Provedtwo-precision-attentionenergyquantizationlaw

Quantized-score error: entrywise |q − q̂| ≤ ε_q, |k − k̂| ≤ ε_k ⇒ |⟨q,k⟩ − ⟨q̂,k̂⟩| ≤ d(ε_q K∞ + Q∞ ε_k)

For all $\htmlData{mx=1}{d \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{q, \mathit{qh}, k, \mathit{kh} : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{d}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{\mathit{eq}, \mathit{ek}, \mathit{Kinf}, \mathit{Qinf} \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=10}{\forall i,\ \htmlData{mx=7}{\left\lvert \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{q}(\htmlData{mx=1}{i})} - \htmlData{mx=5}{\htmlData{mx=3}{\mathit{qh}}(\htmlData{mx=4}{i})}} \right\rvert} \mathrel{\htmlData{mx=9}{\leq}} \htmlData{mx=8}{\mathit{eq}}}$ , $\htmlData{mx=10}{\forall i,\ \htmlData{mx=7}{\left\lvert \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{k}(\htmlData{mx=1}{i})} - \htmlData{mx=5}{\htmlData{mx=3}{\mathit{kh}}(\htmlData{mx=4}{i})}} \right\rvert} \mathrel{\htmlData{mx=9}{\leq}} \htmlData{mx=8}{\mathit{ek}}}$ , $\htmlData{mx=6}{\forall i,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{k}(\htmlData{mx=1}{i})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{\mathit{Kinf}}}$ and $\htmlData{mx=6}{\forall i,\ \htmlData{mx=3}{\left\lvert \htmlData{mx=2}{\htmlData{mx=0}{\mathit{qh}}(\htmlData{mx=1}{i})} \right\rvert} \mathrel{\htmlData{mx=5}{\leq}} \htmlData{mx=4}{\mathit{Qinf}}}$ , then

\begin{aligned} & \htmlData{mx=18}{\left\lvert \htmlData{mx=17}{\htmlData{mx=8}{\sum_{i} \htmlData{mx=7}{\htmlData{mx=3}{\htmlData{mx=1}{q}(\htmlData{mx=2}{i})} \cdot \htmlData{mx=6}{\htmlData{mx=4}{k}(\htmlData{mx=5}{i})}}} - \htmlData{mx=16}{\sum_{i} \htmlData{mx=15}{\htmlData{mx=11}{\htmlData{mx=9}{\mathit{qh}}(\htmlData{mx=10}{i})} \cdot \htmlData{mx=14}{\htmlData{mx=12}{\mathit{kh}}(\htmlData{mx=13}{i})}}}} \right\rvert} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=27}{\htmlData{mx=19}{d} \cdot (\htmlData{mx=26}{\htmlData{mx=22}{\htmlData{mx=20}{\mathit{eq}} \cdot \htmlData{mx=21}{\mathit{Kinf}}} + \htmlData{mx=25}{\htmlData{mx=23}{\mathit{Qinf}} \cdot \htmlData{mx=24}{\mathit{ek}}}})} \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 d. Lean writes f x; papers write f(x). (Fin d)
⟹Implies: If Fin(d), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin d → ℝ)
x : TVariables: q, qh, k, kh range over Fin d → ℝ: the statement must hold for every choice. ((q qh k kh : Fin d → ℝ))
x ∈ TVariables: eq, ek, Kinf, Qinf range over real numbers: the statement must hold for every choice. ((eq ek Kinf Qinf : ℝ))
∀For all: The statement after the comma holds for every i. (∀ i, |q i - qh i| ≤ eq)
∑Summation: Add up q(i) · k(i) for every i. (∑ i, q i * k i)
|x|Absolute value: The size of q(i) − qh(i) without its sign. (|q i - qh i|)
≤At most: |q(i) − qh(i)| is less than or equal to eq. (|q i - qh i| ≤ eq)
-Subtraction: q(i) minus qh(i). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (q i - qh i)
·Multiplication: q(i) times k(i). (q i * k i)
RReal numbers: The real numbers: every point on the number line. (ℝ)
+Addition: eq · Kinf plus Qinf · ek. (eq * Kinf + Qinf * ek)

theorem tp-quantized-score-error {d : ℕ} (q qh k kh : Fin d → ℝ) (eq ek Kinf Qinf : ℝ) (hq : ∀ i, |q i - qh i| ≤ eq) (hk : ∀ i, |k i - kh i| ≤ ek) (hK : ∀ i, |k i| ≤ Kinf) (hQ : ∀ i, |qh i| ≤ Qinf) :
    |∑ i, q i * k i - ∑ i, qh i * kh i| ≤ d * (eq * Kinf + Qinf * ek)

A deterministic, computable tilt bound for low-precision QKᵀ used as a rejection proposal. Viable for INT8 (η ≈ Q∞K∞); too loose for INT4 at d = 64, where SNIS + MLMC must be used instead. Part of 'Two-Precision Exact Attention: low-precision scores as a proposal, full-precision softmax as the exact target'.

How it was proved

Proved: 4 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: 24 lines of Lean

Quantized-score error: entrywise |q − q̂| ≤ ε_q, |k − k̂| ≤ ε_k ⇒ |⟨q,k⟩ − ⟨q̂,k̂⟩| ≤ d(ε_q K∞ + Q∞ ε_k)

How it was proved

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