Provedquery-cover-lawdecode-attentionsoftmaxkv-cache

Amortized cost of memoized decode attention: (S·n + T·eF²/ε)/T = nS/T + eF²/ε

For all $\htmlData{mx=1}{n, S, T, F, \varepsilon \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{T}$ and $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\varepsilon}$ , then

\htmlData{mx=13}{\frac{\htmlData{mx=11}{\htmlData{mx=2}{\htmlData{mx=0}{S} \cdot \htmlData{mx=1}{n}} + \htmlData{mx=10}{\htmlData{mx=3}{T} \cdot \left(\htmlData{mx=9}{\frac{\htmlData{mx=7}{\htmlData{mx=4}{e^{1}} \cdot \htmlData{mx=6}{\htmlData{mx=5}{F}^{2}}}}{\htmlData{mx=8}{\varepsilon}}}\right)}}}{\htmlData{mx=12}{T}}} \mathrel{\htmlData{mx=26}{=}} \htmlData{mx=25}{\htmlData{mx=18}{\frac{\htmlData{mx=16}{\htmlData{mx=14}{n} \cdot \htmlData{mx=15}{S}}}{\htmlData{mx=17}{T}}} + \htmlData{mx=24}{\frac{\htmlData{mx=22}{\htmlData{mx=19}{e^{1}} \cdot \htmlData{mx=21}{\htmlData{mx=20}{F}^{2}}}}{\htmlData{mx=23}{\varepsilon}}}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: n, S, T, F, ε range over real numbers: the statement must hold for every choice. ((n S T F ε : ℝ))
<Less than: 0 is strictly smaller than T. (0 < T)
=Equality: (S · n + T · (exp(1) · F^2/ε))/T and n · S/T + exp(1) · F^2/ε are the same. ((S * n + T * (Real.exp 1 * F ^ 2 / ε)) …)
·Multiplication: S times n. (S * n)
+Addition: S · n plus T · (exp(1) · F^2/ε). (S * n + T * (Real.exp 1 * F ^ 2 / ε))
eˣExponential function: e raised to the power 1, where e ≈ 2.718. (Real.exp 1)
a/bDivision: exp(1) · F^2 divided by ε. In Lean, dividing by 0 gives 0. (Real.exp 1 * F ^ 2 / ε)
aᵇPower: F raised to the power 2. (F ^ 2)

theorem qc-amortized-cost (n S T F ε : ℝ) (hT : 0 < T) (hε : 0 < ε) :
    (S * n + T * (Real.exp 1 * F ^ 2 / ε)) / T = n * S / T + Real.exp 1 * F ^ 2 / ε

S snapshots (one full pass each) plus T tokens at eF²/ε proposals each. The context length n enters only through the snapshot rate S/T, which is the covering number of the query trajectory. Part of 'The Query-Cover Law: decode attention cost is set by where the queries go, not by how long the context is'.

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: 10 lines of Lean

Amortized cost of memoized decode attention: (S·n + T·eF²/ε)/T = nS/T + eF²/ε

How it was proved

More proved statements