proofs
Proveddrift-lawdecodesnapshot-reuseattention-compute-law

Refresh optimum: T = √(n/K) attains n/T + K T = 2√(nK)

For all n,K∈R\htmlData{mx=1}{n, K \in \htmlData{mx=0}{\mathbb{R}}}, if 0<n0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n} and 0<K0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{K}, then

nnK+K⋅nK=2n⋅K\htmlData{mx=12}{\htmlData{mx=5}{\frac{\htmlData{mx=0}{n}}{\htmlData{mx=4}{\sqrt{\htmlData{mx=3}{\frac{\htmlData{mx=1}{n}}{\htmlData{mx=2}{K}}}}}}} + \htmlData{mx=11}{\htmlData{mx=6}{K} \cdot \htmlData{mx=10}{\sqrt{\htmlData{mx=9}{\frac{\htmlData{mx=7}{n}}{\htmlData{mx=8}{K}}}}}}} \mathrel{\htmlData{mx=18}{=}} \htmlData{mx=17}{2\htmlData{mx=16}{\sqrt{\htmlData{mx=15}{\htmlData{mx=13}{n} \cdot \htmlData{mx=14}{K}}}}}
What the notation means · or point at any symbol
RReal numbers
The real numbers: every point on the number line. (ℝ)
x ∈ TVariables
n, K range over real numbers: the statement must hold for every choice. ((n K : ℝ))
<Less than
0 is strictly smaller than n. (0 < n)
+Addition
n/√(n/K) plus K · √(n/K). (n / Real.sqrt (n / K) + K * Real.sqrt (…)
=Equality
n/√(n/K) + K · √(n/K) and 2 · √(n · K) are the same. (n / Real.sqrt (n / K) + K * Real.sqrt (…)
√Square root
The non-negative number whose square is n/K. (Real.sqrt (n / K))
a/bDivision
n divided by K. In Lean, dividing by 0 gives 0. (n / K)
·Multiplication
K times √(n/K). (K * Real.sqrt (n / K))

The optimal refresh period in the transient (chord) regime. Part of 'The Drift Law: decode K/V traffic is set by how far the query drifts' (exact snapshot reuse at decode).

How it was proved

Proved
5 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
Sign in to see the proofThe proof, its signed report and its history.

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