proofs
Proveddrift-lawdecodechi-squarefirst-principles

Stratified snapshot reuse: off an exceptional set O, if the drift t − s has oscillation ≤ Δ, then (Σ_{∉O} e^{2t−s})(Σ_{∉O} e^{s}) ≤ cosh²(Δ/2)(Σ_{∉O} e^{t})²

For all J∈Type⁡\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}}, s,t:J→R\htmlData{mx=3}{s, t : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}, O∈Finset⁡ J\htmlData{mx=3}{O \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Finset}}\,\htmlData{mx=1}{J}}} and Δ∈R\htmlData{mx=1}{\Delta \in \htmlData{mx=0}{\mathbb{R}}}, if ∀i,j, i∉O  ⟹  j∉O  ⟹  t(i)−s(i)−(t(j)−s(j))≤Δ\htmlData{mx=25}{\forall i, j,\ \htmlData{mx=0}{i} \mathrel{\htmlData{mx=2}{\notin}} \htmlData{mx=1}{O} \mathrel{\htmlData{mx=24}{\implies}} \htmlData{mx=3}{j} \mathrel{\htmlData{mx=5}{\notin}} \htmlData{mx=4}{O} \mathrel{\htmlData{mx=23}{\implies}} \htmlData{mx=20}{\htmlData{mx=12}{\htmlData{mx=8}{\htmlData{mx=6}{t}(\htmlData{mx=7}{i})} - \htmlData{mx=11}{\htmlData{mx=9}{s}(\htmlData{mx=10}{i})}} - (\htmlData{mx=19}{\htmlData{mx=15}{\htmlData{mx=13}{t}(\htmlData{mx=14}{j})} - \htmlData{mx=18}{\htmlData{mx=16}{s}(\htmlData{mx=17}{j})}})} \mathrel{\htmlData{mx=22}{\leq}} \htmlData{mx=21}{\Delta}} and ∃j, j∉O\htmlData{mx=3}{\exists j,\ \htmlData{mx=0}{j} \mathrel{\htmlData{mx=2}{\notin}} \htmlData{mx=1}{O}} (given Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}} and DecidableEq⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{DecidableEq}}\,\htmlData{mx=1}{J}}), then

(∑j∈filter⁡(j↦j∉O)e2t(j)−s(j))⋅(∑j∈filter⁡(j↦j∉O)es(j))≤cosh⁡(Δ2)2⋅(∑j∈filter⁡(j↦j∉O)et(j))2\begin{aligned} & \htmlData{mx=28}{\left(\htmlData{mx=16}{\sum_{j \in \htmlData{mx=6}{\htmlData{mx=1}{\operatorname{filter}}(\htmlData{mx=5}{j \mapsto \htmlData{mx=2}{j} \mathrel{\htmlData{mx=4}{\notin}} \htmlData{mx=3}{O}})}} \htmlData{mx=15}{e^{\htmlData{mx=14}{\htmlData{mx=10}{2\htmlData{mx=9}{\htmlData{mx=7}{t}(\htmlData{mx=8}{j})}} - \htmlData{mx=13}{\htmlData{mx=11}{s}(\htmlData{mx=12}{j})}}}}}\right) \cdot \left(\htmlData{mx=27}{\sum_{j \in \htmlData{mx=22}{\htmlData{mx=17}{\operatorname{filter}}(\htmlData{mx=21}{j \mapsto \htmlData{mx=18}{j} \mathrel{\htmlData{mx=20}{\notin}} \htmlData{mx=19}{O}})}} \htmlData{mx=26}{e^{\htmlData{mx=25}{\htmlData{mx=23}{s}(\htmlData{mx=24}{j})}}}}\right)} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=45}{\htmlData{mx=32}{\htmlData{mx=31}{\cosh \left(\htmlData{mx=30}{\frac{\htmlData{mx=29}{\Delta}}{2}}\right)}^{2}} \cdot \htmlData{mx=44}{\left(\htmlData{mx=43}{\sum_{j \in \htmlData{mx=38}{\htmlData{mx=33}{\operatorname{filter}}(\htmlData{mx=37}{j \mapsto \htmlData{mx=34}{j} \mathrel{\htmlData{mx=36}{\notin}} \htmlData{mx=35}{O}})}} \htmlData{mx=42}{e^{\htmlData{mx=41}{\htmlData{mx=39}{t}(\htmlData{mx=40}{j})}}}}\right)^{2}}} \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)
DecidableEq
DecidableEq: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (DecidableEq)
⟹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))
Finset
Finset: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset)
∀For all
The statement after the comma holds for every i, j. (∀ i j, i ∉ O → j ∉ O → (t i - s i) - (t…)
∃There exists
At least one j makes the statement after the comma true. (∃ j, j ∉ O)
∑Summation
Add up exp(2 · t(j) − s(j)) for each j in filter(j ↦ j ∉ O). (∑ j ∈ Finset.univ.filter (fun j => j ∉ …)
RReal numbers
The real numbers: every point on the number line. (ℝ)
∉Not an element of
i is not a member of O. (i ∉ O)
filter
Finset.univ.filter: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ.filter)
≤At most
t(i) − s(i) − (t(j) − s(j)) is less than or equal to Δ. ((t i - s i) - (t j - s j) ≤ Δ)
-Subtraction
t(i) minus s(i). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (t i - s i)
x ↦ f(x)Function (λ-abstraction)
The function sending j to j ∉ O. Lean writes fun x => …; papers write x ↦ …. (fun j => j ∉ O)
eˣExponential function
e raised to the power 2 · t(j) − s(j), where e ≈ 2.718. (Real.exp (2 * t j - s j))
·Multiplication
2 times t(j). (2 * t j)
coshHyperbolic cosine
(eˣ + e⁻ˣ)/2 at Δ/2. (Real.cosh (Δ / 2))
aᵇPower
cosh(Δ/2) raised to the power 2. (Real.cosh (Δ / 2) ^ 2)
a/bDivision
Δ divided by 2. In Lean, dividing by 0 gives 0. (Δ / 2)

The importance-sampling second-moment factor of the stale snapshot restricted to the keys outside an exceptional set, relative to the current mass of those keys. Keys in O are read exactly; the rest are estimated by importance sampling from the snapshot with a factor governed only by the oscillation of the drift on the complement. Proof: Kantorovich range inequality with the weights zeroed on O. First-principles reformulation, 2026-10-05.

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
85 lines of Lean
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