Provedchi-squareproposaltemperingfirst-principles

Temperature-tilt proposals: for scores of range ≤ R and θ ∈ [0,1], the proposal softmax(θ s) has cost factor ≤ cosh²((1−θ)R/2) and envelope max π/r ≤ e^{(1−θ)R}

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{s : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=1}{R, \theta \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{n}$ , $\htmlData{mx=9}{\forall i, j,\ \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{i})} - \htmlData{mx=5}{\htmlData{mx=3}{s}(\htmlData{mx=4}{j})}} \mathrel{\htmlData{mx=8}{\leq}} \htmlData{mx=7}{R}}$ , $0 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{\theta}$ and $\htmlData{mx=0}{\theta} \mathrel{\htmlData{mx=1}{\leq}} 1$ , then

\begin{aligned} & \left(\right. \htmlData{mx=28}{\sum_{j} \htmlData{mx=27}{\frac{\htmlData{mx=12}{\left(\htmlData{mx=11}{\frac{\htmlData{mx=5}{e^{\htmlData{mx=4}{\htmlData{mx=2}{s}(\htmlData{mx=3}{j})}}}}{\htmlData{mx=10}{\sum_{k} \htmlData{mx=9}{e^{\htmlData{mx=8}{\htmlData{mx=6}{s}(\htmlData{mx=7}{k})}}}}}}\right)^{2}}}{\htmlData{mx=26}{\frac{\htmlData{mx=18}{e^{\htmlData{mx=17}{\htmlData{mx=13}{\theta} \cdot \htmlData{mx=16}{\htmlData{mx=14}{s}(\htmlData{mx=15}{j})}}}}}{\htmlData{mx=25}{\sum_{k} \htmlData{mx=24}{e^{\htmlData{mx=23}{\htmlData{mx=19}{\theta} \cdot \htmlData{mx=22}{\htmlData{mx=20}{s}(\htmlData{mx=21}{k})}}}}}}}}}} \\ & \quad \htmlData{mx=1}{\leq} \htmlData{mx=35}{\htmlData{mx=34}{\cosh \left(\htmlData{mx=33}{\frac{\htmlData{mx=32}{(\htmlData{mx=30}{1 - \htmlData{mx=29}{\theta}}) \cdot \htmlData{mx=31}{R}}}{2}}\right)}^{2}} \left.\right) \\ & \htmlData{mx=0}{\land} \left(\right. \htmlData{mx=36}{\forall} j, \\ & \quad \htmlData{mx=62}{\frac{\htmlData{mx=47}{\frac{\htmlData{mx=41}{e^{\htmlData{mx=40}{\htmlData{mx=38}{s}(\htmlData{mx=39}{j})}}}}{\htmlData{mx=46}{\sum_{k} \htmlData{mx=45}{e^{\htmlData{mx=44}{\htmlData{mx=42}{s}(\htmlData{mx=43}{k})}}}}}}}{\htmlData{mx=61}{\frac{\htmlData{mx=53}{e^{\htmlData{mx=52}{\htmlData{mx=48}{\theta} \cdot \htmlData{mx=51}{\htmlData{mx=49}{s}(\htmlData{mx=50}{j})}}}}}{\htmlData{mx=60}{\sum_{k} \htmlData{mx=59}{e^{\htmlData{mx=58}{\htmlData{mx=54}{\theta} \cdot \htmlData{mx=57}{\htmlData{mx=55}{s}(\htmlData{mx=56}{k})}}}}}}}}} \\ & \quad \quad \htmlData{mx=37}{\leq} \htmlData{mx=67}{e^{\htmlData{mx=66}{(\htmlData{mx=64}{1 - \htmlData{mx=63}{\theta}}) \cdot \htmlData{mx=65}{R}}}} \left.\right) \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
<Less than: 0 is strictly smaller than n. (0 < n)
Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
x ∈ TVariables: R, θ range over real numbers: the statement must hold for every choice. ((R θ : ℝ))
∀For all: The statement after the comma holds for every i, j. (∀ i j, s i - s j ≤ R)
∧And: Both hold: (∑ j, (exp(s(j))/(∑ k, exp(s(k))))^2/(exp(θ · s(j))/(∑ k, …, and ∀ j, exp(s(j))/(∑ k, exp(s(k)))/(exp(θ · s(j))/(∑ k, exp(θ…. ((∑ j, (Real.exp (s j) / ∑ k, Real.exp (…)
aᵇPower: exp(s(j))/(∑ k, exp(s(k))) raised to the power 2. ((Real.exp (s j) / ∑ k, Real.exp (s k)) …)
-Subtraction: s(i) minus s(j). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (s i - s j)
≤At most: s(i) − s(j) is less than or equal to R. (s i - s j ≤ R)
eˣExponential function: e raised to the power s(j), where e ≈ 2.718. (Real.exp (s j))
a/bDivision: exp(s(j)) divided by ∑ k, exp(s(k)). In Lean, dividing by 0 gives 0. (Real.exp (s j) / ∑ k, Real.exp (s k))
RReal numbers: The real numbers: every point on the number line. (ℝ)
∑Summation: Add up exp(s(k)) for every k. (∑ k, Real.exp (s k))
·Multiplication: θ times s(j). (θ * s j)
coshHyperbolic cosine: (eˣ + e⁻ˣ)/2 at (1 − θ) · R/2. (Real.cosh ((1 - θ) * R / 2))

theorem ua-tilt-family-cost-bound {n : ℕ} (hn : 0 < n) (s : Fin n → ℝ) (R θ : ℝ) (hR : ∀ i j, s i - s j ≤ R) (hθ0 : 0 ≤ θ) (hθ1 : θ ≤ 1) :
    (∑ j, (Real.exp (s j) / ∑ k, Real.exp (s k)) ^ 2 / (Real.exp (θ * s j) / ∑ k, Real.exp (θ * s k)) ≤ Real.cosh ((1 - θ) * R / 2) ^ 2) ∧ (∀ j, (Real.exp (s j) / ∑ k, Real.exp (s k)) / (Real.exp (θ * s j) / ∑ k, Real.exp (θ * s k)) ≤ Real.exp ((1 - θ) * R))

A proposal at reduced temperature θ. θ = 0 is uniform (cosh²(R/2)), θ = 1 is the row itself (1), and θ = 1/2 is the half-temperature proposal with cost ≤ cosh²(R/4) ≈ e^{R/2}/4: tempering halves the exponent. Hashing-based estimators realise sampling from the half-temperature tilt for hashable kernels, which is why they beat blind sampling by a square root. Proof: the Kantorovich range inequality with weights e^{θ s} and values e^{(1−θ)s}. 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: 92 lines of Lean

Temperature-tilt proposals: for scores of range ≤ R and θ ∈ [0,1], the proposal softmax(θ s) has cost factor ≤ cosh²((1−θ)R/2) and envelope max π/r ≤ e^{(1−θ)R}

How it was proved

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