Near $T_c$ : $\Delta(T)/\Delta(0) \approx 1.74\sqrt{1-T/T_c}$

Proved

BCSTheory.gap_near_Tc_weak_coupling

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

bcs-theorysuperconductivity

For fixed $k_B,\hbar\omega_D>0$ : for every coupling $\lambda>0$ the limit $L(\lambda) = \lim_{T\to T_c^-} \Delta(T)^2/(1-T/T_c)$ exists, and in the weak-coupling limit

\lim_{\lambda\to0^+}\frac{L(\lambda)}{\Delta(0)^2} = \frac{8e^{2\gamma}}{7\zeta(3)} = (1.7367\ldots)^2,

i.e. $\Delta(T)/\Delta(0)\approx 1.74\sqrt{1-T/T_c}$ for $T\approx T_c$ .

Preamble

import Mathlib
import Definitions.Def_BCSTheory_GapEquation
open Filter Topology

Formal statement

namespace BCSTheory

theorem gap_near_Tc_weak_coupling (kB hbarOmegaD : ℝ) (hkB : 0 < kB)
    (hω : 0 < hbarOmegaD) :
    ∃ L : ℝ → ℝ,
      (∀ coupling : ℝ, 0 < coupling →
        Tendsto (fun T : ℝ => gap kB hbarOmegaD coupling T ^ 2 /
            (1 - T / criticalTemperature kB hbarOmegaD coupling))
          (𝓝[<] (criticalTemperature kB hbarOmegaD coupling)) (𝓝 (L coupling))) ∧
      Tendsto (fun coupling : ℝ => L coupling / zeroTemperatureGap hbarOmegaD coupling ^ 2)
        (𝓝[>] 0)
        (𝓝 (8 * Real.exp (2 * Real.eulerMascheroniConstant) / (7 * (riemannZeta 3).re))) := 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

\Delta(T)/\Delta(0)\approx1{,}74\sqrt{1-T/T_c}

für

T\approx T_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 $k_B>0$ , $\hbar\omega_D>0$ . For $\lambda\in\mathbb R$ write $T_c(\lambda) = T_c(k_B,\hbar\omega_D,\lambda)$ , $\Delta_\lambda(T) = \Delta(k_B,\hbar\omega_D,\lambda,T)$ and $\Delta_0(\lambda) = \Delta_0(\hbar\omega_D,\lambda)$ (the supremum-based critical temperature, temperature-dependent gap and zero-temperature gap of the definition file). The statement asserts that there is a function $L:\mathbb R\to\mathbb R$ such that

for every $\lambda>0$ ,

\lim_{T\to T_c(\lambda)^-}\ \frac{\Delta_\lambda(T)^2}{1 - T/T_c(\lambda)} = L(\lambda),

the limit being taken as $T$ approaches $T_c(\lambda)$ through values strictly less than $T_c(\lambda)$ ; 2.

\lim_{\lambda\to0^+}\frac{L(\lambda)}{\Delta_0(\lambda)^2} = \frac{8\,e^{2\gamma}}{7\,\zeta(3)},

where $\gamma$ is the Euler–Mascheroni constant and $\zeta(3)$ is the real part of the Riemann zeta function at $3$ (i.e. $\sum_{n\ge1}n^{-3}$ ).

Numerically $8e^{2\gamma}/(7\zeta(3))\approx 3.016\approx 1.7367^2$ . Division by zero is read as $0$ (relevant only if $T_c(\lambda)=0$ or $\Delta_0(\lambda)=0$ ). The values of $L$ at $\lambda\le0$ are unconstrained.

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

Near TcT_cTc​: Δ(T)/Δ(0)≈1.741−T/Tc\Delta(T)/\Delta(0) \approx 1.74\sqrt{1-T/T_c}Δ(T)/Δ(0)≈1.741−T/Tc​​

Near $T_c$ : $\Delta(T)/\Delta(0) \approx 1.74\sqrt{1-T/T_c}$