Provedalphabaguettebaguette-zschedule

Int64 headroom: 2¹⁶ terms of magnitude ≤ 2⁴⁰ sum to at most 2⁵⁶ < 2⁶³

For all $\htmlData{mx=4}{q : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Fin}}\,65536} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{Z}}}$ , if $\htmlData{mx=6}{\forall i,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{q}(\htmlData{mx=1}{i})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{2^{40}}}$ , then

\htmlData{mx=4}{\left\lvert \htmlData{mx=3}{\sum_{i} \htmlData{mx=2}{\htmlData{mx=0}{q}(\htmlData{mx=1}{i})}} \right\rvert} \mathrel{\htmlData{mx=6}{\leq}} \htmlData{mx=5}{2^{56}} \mathbin{\htmlData{mx=11}{\land}} \htmlData{mx=8}{\htmlData{mx=7}{2}^{56}} \mathrel{\htmlData{mx=10}{<}} \htmlData{mx=9}{2^{63}}

What the notation means · or point at any symbol

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 65536. Lean writes f x; papers write f(x). (Fin 65536)
⟹Implies: If Fin(65536), then ℤ. (Between types, A → B is the type of functions from A to B.) (Fin 65536 → ℤ)
∀For all: The statement after the comma holds for every i. (∀ i, |q i| ≤ 2 ^ 40)
∧And: Both hold: |∑ i, q(i)| ≤ 2^56, and 2^56 < 2^63. (|∑ i, q i| ≤ 2 ^ 56 ∧ (2:ℤ) ^ 56 < 2 ^ …)
∑Summation: Add up q(i) for every i. (∑ i, q i)
|x|Absolute value: The size of q(i) without its sign. (|q i|)
≤At most: |q(i)| is less than or equal to 2^40. (|q i| ≤ 2 ^ 40)
ZIntegers: The integers …, −2, −1, 0, 1, 2, …. (ℤ)
aᵇPower: 2 raised to the power 40. (2 ^ 40)
(x : T)As integers: 2 read as an element of integers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((2:ℤ))
<Less than: 2^56 is strictly smaller than 2^63. ((2:ℤ) ^ 56 < 2 ^ 63)

theorem bz-int64-headroom (q : Fin 65536 → ℤ) (h : ∀ i, |q i| ≤ 2 ^ 40) :
    |∑ i, q i| ≤ 2 ^ 56 ∧ (2:ℤ) ^ 56 < 2 ^ 63

Baguette's largest reductions have at most 65,536 terms (vocabulary rows, or 4,096 keys × 9 heads ≤ 2¹⁶). With per-term codes bounded by 2⁴⁰, an int64 accumulator cannot overflow in any order. 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: 17 lines of Lean

Int64 headroom: 2¹⁶ terms of magnitude ≤ 2⁴⁰ sum to at most 2⁵⁶ < 2⁶³

How it was proved

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