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

Sampleability loop 3: the certificate's variance is Σπ⁴/r³ − (Σπ²/r)² (variance of (π/r)² under a proposal r)

For all J∈Type⁡\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}} and π,r:J→R\htmlData{mx=3}{\pi, r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}, if ∀j, 0<r(j)\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} and ∑jr(j)=1\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1 (given Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}), then

∑jr(j)⋅((π(j)r(j))2−∑kπ(k)2r(k))2=∑jπ(j)4r(j)3−(∑jπ(j)2r(j))2\begin{aligned} & \htmlData{mx=24}{\sum_{j} \htmlData{mx=23}{\htmlData{mx=3}{\htmlData{mx=1}{r}(\htmlData{mx=2}{j})} \cdot \htmlData{mx=22}{\left(\htmlData{mx=21}{\htmlData{mx=11}{\left(\htmlData{mx=10}{\frac{\htmlData{mx=6}{\htmlData{mx=4}{\pi}(\htmlData{mx=5}{j})}}{\htmlData{mx=9}{\htmlData{mx=7}{r}(\htmlData{mx=8}{j})}}}\right)^{2}} - \htmlData{mx=20}{\sum_{k} \htmlData{mx=19}{\frac{\htmlData{mx=15}{\htmlData{mx=14}{\htmlData{mx=12}{\pi}(\htmlData{mx=13}{k})}^{2}}}{\htmlData{mx=18}{\htmlData{mx=16}{r}(\htmlData{mx=17}{k})}}}}}\right)^{2}}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=45}{\htmlData{mx=34}{\sum_{j} \htmlData{mx=33}{\frac{\htmlData{mx=28}{\htmlData{mx=27}{\htmlData{mx=25}{\pi}(\htmlData{mx=26}{j})}^{4}}}{\htmlData{mx=32}{\htmlData{mx=31}{\htmlData{mx=29}{r}(\htmlData{mx=30}{j})}^{3}}}}} - \htmlData{mx=44}{\left(\htmlData{mx=43}{\sum_{j} \htmlData{mx=42}{\frac{\htmlData{mx=38}{\htmlData{mx=37}{\htmlData{mx=35}{\pi}(\htmlData{mx=36}{j})}^{2}}}{\htmlData{mx=41}{\htmlData{mx=39}{r}(\htmlData{mx=40}{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
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 r(j) for every j. (∑ j, r j)
=Equality
∑ j, r(j) and 1 are the same. (∑ j, r 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)
·Multiplication
r(j) times ((π(j)/r(j))^2 − (∑ k, π(k)^2/r(k)))^2. (r j * ((π j / r j) ^ 2 - ∑ k, π k ^ 2 /…)
aᵇPower
π(j)/r(j) raised to the power 2. ((π j / r j) ^ 2)
-Subtraction
(π(j)/r(j))^2 minus ∑ k, π(k)^2/r(k). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. ((π j / r j) ^ 2 - ∑ k, π k ^ 2 / r k)
πPi
The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
a/bDivision
π(j) divided by r(j). In Lean, dividing by 0 gives 0. (π j / r j)

The self-certificate is a sample mean of (π/r)² under r; its variance is the centred fourth-moment expression. Reliability of the certificate reduces to controlling Σπ⁴/r³. 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
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