Provedfisher-cover-lawdecode-attentionexponential-familylaw

ESS fraction of normalised weights: 1 ≤ N Σ w_i², i.e. 1/(N Σ w²) ≤ 1 with equality iff uniform

For all $\htmlData{mx=1}{N \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{w : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{N}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{N}$ and $\htmlData{mx=3}{\sum_{i} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{i})}} \mathrel{\htmlData{mx=4}{=}} 1$ , then

1 \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\htmlData{mx=1}{\htmlData{mx=0}{N}} \cdot \htmlData{mx=6}{\sum_{i} \htmlData{mx=5}{\htmlData{mx=4}{\htmlData{mx=2}{w}(\htmlData{mx=3}{i})}^{2}}}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to N. Lean writes f x; papers write f(x). (Fin N)
⟹Implies: If Fin(N), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin N → ℝ)
<Less than: 0 is strictly smaller than N. (0 < N)
∑Summation: Add up w(i) for every i. (∑ i, w i)
=Equality: ∑ i, w(i) and 1 are the same. (∑ i, w i = 1)
≤At most: 1 is less than or equal to N · (∑ i, w(i)^2). (1 ≤ (N : ℝ) * ∑ i, w i ^ 2)
(x : T)As real numbers: N read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((N : ℝ))
·Multiplication: N times ∑ i, w(i)^2. ((N : ℝ) * ∑ i, w i ^ 2)
RReal numbers: The real numbers: every point on the number line. (ℝ)
aᵇPower: w(i) raised to the power 2. (w i ^ 2)

theorem fc-ess-fraction-le-one {N : ℕ} (w : Fin N → ℝ) (hN : 0 < N) (hsum : ∑ i, w i = 1) :
    1 ≤ (N : ℝ) * ∑ i, w i ^ 2

The observable effective sample size of the SNIS weights estimates 1/(1+χ²), giving a self-certifying (heuristic) snapshot-refresh rule with no dual-norm oracle. Part of 'Fisher-Cover Law: snapshot reuse is governed by the Fisher geometry of the attention exponential family'.

How it was proved

Proved: 4 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: 14 lines of Lean

ESS fraction of normalised weights: 1 ≤ N Σ w_i², i.e. 1/(N Σ w²) ≤ 1 with equality iff uniform

How it was proved

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