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

Sampleability loop 4: if the row moves by a pointwise factor c (π′ ≤ c π), a stale proposal's cost factor grows by at most c²

For all J∈Type⁡\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}, π,π′,r:J→R\htmlData{mx=3}{\pi, \pi', r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}} and c∈R\htmlData{mx=1}{c \in \htmlData{mx=0}{\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})}}, ∀j, 0≤π′(j)\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\pi'}(\htmlData{mx=1}{j})}} and ∀j, π′(j)≤c⋅π(j)\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}{c} \cdot \htmlData{mx=6}{\htmlData{mx=4}{\pi}(\htmlData{mx=5}{j})}}} (given Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}), then

∑jπ′(j)2r(j)≤c2⋅∑jπ(j)2r(j)\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=21}{\leq}} \htmlData{mx=20}{\htmlData{mx=10}{\htmlData{mx=9}{c}^{2}} \cdot \htmlData{mx=19}{\sum_{j} \htmlData{mx=18}{\frac{\htmlData{mx=14}{\htmlData{mx=13}{\htmlData{mx=11}{\pi}(\htmlData{mx=12}{j})}^{2}}}{\htmlData{mx=17}{\htmlData{mx=15}{r}(\htmlData{mx=16}{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, π', r range over J → ℝ: the statement must hold for every choice. ((π π' r : J → ℝ))
∀For all
The statement after the comma holds for every j. (∀ j, 0 < r j)
∑Summation
Add up π'(j)^2/r(j) for every j. (∑ j, π' j ^ 2 / r j)
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 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)
·Multiplication
c times π(j). (c * π j)
πPi
The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)

Non-stationarity bound: with c = e^{2η} for a query drift η in the key-dual norm, an untracked drift costs at most e^{4η}; the proposal only needs to track the slow component. 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
15 lines of Lean
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