Provedsampleability-lawself-certifying-proposalimportance-samplingattention-compute-law

Sampleability loop 3: a ratio-bounded proposal (π ≤ K r) has Σπ⁴/r³ ≤ K³, so the certificate has bounded variance

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{\pi, r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{K \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}}$ , $\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1$ and $\htmlData{mx=9}{\forall j,\ \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{\htmlData{mx=3}{K} \cdot \htmlData{mx=6}{\htmlData{mx=4}{r}(\htmlData{mx=5}{j})}}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ ), then

\htmlData{mx=9}{\sum_{j} \htmlData{mx=8}{\frac{\htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}^{4}}}{\htmlData{mx=7}{\htmlData{mx=6}{\htmlData{mx=4}{r}(\htmlData{mx=5}{j})}^{3}}}}} \mathrel{\htmlData{mx=12}{\leq}} \htmlData{mx=11}{\htmlData{mx=10}{K}^{3}}

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: π, r range over J → ℝ: the statement must hold for every choice. ((π r : J → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 < r j)
∑Summation: Add up π(j) for every j. (∑ j, π j)
=Equality: ∑ j, π(j) and 1 are the same. (∑ j, π j = 1)
RReal numbers: The real numbers: every point on the number line. (ℝ)
<Less than: 0 is strictly smaller than r(j). (0 < r j)
≤At most: 0 is less than or equal to π(j). (0 ≤ π j)
aᵇPower: π(j) raised to the power 4. (π j ^ 4)
a/bDivision: π(j)^4 divided by r(j)^3. In Lean, dividing by 0 gives 0. (π j ^ 4 / r j ^ 3)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
·Multiplication: K times r(j). (K * r j)

theorem sl-certificate-fourth-moment-bound {J : Type} [Fintype J] (π r : J → ℝ) (K : ℝ) (hr : ∀ j, 0 < r j) (hπ : ∀ j, 0 ≤ π j) (hπs : ∑ j, π j = 1) (hK : ∀ j, π j ≤ K * r j) :
    ∑ j, π j ^ 4 / r j ^ 3 ≤ K ^ 3

Whenever the importance ratio is bounded by K (as in the Conversion construction, K ≤ e²), the certificate's variance is ≤ K³; learned proposals must be ratio-bounded by design to avoid PSIS-type silent under-reporting. Part of 'The Sampleability Law: a transformer can be trained to make its own attention cheap to sample, and the samples certify the saving' (self-certifying learned proposals inside the Attention Compute Law).

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: 16 lines of Lean

Sampleability loop 3: a ratio-bounded proposal (π ≤ K r) has Σπ⁴/r³ ≤ K³, so the certificate has bounded variance

How it was proved

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