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

The certified cost factor equals 1 only for the exact proposal: Σπ²/r = 1 implies π = r pointwise

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

\htmlData{mx=7}{\forall j,\ \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{\htmlData{mx=3}{r}(\htmlData{mx=4}{j})}}

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)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)
aᵇPower: π(j) raised to the power 2. (π j ^ 2)
a/bDivision: π(j)^2 divided by r(j). In Lean, dividing by 0 gives 0. (π j ^ 2 / r j)

theorem sc-cost-factor-eq-one-iff {J : Type} [Fintype J] (π r : J → ℝ) (hr : ∀ j, 0 < r j) (hrs : ∑ j, r j = 1) (hπs : ∑ j, π j = 1) (h : ∑ j, π j ^ 2 / r j = 1) :
    ∀ j, π j = r j

Σ(π−r)²/r = Σπ²/r − 2 + 1 vanishes only when every term does; the certificate is a genuine distance to the attention row. 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: 24 lines of Lean

The certified cost factor equals 1 only for the exact proposal: Σπ²/r = 1 implies π = r pointwise

How it was proved

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