%
%
%
%
%
%
%
%

%
%
\providecommand{\ZZ}{\mathbb{Z}}
\providecommand{\RR}{\mathbb{R}}
\providecommand{\NN}{\mathbb{N}}
\providecommand{\CC}{\mathbb{C}}
\providecommand{\Zthree}{\mathbb{Z}^{3}}
\providecommand{\EE}{\mathbb{E}}
\providecommand{\PP}{\mathbb{P}}
\providecommand{\Ber}{\mathrm{Ber}}
\providecommand{\eps}{\varepsilon}
\providecommand{\Ord}{\mathcal{O}}
\providecommand{\one}{\mathbf{1}}
\providecommand{\supp}{\operatorname{supp}}
\providecommand{\dist}{\operatorname{dist}}
\providecommand{\diam}{\operatorname{diam}}
\providecommand{\Tr}{\operatorname{Tr}}
\providecommand{\avg}{\operatorname{avg}}
\providecommand{\Var}{\operatorname{Var}}
\providecommand{\ReP}{\operatorname{Re}}               %
\providecommand{\la}{\langle}
\providecommand{\ra}{\rangle}
\providecommand{\abs}[1]{\lvert #1\rvert}
\providecommand{\norm}[1]{\lVert #1\rVert}
\providecommand{\ex}[1]{\langle #1\rangle}   %

%
%
\providecommand{\Lam}{\Lambda}
\providecommand{\Lamo}{\Lambda^{\circ}}             %
\providecommand{\bdtwo}{\partial_{2}}                %
\providecommand{\Gp}{G'}                             %
\providecommand{\GpD}{G'(D)}
\providecommand{\XX}{\mathbb{X}}                     %
\providecommand{\MM}{\mathbb{M}}                     %
\providecommand{\KK}{\mathfrak{K}}                   %
\providecommand{\KKh}{\widehat{\mathfrak{K}}}        %
\providecommand{\KKb}{\mathfrak{K}^{\flat}}          %
\providecommand{\FF}{\mathfrak{F}}                   %
\providecommand{\FS}{F_{S}}                          %
\providecommand{\Dep}{\mathfrak{D}}                  %
\providecommand{\cQ}{\mathcal{Q}}                    %
\providecommand{\Qh}{\widehat{Q}}
\providecommand{\cU}{\mathcal{U}}                    %
\providecommand{\cP}{\mathcal{P}}
\providecommand{\Pad}{\mathcal{P}^{\mathrm{ad}}}     %
\providecommand{\Grid}{\mathfrak{G}}                 %
\providecommand{\Col}{\mathrm{Col}}                  %
%
\providecommand{\Sigo}{\Sigma^{\circ}}               %
\providecommand{\aSig}{a_{\Sigma}}
\providecommand{\Sst}[1]{\mathfrak{S}^{*}_{#1}}      %
\providecommand{\Sss}[1]{\mathfrak{S}^{**}_{#1}}
\providecommand{\Bpay}{B_{\mathrm{pay}}}
\providecommand{\Qatt}{\mathcal{Q}_{\mathrm{att}}}
\providecommand{\ncut}{n_{\mathrm{cut}}}
\providecommand{\Scut}{S_{\mathrm{cut}}}

%
\providecommand{\gfix}{g^{\mathrm{fix}}}             %
\providecommand{\WG}{W_{G'}}                         %
\providecommand{\AG}{A_{G'}}                         %
\providecommand{\vt}{\tilde v}                       %
\providecommand{\omb}{\omega_{\beta}}                %
\providecommand{\thJ}{\vartheta_{J}}                 %
\providecommand{\Rone}{R_{1}}                        %

%
%
\providecommand{\bGM}{\beta_{\mathrm{GM}}}           %
\providecommand{\bGMo}{\beta^{0}_{\mathrm{GM}}}     %
\providecommand{\Kst}{K_{*}}                         %
\providecommand{\cdel}{c_{\delta}}                   %
\providecommand{\cdelsp}{c_{\delta}^{\mathrm{sp}}}
\providecommand{\astar}{a_{*}}
\providecommand{\qm}{q_{m}}
\providecommand{\mD}{m_{D}}
\providecommand{\mt}{\tilde m}
\providecommand{\lPV}{\lambda_{\mathrm{PV}}}
\providecommand{\MY}{M_{Y}}
\providecommand{\bD}{b_{D}}                          %
\providecommand{\dGM}{\delta_{\mathrm{GM}}}

