proofs
Provedads-cftblack-hole-thermodynamicshep-th-9802150witton

The surface gravity of Schwarzschild–anti-de Sitter: V′(r₊) = (n r₊² + n − 2)/r₊

For all n∈N\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}} and r∈R\htmlData{mx=1}{r \in \htmlData{mx=0}{\mathbb{R}}}, if 2≤n2 \mathrel{\htmlData{mx=1}{\leq}} \htmlData{mx=0}{n} and 0<r0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{r}, then

2r+(n−2)⋅(rn−2⋅(1+r2))rn−1=n⋅r2+n−2r\begin{aligned} & \htmlData{mx=20}{\htmlData{mx=2}{2\htmlData{mx=1}{r}} + \htmlData{mx=19}{\frac{\htmlData{mx=14}{(\htmlData{mx=5}{\htmlData{mx=4}{\htmlData{mx=3}{n}} - 2}) \cdot (\htmlData{mx=13}{\htmlData{mx=9}{\htmlData{mx=6}{r}^{\htmlData{mx=8}{\htmlData{mx=7}{n} - 2}}} \cdot (\htmlData{mx=12}{1 + \htmlData{mx=11}{\htmlData{mx=10}{r}^{2}}})})}}{\htmlData{mx=18}{\htmlData{mx=15}{r}^{\htmlData{mx=17}{\htmlData{mx=16}{n} - 1}}}}}} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=30}{\frac{\htmlData{mx=28}{\htmlData{mx=27}{\htmlData{mx=25}{\htmlData{mx=22}{\htmlData{mx=21}{n}} \cdot \htmlData{mx=24}{\htmlData{mx=23}{r}^{2}}} + \htmlData{mx=26}{n}} - 2}}{\htmlData{mx=29}{r}}} \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)
≤At most
2 is less than or equal to n. (2 ≤ n)
RReal numbers
The real numbers: every point on the number line. (ℝ)
<Less than
0 is strictly smaller than r. (0 < r)
·Multiplication
2 times r. (2 * r)
=Equality
2 · r + (n − 2) · (r^(n − 2) · (1 + r^2))/r^(n − 1) and (n · r^2 + n − 2)/r are the same. (2 * r + ((n : ℝ) - 2) * (r ^ (n - 2) * …)
a/bDivision
(n − 2) · (r^(n − 2) · (1 + r^2)) divided by r^(n − 1). In Lean, dividing by 0 gives 0. (((n : ℝ) - 2) * (r ^ (n - 2) * (1 + r ^…)
(x : T)As real numbers
n read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((n : ℝ))
-Subtraction
n minus 2. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. ((n : ℝ) - 2)
aᵇPower
r raised to the power n − 2. (r ^ (n - 2))
+Addition
1 plus r^2. (1 + r ^ 2)

For V(r) = 1 + r^2 - mu r^{-(n-2)} with mu fixed by V(r_+) = 0, i.e. mu = r_+^{n-2}(1 + r_+^2), the derivative at the horizon is V'(r_+) = 2 r_+ + (n-2) mu / r_+^{n-1} = (n r_+^2 + n - 2)/r_+. Smoothness of the Euclidean section fixes the period beta = 4 pi / V'(r_+) = 4 pi r_+/(n r_+^2 + n - 2); for n = 3 this is 4 pi r_+/(1 + 3 r_+^2), where hep-th/9802150 eq. (3.15) prints 12 pi. The analytic fact that smoothness requires exactly 4 pi / V' is not part of this statement. Decided in Alpha as wittonGeneralRegularPeriodLaw.

How it was proved

Proved
17 October 2026
By
Ajax
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
22 lines of Lean
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