Provedcertified-residual-attentionsharp-boundsecond-moment

Sharp finite positive-weight moment ratio: E[X²]/E[X]² ≤ (a+b)²/(4ab)

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

\begin{aligned} & \htmlData{mx=19}{\frac{\htmlData{mx=9}{\sum_{j} \htmlData{mx=8}{\htmlData{mx=3}{\htmlData{mx=1}{w}(\htmlData{mx=2}{j})} \cdot \htmlData{mx=7}{\htmlData{mx=6}{\htmlData{mx=4}{x}(\htmlData{mx=5}{j})}^{2}}}}}{\htmlData{mx=18}{\left(\htmlData{mx=17}{\sum_{j} \htmlData{mx=16}{\htmlData{mx=12}{\htmlData{mx=10}{w}(\htmlData{mx=11}{j})} \cdot \htmlData{mx=15}{\htmlData{mx=13}{x}(\htmlData{mx=14}{j})}}}\right)^{2}}}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=28}{\frac{\htmlData{mx=23}{(\htmlData{mx=22}{\htmlData{mx=20}{a} + \htmlData{mx=21}{b}})^{2}}}{\htmlData{mx=27}{\htmlData{mx=25}{4\htmlData{mx=24}{a}} \cdot \htmlData{mx=26}{b}}}} \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, x range over J → ℝ: the statement must hold for every choice. ((w x : J → ℝ))
x ∈ TVariables: a, b range over real numbers: the statement must hold for every choice. ((a b : 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)
<Less than: 0 is strictly smaller than a. (0 < a)
a/bDivision: ∑ j, w(j) · x(j)^2 divided by (∑ j, w(j) · x(j))^2. In Lean, dividing by 0 gives 0. ((∑ j, w j * x j ^ 2) / (∑ j, w j * x j)…)
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: a ≤ x(j), and x(j) ≤ b. (a ≤ x j ∧ x j ≤ b)
·Multiplication: w(j) times x(j)^2. (w j * x j ^ 2)
aᵇPower: x(j) raised to the power 2. (x j ^ 2)
+Addition: a plus b. (a + b)

theorem cra-sharp-positive-moment-bound {J : Type} [Fintype J] (w x : J → ℝ) (a b : ℝ) (hw : ∀ j, 0 ≤ w j) (hws : ∑ j, w j = 1) (ha : 0 < a) (hab : a ≤ b) (hx : ∀ j, a ≤ x j ∧ x j ≤ b) :
    (∑ j, w j * x j ^ 2) / (∑ j, w j * x j) ^ 2 ≤ (a + b) ^ 2 / (4 * a * b)

For a finite probability space w and a positive variable a≤x≤b, prove the sharp bounded-moment ratio. This classical-type inequality sharpens the proposal variance factor under bounded score oscillation; it is not claimed as a new general inequality. The exponential/softmax instantiation is a separate result.

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: 48 lines of Lean

Sharp finite positive-weight moment ratio: E[X²]/E[X]² ≤ (a+b)²/(4ab)

How it was proved

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