Proveddrift-lawdecodesnapshot-reuseattention-compute-law

Tilt ratio bounds: if every score moved by at most η then e^{−η} ≤ Z_t/Z_s ≤ e^{η}

For all $\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{s, t : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{\eta \in \htmlData{mx=0}{\mathbb{R}}}$ , if $\htmlData{mx=10}{\forall j,\ \htmlData{mx=7}{\left\lvert \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{t}(\htmlData{mx=1}{j})} - \htmlData{mx=5}{\htmlData{mx=3}{s}(\htmlData{mx=4}{j})}} \right\rvert} \mathrel{\htmlData{mx=9}{\leq}} \htmlData{mx=8}{\eta}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{J}}$ ), then

\begin{aligned} & \htmlData{mx=3}{e^{\htmlData{mx=2}{-\htmlData{mx=1}{\eta}}}} \mathrel{\htmlData{mx=15}{\leq}} \htmlData{mx=14}{\frac{\htmlData{mx=8}{\sum_{k} \htmlData{mx=7}{e^{\htmlData{mx=6}{\htmlData{mx=4}{t}(\htmlData{mx=5}{k})}}}}}{\htmlData{mx=13}{\sum_{k} \htmlData{mx=12}{e^{\htmlData{mx=11}{\htmlData{mx=9}{s}(\htmlData{mx=10}{k})}}}}}} \\ & \htmlData{mx=0}{\land} \htmlData{mx=26}{\frac{\htmlData{mx=20}{\sum_{k} \htmlData{mx=19}{e^{\htmlData{mx=18}{\htmlData{mx=16}{t}(\htmlData{mx=17}{k})}}}}}{\htmlData{mx=25}{\sum_{k} \htmlData{mx=24}{e^{\htmlData{mx=23}{\htmlData{mx=21}{s}(\htmlData{mx=22}{k})}}}}}} \mathrel{\htmlData{mx=29}{\leq}} \htmlData{mx=28}{e^{\htmlData{mx=27}{\eta}}} \end{aligned}

What the notation means · or point at any symbol

Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to J. Lean writes f x; papers write f(x). (Fintype J)
Nonempty: Nonempty: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Nonempty)
⟹Implies: If J, then ℝ. (Between types, A → B is the type of functions from A to B.) (J → ℝ)
x : TVariables: s, t range over J → ℝ: the statement must hold for every choice. ((s t : J → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, |t j - s j| ≤ η)
eˣExponential function: e raised to the power −η, where e ≈ 2.718. (Real.exp (-η))
∧And: Both hold: exp(−η) ≤ (∑ k, exp(t(k)))/(∑ k, exp(s(k))), and (∑ k, exp(t(k)))/(∑ k, exp(s(k))) ≤ exp(η). (Real.exp (-η) ≤ (∑ k, Real.exp (t k)) /…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
|x|Absolute value: The size of t(j) − s(j) without its sign. (|t j - s j|)
≤At most: |t(j) − s(j)| is less than or equal to η. (|t j - s j| ≤ η)
-Subtraction: t(j) minus s(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (t j - s j)
−xNegation: The negative of η. (-η)
a/bDivision: ∑ k, exp(t(k)) divided by ∑ k, exp(s(k)). In Lean, dividing by 0 gives 0. ((∑ k, Real.exp (t k)) / ∑ k, Real.exp (…)
∑Summation: Add up exp(t(k)) for every k. (∑ k, Real.exp (t k))

theorem rl-tilt-ratio-bounds {J : Type} [Fintype J] [Nonempty J] (s t : J → ℝ) (η : ℝ) (hη : ∀ j, |t j - s j| ≤ η) :
    Real.exp (-η) ≤ (∑ k, Real.exp (t k)) / ∑ k, Real.exp (s k) ∧ (∑ k, Real.exp (t k)) / ∑ k, Real.exp (s k) ≤ Real.exp η

The normaliser-ratio estimator has bounded relative variance, so the new-key combination is debiasable by MLMC at constant cost. 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: 21 lines of Lean

Tilt ratio bounds: if every score moved by at most η then e^{−η} ≤ Z_t/Z_s ≤ e^{η}

How it was proved

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