmlData{mx=8}{u}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

\begin{aligned} & \htmlData{mx=21}{\frac{\htmlData{mx=10}{\sum_{j} \htmlData{mx=9}{\htmlData{mx=3}{\htmlData{mx=1}{w}(\htmlData{mx=2}{j})} \cdot \htmlData{mx=8}{(\htmlData{mx=7}{e^{\htmlData{mx=6}{\htmlData{mx=4}{\delta}(\htmlData{mx=5}{j})}}})^{2}}}}}{\htmlData{mx=20}{\left(\htmlData{mx=19}{\sum_{j} \htmlData{mx=18}{\htmlData{mx=13}{\htmlData{mx=11}{w}(\htmlData{mx=12}{j})} \cdot \htmlData{mx=17}{e^{\htmlData{mx=16}{\htmlData{mx=14}{\delta}(\htmlData{mx=15}{j})}}}}}\right)^{2}}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=34}{\frac{\htmlData{mx=27}{(\htmlData{mx=26}{\htmlData{mx=23}{e^{\htmlData{mx=22}{l}}} + \htmlData{mx=25}{e^{\htmlData{mx=24}{u}}}})^{2}}}{\htmlData{mx=33}{\htmlData{mx=30}{4\htmlData{mx=29}{e^{\htmlData{mx=28}{l}}}} \cdot \htmlData{mx=32}{e^{\htmlData{mx=31}{u}}}}}} \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: w, δ range over J → ℝ: the statement must hold for every choice. ((w δ : J → ℝ))
x ∈ TVariables: l, u range over real numbers: the statement must hold for every choice. ((l u : ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 ≤ w j)
∑Summation: Add up w(j) for every j. (∑ j, w j)
=Equality: ∑ j, w(j) and 1 are the same. (∑ j, w j = 1)
a/bDivision: ∑ j, w(j) · exp(δ(j))^2 divided by (∑ j, w(j) · exp(δ(j)))^2. In Lean, dividing by 0 gives 0. ((∑ j, w j * (Real.exp (δ j)) ^ 2) / (∑ …)
RReal numbers: The real numbers: every point on the number line. (ℝ)
≤At most: 0 is less than or equal to w(j). (0 ≤ w j)
∧And: Both hold: l ≤ δ(j), and δ(j) ≤ u. (l ≤ δ j ∧ δ j ≤ u)
·Multiplication: w(j) times exp(δ(j))^2. (w j * (Real.exp (δ j)) ^ 2)
aᵇPower: exp(δ(j)) raised to the power 2. ((Real.exp (δ j)) ^ 2)
eˣExponential function: e raised to the power δ(j), where e ≈ 2.718. (Real.exp (δ j))
+Addition: exp(l) plus exp(u). (Real.exp l + Real.exp u)

theorem cra-exponential-tilt-sharp-bound {J : Type} [Fintype J] (w δ : J → ℝ) (l u : ℝ) (hw : ∀ j, 0 ≤ w j) (hws : ∑ j, w j = 1) (hlu : l ≤ u) (hδ : ∀ j, l ≤ δ j ∧ δ j ≤ u) :
    (∑ j, w j * (Real.exp (δ j)) ^ 2) / (∑ j, w j * Real.exp (δ j)) ^ 2 ≤
  (Real.exp l + Real.exp u) ^ 2 / (4 * Real.exp l * Real.exp u)

For nonnegative normalized base weights w and score increments δ_j in [l,u], the normalized exponential-tilt second moment is at most (e^l+e^u)^2/(4e^{l+u}). This is the finite probability-space form behind the cosh²((u-l)/2) oscillation bound; it sharpens the crude exp(u-l) factor. The general bounded-moment inequality is separately verified.

How it was proved

Proved: 5 October 2026
By: Ajax
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: 14 lines of Lean

Sharp exponential-tilt second-moment bound from score oscillation

How it was proved

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