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Range bound for second moments: if every x_j lies in [a,b] with a > 0, then 4ab·(Σw)(Σw x²) ≤ (a+b)²(Σw x)² for nonnegative weights w

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 $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{a}$ , $\htmlData{mx=0}{a} \mathrel{\htmlData{mx=2}{\leq}} \htmlData{mx=1}{b}$ , $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{j})}}$ 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}{\htmlData{mx=4}{\htmlData{mx=2}{4\htmlData{mx=1}{a}} \cdot \htmlData{mx=3}{b}} \cdot \left(\htmlData{mx=18}{\left(\htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\htmlData{mx=5}{w}(\htmlData{mx=6}{j})}}\right) \cdot \htmlData{mx=17}{\sum_{j} \htmlData{mx=16}{\htmlData{mx=11}{\htmlData{mx=9}{w}(\htmlData{mx=10}{j})} \cdot \htmlData{mx=15}{\htmlData{mx=14}{\htmlData{mx=12}{x}(\htmlData{mx=13}{j})}^{2}}}}}\right)} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=33}{\htmlData{mx=23}{(\htmlData{mx=22}{\htmlData{mx=20}{a} + \htmlData{mx=21}{b}})^{2}} \cdot \htmlData{mx=32}{\left(\htmlData{mx=31}{\sum_{j} \htmlData{mx=30}{\htmlData{mx=26}{\htmlData{mx=24}{w}(\htmlData{mx=25}{j})} \cdot \htmlData{mx=29}{\htmlData{mx=27}{x}(\htmlData{mx=28}{j})}}}\right)^{2}}} \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))
<Less than: 0 is strictly smaller than a. (0 < a)
≤At most: a is less than or equal to b. (a ≤ b)
∀For all: The statement after the comma holds for every j. (∀ j, 0 ≤ w j)
·Multiplication: 4 times a. (4 * a)
RReal numbers: The real numbers: every point on the number line. (ℝ)
∧And: Both hold: a ≤ x(j), and x(j) ≤ b. (a ≤ x j ∧ x j ≤ b)
∑Summation: Add up w(j) for every j. (∑ j, w j)
aᵇPower: x(j) raised to the power 2. (x j ^ 2)
+Addition: a plus b. (a + b)

theorem ua-kantorovich-range-bound {J : Type} [Fintype J] (w x : J → ℝ) (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) (hw : ∀ j, 0 ≤ w j) (hx : ∀ j, a ≤ x j ∧ x j ≤ b) :
    4 * a * b * ((∑ j, w j) * ∑ j, w j * x j ^ 2) ≤ (a + b) ^ 2 * (∑ j, w j * x j) ^ 2

A Kantorovich-type bounded-ratio inequality: Σw(x−a)(x−b) ≤ 0 gives Σwx² ≤ (a+b)Σwx − abΣw, and the square ((a+b)μ − 2abW)² ≥ 0 finishes. It is the engine of the cosh² bounds for uniform proposals and for drift. Classical-type; added after an external review (2026-10-05) pointed out that the e^R bound can be sharpened.

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: 24 lines of Lean

Range bound for second moments: if every x_j lies in [a,b] with a > 0, then 4ab·(Σw)(Σw x²) ≤ (a+b)²(Σw x)² for nonnegative weights w

How it was proved

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