% Shared macros. One object, one macro. Sections add nothing here; local symbols stay local to a proof.
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\E}{\mathbb{E}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\dd}{\,\mathrm{d}}
\newcommand{\one}{\mathbf{1}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\Hess}{\operatorname{Hess}}
\newcommand{\Var}{\operatorname{Var}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Lip}{\operatorname{Lip}}
\newcommand{\atanh}{\operatorname{atanh}}
\newcommand{\erfinv}{\operatorname{erfinv}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\Law}{\mathcal{L}}          % law of a path: \Law_x
\newcommand{\Cube}{\Sigma_N}            % {-1,1}^N
\newcommand{\Slice}{S_N}                % S_N(k): configurations with k plus signs
\newcommand{\Eeta}{\E_{\eta}}           % average over eta uniform on {-1,1}^{r-1}
\newcommand{\tv}{\boldsymbol{t}}        % multi-type magnetization (t_eta)
\newcommand{\ov}{\boldsymbol{\omega}}   % overlaps with the further retained patterns
% free energies at fixed overlap
\newcommand{\Lam}{\Lambda}              % Gaussian rows, one binary pattern
\newcommand{\LamR}{\Lambda^{\mathrm{R}}}% +-1 patterns
\newcommand{\eR}{e^{\mathrm{R}}}        % T = 0, +-1 patterns
\newcommand{\Phimix}{\Phi^{\mathrm{mix}}}
\newcommand{\Sbeta}{\mathcal{S}_{\beta}}% (beta/2) int_0^1 dw/D_x(w)
\newcommand{\Fr}{\mathcal{F}_{r}}       % beta V^2/2 + Phi^{(r)}(t)
% capacity
\newcommand{\Well}{\mathcal{W}}
\newcommand{\WellR}{\mathcal{W}^{\mathrm{R}}}
\newcommand{\alc}{\alpha_{\mathrm{c}}}
\newcommand{\alcR}{\alpha_{\mathrm{c}}^{\mathrm{R}}}
\newcommand{\alo}{\underline{\alpha}}   % 0.138198129
\newcommand{\ahi}{\overline{\alpha}}    % 0.13820106
% Parisi and sphere functionals
\newcommand{\Qone}{\mathcal{Q}_{1}}     % nondecreasing right-continuous paths [0,1) -> [0,1]
\newcommand{\Qinf}{\mathcal{Q}_{\infty}}% bounded paths with values in R_+
\newcommand{\psiS}{\psi_{\mathrm{S}}}
\newcommand{\psiSs}{\psi_{\mathrm{S}}^{*}}
\newcommand{\resp}{\mathcal{M}}         % magnetization response \resp_\nu(u) = d_x phi_nu(0,u)
\newcommand{\rad}{\operatorname{rad}}   % rad(v) = (alpha/2)(1 + log v - v)
\newcommand{\Nd}{\mathsf{g}}            % standard normal density
\newcommand{\Nc}{\mathsf{G}}            % standard normal distribution function
\newcommand{\wf}{\mathsf{w}}            % w(m) = 2 g(lambda_0(m))