Provedfisher-cover-lawdecode-attentionexponential-familylaw

Log-convexity of the partition function: Z(q+Δ)² ≤ Z(q) Z(q+2Δ)

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ and $\htmlData{mx=5}{a, d : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ :

\begin{aligned} & \htmlData{mx=10}{\left(\htmlData{mx=9}{\sum_{k} \htmlData{mx=8}{e^{\htmlData{mx=7}{\htmlData{mx=3}{\htmlData{mx=1}{a}(\htmlData{mx=2}{k})} + \htmlData{mx=6}{\htmlData{mx=4}{d}(\htmlData{mx=5}{k})}}}}}\right)^{2}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=26}{\left(\htmlData{mx=15}{\sum_{k} \htmlData{mx=14}{e^{\htmlData{mx=13}{\htmlData{mx=11}{a}(\htmlData{mx=12}{k})}}}}\right) \cdot \left(\htmlData{mx=25}{\sum_{k} \htmlData{mx=24}{e^{\htmlData{mx=23}{\htmlData{mx=18}{\htmlData{mx=16}{a}(\htmlData{mx=17}{k})} + \htmlData{mx=22}{2\htmlData{mx=21}{\htmlData{mx=19}{d}(\htmlData{mx=20}{k})}}}}}}\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)
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: a, d range over Fin n → ℝ: the statement must hold for every choice. ((a d : Fin n → ℝ))
aᵇPower: ∑ k, exp(a(k) + d(k)) raised to the power 2. ((∑ k, Real.exp (a k + d k)) ^ 2)
≤At most: (∑ k, exp(a(k) + d(k)))^2 is less than or equal to (∑ k, exp(a(k))) · (∑ k, exp(a(k) + 2 · d(k))). ((∑ k, Real.exp (a k + d k)) ^ 2 ≤ (∑ k,…)
∑Summation: Add up exp(a(k) + d(k)) for every k. (∑ k, Real.exp (a k + d k))
eˣExponential function: e raised to the power a(k) + d(k), where e ≈ 2.718. (Real.exp (a k + d k))
RReal numbers: The real numbers: every point on the number line. (ℝ)
+Addition: a(k) plus d(k). (a k + d k)
·Multiplication: 2 times d(k). (2 * d k)

theorem fc-partition-log-convex {n : ℕ} (a d : Fin n → ℝ) :
    (∑ k, Real.exp (a k + d k)) ^ 2 ≤ (∑ k, Real.exp (a k)) * (∑ k, Real.exp (a k + 2 * d k))

Cauchy–Schwarz; equivalently 1 + χ² ≥ 1. Part of 'Fisher-Cover Law: snapshot reuse is governed by the Fisher geometry of the attention exponential family'.

How it was proved

Proved: 4 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: 19 lines of Lean

Log-convexity of the partition function: Z(q+Δ)² ≤ Z(q) Z(q+2Δ)

How it was proved

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