Provedalphabaguettebaguette-zv2

The cotangent reaching key position j is bounded by (T − j)·G, maximal at position 0

For all $\htmlData{mx=1}{T, j \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=3}{w, g : \htmlData{mx=0}{\mathbb{N}} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{G \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall i,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{i})}}$ , $\htmlData{mx=4}{\forall i,\ \htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{i})} \mathrel{\htmlData{mx=3}{\leq}} 1}$ and $\htmlData{mx=6}{\forall i,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{g}(\htmlData{mx=1}{i})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{G}}$ , then

\htmlData{mx=18}{\left\lvert \htmlData{mx=17}{\sum_{i \in \htmlData{mx=9}{\htmlData{mx=4}{\operatorname{filter}\left(\htmlData{mx=3}{\{0, \dots, \htmlData{mx=2}{T} - 1\}}\right)}(\htmlData{mx=8}{i \mapsto \htmlData{mx=5}{j} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{i}})}} \htmlData{mx=16}{\htmlData{mx=12}{\htmlData{mx=10}{w}(\htmlData{mx=11}{i})} \cdot \htmlData{mx=15}{\htmlData{mx=13}{g}(\htmlData{mx=14}{i})}}} \right\rvert} \mathrel{\htmlData{mx=26}{\leq}} \htmlData{mx=25}{\htmlData{mx=23}{\htmlData{mx=22}{\htmlData{mx=21}{\htmlData{mx=19}{T} - \htmlData{mx=20}{j}}}} \cdot \htmlData{mx=24}{G}}

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: T, j range over natural numbers: the statement must hold for every choice. ((T j : ℕ))
⟹Implies: If ℕ, then ℝ. (Between types, A → B is the type of functions from A to B.) (ℕ → ℝ)
x : TVariables: w, g range over ℕ → ℝ: the statement must hold for every choice. ((w g : ℕ → ℝ))
∀For all: The statement after the comma holds for every i. (∀ i, 0 ≤ w i)
∑Summation: Add up w(i) · g(i) for each i in filter(range(T))(i ↦ j ≤ i). (∑ i ∈ (Finset.range T).filter (fun i =>…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
≤At most: 0 is less than or equal to w(i). (0 ≤ w i)
|x|Absolute value: The size of g(i) without its sign. (|g i|)
e.ffilter: filter of range(T): Lean's dot notation e.filter means filter applied to e. ((Finset.range T).filter)
{0,…,n−1}Range: The numbers 0, 1, …, T − 1 (T of them). (Finset.range T)
f(x)Function application: w applied to i. Lean writes f x; papers write f(x). (w i)
x ↦ f(x)Function (λ-abstraction): The function sending i to j ≤ i. Lean writes fun x => …; papers write x ↦ …. (fun i => j ≤ i)
·Multiplication: w(i) times g(i). (w i * g i)
(x : T)As real numbers: T − j read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. (((T - j : ℕ) : ℝ))
(x : T)As natural numbers: T − j read as an element of natural numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((T - j : ℕ))
-Subtraction: T minus j. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (T - j)

theorem bz2-fan-in-cotangent-bound (T j : ℕ) (w g : ℕ → ℝ) (hw0 : ∀ i, 0 ≤ w i) (hw1 : ∀ i, w i ≤ 1) (G : ℝ) (hg : ∀ i, |g i| ≤ G) :
    |∑ i ∈ (Finset.range T).filter (fun i => j ≤ i), w i * g i| ≤ ((T - j : ℕ) : ℝ) * G

With attention weights in [0,1] and per-query cotangents |gᵢ| ≤ G, the total cotangent position j receives from the causal queries i ≥ j is at most (T − j)·G. At Baguette's T = 4096 the first token can collect up to 4096× a single row's cotangent. That matches the recorded position-0 embedding blow-ups and motivates sink-anchored windows. Part of 'Baguette-Z v2' (revision after the hostile review).

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: 25 lines of Lean

The cotangent reaching key position j is bounded by (T − j)·G, maximal at position 0

How it was proved

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