proofs
Proveddrift-lawdecodechi-squareheadline-theorem

Drift cost by oscillation: if the score drift δ = t − s has oscillation ≤ Δ then 1+χ²(π_t‖π_s) ≤ cosh²(Δ/2); under |δ| ≤ η this is ≤ cosh²η

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}}} and Δ,η∈R\htmlData{mx=1}{\Delta, \eta \in \htmlData{mx=0}{\mathbb{R}}}, if ∀i,j, t(i)−s(i)−(t(j)−s(j))≤Δ\htmlData{mx=17}{\forall i, j,\ \htmlData{mx=14}{\htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{t}(\htmlData{mx=1}{i})} - \htmlData{mx=5}{\htmlData{mx=3}{s}(\htmlData{mx=4}{i})}} - (\htmlData{mx=13}{\htmlData{mx=9}{\htmlData{mx=7}{t}(\htmlData{mx=8}{j})} - \htmlData{mx=12}{\htmlData{mx=10}{s}(\htmlData{mx=11}{j})}})} \mathrel{\htmlData{mx=16}{\leq}} \htmlData{mx=15}{\Delta}} (given Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}} and Nonempty⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{J}}), then

∑j(et(j)∑ket(k))2es(j)∑kes(k)≤cosh⁡(Δ2)2∧((∀j, ∣t(j)−s(j)∣≤η)  ⟹  ∑j(et(j)∑ket(k))2es(j)∑kes(k)≤cosh⁡η2)\begin{aligned} & \htmlData{mx=24}{\sum_{j} \htmlData{mx=23}{\frac{\htmlData{mx=12}{\left(\htmlData{mx=11}{\frac{\htmlData{mx=5}{e^{\htmlData{mx=4}{\htmlData{mx=2}{t}(\htmlData{mx=3}{j})}}}}{\htmlData{mx=10}{\sum_{k} \htmlData{mx=9}{e^{\htmlData{mx=8}{\htmlData{mx=6}{t}(\htmlData{mx=7}{k})}}}}}}\right)^{2}}}{\htmlData{mx=22}{\frac{\htmlData{mx=16}{e^{\htmlData{mx=15}{\htmlData{mx=13}{s}(\htmlData{mx=14}{j})}}}}{\htmlData{mx=21}{\sum_{k} \htmlData{mx=20}{e^{\htmlData{mx=19}{\htmlData{mx=17}{s}(\htmlData{mx=18}{k})}}}}}}}}} \\ & \quad \htmlData{mx=1}{\leq} \htmlData{mx=28}{\htmlData{mx=27}{\cosh \left(\htmlData{mx=26}{\frac{\htmlData{mx=25}{\Delta}}{2}}\right)}^{2}} \\ & \htmlData{mx=0}{\land} \left(\right. (\htmlData{mx=40}{\forall j,\ \htmlData{mx=37}{\left\lvert \htmlData{mx=36}{\htmlData{mx=32}{\htmlData{mx=30}{t}(\htmlData{mx=31}{j})} - \htmlData{mx=35}{\htmlData{mx=33}{s}(\htmlData{mx=34}{j})}} \right\rvert} \mathrel{\htmlData{mx=39}{\leq}} \htmlData{mx=38}{\eta}}) \\ & \htmlData{mx=29}{\implies} \htmlData{mx=64}{\sum_{j} \htmlData{mx=63}{\frac{\htmlData{mx=52}{\left(\htmlData{mx=51}{\frac{\htmlData{mx=45}{e^{\htmlData{mx=44}{\htmlData{mx=42}{t}(\htmlData{mx=43}{j})}}}}{\htmlData{mx=50}{\sum_{k} \htmlData{mx=49}{e^{\htmlData{mx=48}{\htmlData{mx=46}{t}(\htmlData{mx=47}{k})}}}}}}\right)^{2}}}{\htmlData{mx=62}{\frac{\htmlData{mx=56}{e^{\htmlData{mx=55}{\htmlData{mx=53}{s}(\htmlData{mx=54}{j})}}}}{\htmlData{mx=61}{\sum_{k} \htmlData{mx=60}{e^{\htmlData{mx=59}{\htmlData{mx=57}{s}(\htmlData{mx=58}{k})}}}}}}}}} \\ & \quad \htmlData{mx=41}{\leq} \htmlData{mx=67}{\htmlData{mx=66}{\cosh \htmlData{mx=65}{\eta}}^{2}} \left.\right) \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))
x ∈ TVariables
Δ, η range over real numbers: the statement must hold for every choice. ((Δ η : ℝ))
∀For all
The statement after the comma holds for every i, j. (∀ i j, (t i - s i) - (t j - s j) ≤ Δ)
∧And
Both hold: (∑ j, (exp(t(j))/(∑ k, exp(t(k))))^2/(exp(s(j))/(∑ k, exp(…, and (∀ j, |t(j) − s(j)| ≤ η) → (∑ j, (exp(t(j))/(∑ k, exp(t(k)…. (∑ j, (Real.exp (t j) / ∑ k, Real.exp (t…)
RReal numbers
The real numbers: every point on the number line. (ℝ)
aᵇPower
exp(t(j))/(∑ k, exp(t(k))) raised to the power 2. ((Real.exp (t j) / ∑ k, Real.exp (t k)) …)
eˣExponential function
e raised to the power t(j), where e ≈ 2.718. (Real.exp (t j))
a/bDivision
exp(t(j)) divided by ∑ k, exp(t(k)). In Lean, dividing by 0 gives 0. (Real.exp (t j) / ∑ k, Real.exp (t k))
≤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)
∑Summation
Add up exp(t(k)) for every k. (∑ k, Real.exp (t k))
coshHyperbolic cosine
(eˣ + e⁻ˣ)/2 at Δ/2. (Real.cosh (Δ / 2))
|x|Absolute value
The size of t(j) − s(j) without its sign. (|t j - s j|)

A common score shift changes no softmax row, so the right drift measure is the oscillation max δ − min δ. This sharpens the SNIS factor bound e^{2η} of drift-snis-dominates-rejection to cosh²η (1.27 instead of 2.72 at η = 0.5). It bounds the importance-sampling second-moment factor only; the rejection envelope (expected draws) is a different quantity. It also gives the operational sampleability condition: a score approximation s̃ with osc(s − s̃) ≤ Δ has C(softmax s ‖ softmax s̃) ≤ cosh²(Δ/2). From an external review, 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
73 lines of Lean
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