Weak coupling: $k_BT_c \approx 1.14\,\hbar\omega_D\,e^{-1/V_0Z(E_F)}$

Proved

BCSTheory.criticalTemperature_weak_coupling

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

bcs-theorysuperconductivity

For fixed $k_B>0$ and $\hbar\omega_D>0$ , as $\lambda=V_0Z(E_F)\to0^+$ ,

\frac{k_BT_c}{\hbar\omega_D\,e^{-1/\lambda}} \to \frac{2e^{\gamma}}{\pi} = 1.1338\ldots,

where $\gamma$ is the Euler–Mascheroni constant.

Preamble

import Mathlib
import Definitions.Def_BCSTheory_GapEquation
open Filter Topology

Formal statement

namespace BCSTheory

theorem criticalTemperature_weak_coupling (kB hbarOmegaD : ℝ) (hkB : 0 < kB)
    (hω : 0 < hbarOmegaD) :
    Tendsto (fun coupling : ℝ =>
        kB * criticalTemperature kB hbarOmegaD coupling /
          (hbarOmegaD * Real.exp (-1 / coupling)))
      (𝓝[>] 0) (𝓝 (2 * Real.exp Real.eulerMascheroniConstant / Real.pi)) := by sorry

end BCSTheory

Source

Wikipedia (de), Artikel „BCS-Theorie“ (PDF-Ausdruck vom 05.10.2026), Abschnitt „Erfolge der BCS-Theorie“, Unterabschnitt „Existenz einer Energielücke“, Formel

k_BT_c = 1{,}14\,\hbar\omega_D e^{-1/V_0Z(E_F)}

(exakte Konstante

2e^\gamma/\pi

folgt aus

\Delta(0) = 2\hbar\omega_D e^{-1/V_0Z(E_F)} = (\pi/e^\gamma) k_BT_c

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements, with full knowledge of the source text and of the intended meaning. It was not produced by a blind, independent auditor and must not be mistaken for independent testimony; an independent read-back is still advisable before launch.

Let $k_B, \hbar\omega_D$ be real numbers with $k_B>0$ and $\hbar\omega_D>0$ . Then, as $\lambda\to0^+$ (through strictly positive values),

\frac{k_B\,T_c(k_B,\hbar\omega_D,\lambda)}{\hbar\omega_D\,e^{-1/\lambda}} \longrightarrow \frac{2e^{\gamma}}{\pi},

where $\gamma$ is the Euler–Mascheroni constant and $T_c(k_B,\hbar\omega_D,\lambda)$ is the critical temperature of the definition file: the supremum of all $T>0$ with $\int_0^{\hbar\omega_D}\tanh(|\eta|/(2k_BT))/|\eta|\,d\eta = 1/\lambda$ (supremum $=0$ if this set is empty or unbounded). Numerically $2e^\gamma/\pi\approx1.1339$ ; the statement uses the exact constant, not $1.14$ .

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Weak coupling: kBTc≈1.14 ℏωD e−1/V0Z(EF)k_BT_c \approx 1.14\,\hbar\omega_D\,e^{-1/V_0Z(E_F)}kB​Tc​≈1.14ℏωD​e−1/V0​Z(EF​)

Weak coupling: $k_BT_c \approx 1.14\,\hbar\omega_D\,e^{-1/V_0Z(E_F)}$