             \begin{document}$$\square $$\end{document}$

Universality of the Euclidean Conductivity Matrix {#Sec11}
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Here we prove the universality of the Euclidean conductivity matrix, defined in Eq. ([3.4](#Equ28){ref-type=""}). We restrict to the range of parameters ([2.20](#Equ20){ref-type=""}), as discussed at the beginning of Sect. [2.3.1](#Sec6){ref-type="sec"}. In terms of the renormalized parameters, we restate ([2.20](#Equ20){ref-type=""}) as$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} 0<\phi \leqslant \frac{\pi }{2}\;,\qquad m_{\text {R},+}>|m_{\text {R},-}|\;, \end{aligned}$$\end{document}$$where$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} m_{R,+}:=m_{R,-}+6\sqrt{3}\,t_{2}\sin \phi \;. \end{aligned}$$\end{document}$$A key ingredient in the proof is the following regularity result for the correlation functions.

### Proposition 3.3 {#FPar6}

There exists $\documentclass[12pt]{minimal}
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                \begin{document}$$U_{0}>0$$\end{document}$ such that, for 