Provedgrouped-query-attentiondecodehead-packingsampleability-law

Exact-head packing, lower bound: E heads that must stay exact, in KV groups of size ≤ G, hit at least E/G groups

For all $\htmlData{mx=1}{E, K, G \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=7}{\mathit{grp} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{E}} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{K}}}$ , if $\htmlData{mx=22}{\forall k : \htmlData{mx=21}{\htmlData{mx=19}{\operatorname{Fin}}\,\htmlData{mx=20}{K}},\ \htmlData{mx=16}{\lvert \htmlData{mx=15}{\htmlData{mx=8}{\operatorname{filter}}(\htmlData{mx=14}{i \mapsto \htmlData{mx=11}{\htmlData{mx=9}{\mathit{grp}}(\htmlData{mx=10}{i})} \mathrel{\htmlData{mx=13}{=}} \htmlData{mx=12}{k}})} \rvert} \mathrel{\htmlData{mx=18}{\leq}} \htmlData{mx=17}{G}}$ , then

\htmlData{mx=0}{E} \mathrel{\htmlData{mx=10}{\leq}} \htmlData{mx=9}{\htmlData{mx=1}{G} \cdot \htmlData{mx=8}{\lvert \htmlData{mx=7}{\htmlData{mx=5}{\operatorname{image}}(\htmlData{mx=6}{\mathit{grp}})} \rvert}}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
x ∈ TVariables: E, K, G range over natural numbers: the statement must hold for every choice. ((E K G : 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 E. Lean writes f x; papers write f(x). (Fin E)
⟹Implies: If Fin(E), then Fin(K). (Between types, A → B is the type of functions from A to B.) (Fin E → Fin K)
∀For all: The statement after the comma holds for every k. (∀ k : Fin K, (Finset.univ.filter (fun i…)
·Multiplication: G times card(image(grp)). (G * (Finset.univ.image grp).card)
image: Finset.univ.image: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ.image)
|s|Cardinality: The number of elements of filter(i ↦ grp(i) = k). ((Finset.univ.filter (fun i => grp i = k…)
≤At most: card(filter(i ↦ grp(i) = k)) is less than or equal to G. ((Finset.univ.filter (fun i => grp i = k…)
filter: Finset.univ.filter: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ.filter)
x ↦ f(x)Function (λ-abstraction): The function sending i to grp(i) = k. Lean writes fun x => …; papers write x ↦ …. (fun i => grp i = k)
=Equality: grp(i) and k are the same. (grp i = k)

theorem pk-groups-hit-lower-bound {E K G : ℕ} (grp : Fin E → Fin K) (hcap : ∀ k : Fin K, (Finset.univ.filter (fun i => grp i = k)).card ≤ G) :
    E ≤ G * (Finset.univ.image grp).card

Pigeonhole: under grouped-query attention a KV group's whole cache is read if any of its query heads must stay exact, so the number of fully-read groups is at least ⌈E/G⌉ per layer. Part of 'The Sampleability Law' / 'The Drift Law' (GQA exact-head packing: minimising KV groups that must be read in full).

How it was proved

Proved: 5 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: 13 lines of Lean

Exact-head packing, lower bound: E heads that must stay exact, in KV groups of size ≤ G, hit at least E/G groups

How it was proved

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