Provedattentionresidual-seriesexternal-review-4

Residual series: global certificate controls the second-moment factor

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{r, h : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{\rho \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ , $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\rho}$ and $\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{h}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{\rho}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

0 \mathrel{\htmlData{mx=11}{\leq}} \htmlData{mx=10}{\frac{\htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})} \cdot \htmlData{mx=6}{\htmlData{mx=5}{\htmlData{mx=3}{h}(\htmlData{mx=4}{j})}^{2}}}}}{\htmlData{mx=9}{\rho}}} \mathbin{\htmlData{mx=25}{\land}} \htmlData{mx=22}{\frac{\htmlData{mx=20}{\sum_{j} \htmlData{mx=19}{\htmlData{mx=14}{\htmlData{mx=12}{r}(\htmlData{mx=13}{j})} \cdot \htmlData{mx=18}{\htmlData{mx=17}{\htmlData{mx=15}{h}(\htmlData{mx=16}{j})}^{2}}}}}{\htmlData{mx=21}{\rho}}} \mathrel{\htmlData{mx=24}{\leq}} \htmlData{mx=23}{\rho}

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, h range over J → ℝ: the statement must hold for every choice. ((r h : J → 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 ρ. (0<ρ)
∧And: Both hold: 0 ≤ (∑ j, r(j) · h(j)^2)/ρ, and (∑ j, r(j) · h(j)^2)/ρ ≤ ρ. (0≤(∑ j,r j*h j^2)/ρ ∧ (∑ j,r j*h j^2)/ρ…)
a/bDivision: ∑ j, r(j) · h(j)^2 divided by ρ. In Lean, dividing by 0 gives 0. ((∑ j,r j*h j^2)/ρ)
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)
|x|Absolute value: The size of h(j) without its sign. (|h j|)
·Multiplication: r(j) times h(j)^2. (r j*h j^2)
aᵇPower: h(j) raised to the power 2. (h j^2)

theorem residual-series-second-factor-certificate {J : Type} [Fintype J] (r h : J → ℝ) (ρ : ℝ) (hr : ∀ j,0≤r j) (hrs : ∑ j,r j=1) (hρ : 0<ρ) (hh : ∀ j,|h j|≤ρ) :
    0≤(∑ j,r j*h j^2)/ρ ∧ (∑ j,r j*h j^2)/ρ≤ρ

For a nonnegative normalized finite reference and |h|≤ρ with ρ>0, proves 0≤E_r[h²]/ρ≤ρ. Adding ρ<1 gives the strict contraction used by the paper infinite-series variance argument.

How it was proved

Proved: 6 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: 18 lines of Lean

Residual series: global certificate controls the second-moment factor

How it was proved

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