Provedalphabaguettealgorithm-ideaa2

Replacing attention weights by a position-only kernel costs at most (total variation)·max|v|

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{w, p, v : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{M \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=6}{\forall j,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{v}(\htmlData{mx=1}{j})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{M}}$ , then

\begin{aligned} & \htmlData{mx=18}{\left\lvert \htmlData{mx=17}{\htmlData{mx=8}{\sum_{j} \htmlData{mx=7}{\htmlData{mx=3}{\htmlData{mx=1}{w}(\htmlData{mx=2}{j})} \cdot \htmlData{mx=6}{\htmlData{mx=4}{v}(\htmlData{mx=5}{j})}}} - \htmlData{mx=16}{\sum_{j} \htmlData{mx=15}{\htmlData{mx=11}{\htmlData{mx=9}{p}(\htmlData{mx=10}{j})} \cdot \htmlData{mx=14}{\htmlData{mx=12}{v}(\htmlData{mx=13}{j})}}}} \right\rvert} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=29}{\left(\htmlData{mx=27}{\sum_{j} \htmlData{mx=26}{\left\lvert \htmlData{mx=25}{\htmlData{mx=21}{\htmlData{mx=19}{w}(\htmlData{mx=20}{j})} - \htmlData{mx=24}{\htmlData{mx=22}{p}(\htmlData{mx=23}{j})}} \right\rvert}}\right) \cdot \htmlData{mx=28}{M}} \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (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 ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
x : TVariables: w, p, v range over Fin n → ℝ: the statement must hold for every choice. ((w p v : Fin n → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, |v j| ≤ M)
∑Summation: Add up w(j) · v(j) for every j. (∑ j, w j * v j)
-Subtraction: ∑ j, w(j) · v(j) minus ∑ j, p(j) · v(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (∑ j, w j * v j - ∑ j, p j * v j)
|x|Absolute value: The size of v(j) without its sign. (|v j|)
≤At most: |v(j)| is less than or equal to M. (|v j| ≤ M)
·Multiplication: w(j) times v(j). (w j * v j)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem pos-head-tv-certificate {n : ℕ} (w p v : Fin n → ℝ) (M : ℝ) (hv : ∀ j, |v j| ≤ M) :
    |∑ j, w j * v j - ∑ j, p j * v j| ≤ (∑ j, |w j - p j|) * M

Idea A2. For any attention row w and replacement kernel row p over the same keys, the output difference is at most Σ|wⱼ − pⱼ| times the largest value magnitude. This is the certificate for converting a nearly-positional head: measure Σ|w − p| per row, bound the error.

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: 15 lines of Lean

Replacing attention weights by a position-only kernel costs at most (total variation)·max|v|

How it was proved

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