Provedalphabaguettebaguette-zschedule

Integer reductions are invariant under any split-K / chunking of the index set

For all $\htmlData{mx=1}{n, c \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{q : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{Z}}}$ and $\htmlData{mx=7}{\mathit{chunk} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=6}{\to}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{c}}}$ :

\htmlData{mx=15}{\sum_{j : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{c}}} \htmlData{mx=14}{\sum_{i \in \htmlData{mx=10}{\htmlData{mx=3}{\operatorname{filter}}(\htmlData{mx=9}{i \mapsto \htmlData{mx=6}{\htmlData{mx=4}{\mathit{chunk}}(\htmlData{mx=5}{i})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{j}})}} \htmlData{mx=13}{\htmlData{mx=11}{q}(\htmlData{mx=12}{i})}}} \mathrel{\htmlData{mx=20}{=}} \htmlData{mx=19}{\sum_{i} \htmlData{mx=18}{\htmlData{mx=16}{q}(\htmlData{mx=17}{i})}}

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: n, c range over natural numbers: the statement must hold for every choice. ((n c : 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 n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℤ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℤ)
ZIntegers: The integers …, −2, −1, 0, 1, 2, …. (ℤ)
∑Summation: Add up q(i) for each i in filter(i ↦ chunk(i) = j). (∑ i ∈ Finset.univ.filter (fun i => chun…)
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 chunk(i) = j. Lean writes fun x => …; papers write x ↦ …. (fun i => chunk i = j)
=Equality: chunk(i) and j are the same. (chunk i = j)

theorem bz-split-k-invariance {n c : ℕ} (q : Fin n → ℤ) (chunk : Fin n → Fin c) :
    ∑ j : Fin c, ∑ i ∈ Finset.univ.filter (fun i => chunk i = j), q i = ∑ i, q i

For any assignment of terms to chunks (thread blocks, split-K slices), summing within chunks and then across chunks equals the plain sum, exactly, in ℤ. Kernels may choose any chunking without changing a single bit. Part of the plan 'Baguette-Z: closed-loop certified integer-state training'.

How it was proved

Proved: 3 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: 12 lines of Lean

Integer reductions are invariant under any split-K / chunking of the index set

How it was proved

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