%
%
\providecommand{\etaw}{\eta_{w}}
\providecommand{\aeta}{a_{\eta}}
\providecommand{\beeta}{b_{\eta}}
\providecommand{\kP}{k_{P}}
\providecommand{\aP}{a_{P}}
\providecommand{\rbox}{r_{\square}}
\providecommand{\MQ}{M_{\mathcal{Q}}}
\providecommand{\MLam}{M_{\Lambda}}
\providecommand{\MLamp}{M'_{\Lambda}}
%
\providecommand{\rz}{r_{0}}
\providecommand{\rbz}{\bar r_{0}}
\providecommand{\ellD}{\ell_{D}}
\providecommand{\rsep}{r_{\mathrm{sep}}}
\providecommand{\rhol}{\rho_{\lambda}}
\providecommand{\Rst}{R_{*}}
\providecommand{\RNF}{R_{\mathrm{NF}}}
\providecommand{\cNF}{c_{3}^{\mathrm{NF}}}
\providecommand{\thNF}{\theta_{\mathrm{NF}}}
\providecommand{\Funit}{F_{\mathrm{unit}}}
\providecommand{\Rresp}{R_{\mathrm{resp}}}
\providecommand{\Cresp}{C_{\mathrm{resp}}}
\providecommand{\Rpr}{R'}
\providecommand{\gR}{\gamma_{R}}
\providecommand{\Vst}{V_{*}}
\providecommand{\VU}{V_{\mathcal{U}}}
\providecommand{\csh}{c_{\mathrm{sh}}}
\providecommand{\cs}{c_{s}}
\providecommand{\Cunit}{C_{\mathrm{unit}}}
\providecommand{\nsmall}{n_{1}^{\mathrm{small}}}
\providecommand{\nbig}{n_{1}^{\mathrm{big}}}
\providecommand{\Phireg}{\Phi_{\mathrm{reg}}}
\providecommand{\etaT}{\eta_{T}}
\providecommand{\etaS}{\eta_{S}}
\providecommand{\epsc}{\varepsilon_{c}}
\providecommand{\epspp}{\varepsilon''}
\providecommand{\epsD}{\varepsilon_{D}}
\providecommand{\epscol}{\varepsilon_{\mathrm{col}}}
\providecommand{\cg}{c_{g}}
\providecommand{\cgp}{c_{g}'}
\providecommand{\cL}{c_{L}}
\providecommand{\cI}{c_{I}}
\providecommand{\Cst}{C_{\star}'}
\providecommand{\Cstpp}{C_{\star}''}
\providecommand{\Cfree}{C_{\mathrm{free}}}
\providecommand{\Cvol}{C_{\mathrm{vol}}}
\providecommand{\cT}{c_{T}}
\providecommand{\cTpp}{c_{T}''}
\providecommand{\kpl}{\kappa_{\mathrm{pl}}}
\providecommand{\wst}{w_{*}}
\providecommand{\Hull}[2]{H^{#1}_{\lambda}(#2)}      %
\providecommand{\Fhull}[1]{F^{#1}_{\lambda}}         %

%
%
\providecommand{\bC}{\mathbf{C}}                  %
\providecommand{\thz}{\theta_{0}}
\providecommand{\thp}{\theta'}
\providecommand{\rcov}{r_{\mathrm{cov}}}
\providecommand{\kS}{\kappa_{S}}
\providecommand{\kY}{\kappa_{Y}}
\providecommand{\kz}{\kappa_{0}}
\providecommand{\kW}{\kappa_{W}}
\providecommand{\kNF}{\kappa_{\mathrm{NF}}}
\providecommand{\etaH}{\eta_{H}}
\providecommand{\pz}{p_{0}}
\providecommand{\alD}{\alpha_{D}}
\providecommand{\alLam}{\alpha_{\Lambda}}
\providecommand{\cpr}{c'}
\providecommand{\cprz}{c'_{0}}                     %
\providecommand{\cdep}{c_{\mathrm{dep}}}           %
\providecommand{\cprate}{c'_{\mathrm{rate}}}
\providecommand{\cE}{c_{E}}
\providecommand{\cper}{c_{\mathrm{per}}}
\providecommand{\epsE}{\varepsilon_{E}}
\providecommand{\sigF}{\sigma_{F}}
\providecommand{\sigV}{\sigma_{V}}
%
\providecommand{\csix}{c_{16}}
\providecommand{\CPer}{C_{\mathcal{P}}}
\providecommand{\ME}{M_{E}}
\providecommand{\ThF}{\Theta_{4}}
\providecommand{\Kcell}{K_{\mathrm{cell}}}
\providecommand{\Kcol}{K_{\mathrm{col}}}
\providecommand{\RHK}{R_{\mathrm{HK}}}
\providecommand{\Kz}{K_{0}}
\providecommand{\Rdep}{R_{\mathrm{dep}}}
\providecommand{\Lp}{L'}
\providecommand{\qpin}{q_{\mathrm{pin}}}
\providecommand{\Rf}{R_{f}}
\providecommand{\tC}{t_{C}}
\providecommand{\thP}{\theta_{P}}
\providecommand{\epsHK}{\varepsilon_{1}^{\mathrm{HK}}}
\providecommand{\AAD}{A_{0}^{\mathrm{AD}}}
\providecommand{\Cdep}{C_{*}^{\mathrm{dep}}}
\providecommand{\DelU}{\Delta_{\mathcal{U}}}
\providecommand{\vCb}{v_{\mathrm{Cb}}}
\providecommand{\Fone}{F_{1}^{\mathrm{loc}}}
\providecommand{\CU}{C_{\mathcal{U}}}
\providecommand{\CX}{C_{X}}