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\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.055\textwidth}>{\raggedright\arraybackslash}p{0.045\textwidth}>{\raggedright\arraybackslash}p{0.10\textwidth}>{\raggedright\arraybackslash}p{0.075\textwidth}>{\raggedright\arraybackslash}p{0.245\textwidth}>{\raggedright\arraybackslash}p{0.405\textwidth}@{}}
\caption{Complete stage-typed PC5 census.}\label{tab:pc5-complete-census}\\
\toprule
edge & group & item & owner & domain & equation \\
\midrule
\endfirsthead
\multicolumn{6}{l}{\tablename\ \thetable\ continued}\\
\toprule
edge & group & item & owner & domain & equation \\
\midrule
\endhead
\midrule
\multicolumn{6}{r}{continued on next page}\\
\endfoot
\bottomrule
\endlastfoot
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G1; count=7}\\*
\path|0B| & \path|G1| & \path|E| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|S_0B eps_E,0^cut=eps_E,B^cut;tildeF_0B;all_jets| \\
\path|0B| & \path|G1| & \path|i-| & \path|SP| & \path|D_i^0x[0,eps_face](nonempty)| & \path|S_0B kappa_i,0^-,(3)(z,u)=kappa_i,B^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|0B| & \path|G1| & \path|i+| & \path|SP| & \path|D_i^0x[0,eps_face](nonempty)| & \path|S_0B kappa_i,0^+,(3)(z,u)=kappa_i,B^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|0B| & \path|G1| & \path|j-| & \path|LOC| & \path|D_j^0x[0,eps_face](nonempty)| & \path|L_0B kappa_j,0^-,(3)(z,u)=kappa_j,B^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|0B| & \path|G1| & \path|j+| & \path|LOC| & \path|D_j^0x[0,eps_face](nonempty)| & \path|L_0B kappa_j,0^+,(3)(z,u)=kappa_j,B^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|0B| & \path|G1| & \path|k-| & \path|LOC| & \path|D_k^0x[0,eps_face](nonempty)| & \path|L_0B kappa_k,0^-,(3)(z,u)=kappa_k,B^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|0B| & \path|G1| & \path|k+| & \path|LOC| & \path|D_k^0x[0,eps_face](nonempty)| & \path|L_0B kappa_k,0^+,(3)(z,u)=kappa_k,B^+,(3)(z,u);d_frames_equal;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G2; count=6}\\*
\path|0B| & \path|G2| & \path|s_i| & \path|SP| & \path|(-delta_s_i,delta_s_i)x[0,1]x[0,eps_rib](nonempty)| & \path|S_0B rho_s_i,0^(3)(x,t,u)=rho_s_i,B^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0B| & \path|G2| & \path|c_i| & \path|SP| & \path|(-delta_c_i,delta_c_i)x[0,1]x[0,eps_rib](nonempty)| & \path|S_0B rho_c_i,0^(3)(x,t,u)=rho_c_i,B^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0B| & \path|G2| & \path|s_j| & \path|SPLIT| & \path|(-delta_s_j,delta_s_j)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_0B=S_0B_on_tau_star_L_0B_on_local rho_s_j,0^(3)(x,t,u)=rho_s_j,B^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0B| & \path|G2| & \path|c_j| & \path|LOC| & \path|(-delta_c_j,delta_c_j)x[0,1]x[0,eps_rib](nonempty)| & \path|L_0B rho_c_j,0^(3)(x,t,u)=rho_c_j,B^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0B| & \path|G2| & \path|s_k| & \path|SPLIT| & \path|(-delta_s_k,delta_s_k)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_0B=S_0B_on_tau_star_L_0B_on_local rho_s_k,0^(3)(x,t,u)=rho_s_k,B^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0B| & \path|G2| & \path|c_k| & \path|LOC| & \path|(-delta_c_k,delta_c_k)x[0,1]x[0,eps_rib](nonempty)| & \path|L_0B rho_c_k,0^(3)(x,t,u)=rho_c_k,B^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G3; count=2}\\*
\path|0B| & \path|G3| & \path|tau_j| & \path|SP| & \path|N_tau_j^0x[0,eps_tau](nonempty)| & \path|Xi_0B,0^-1 tau_0B,j,0=tau_0B,j^sp=Xi_0B,B^-1 tau_0B,j,B;S_0B tau_0B,j,0=tau_0B,j,B;full_frame;all_jets| \\
\path|0B| & \path|G3| & \path|tau_k| & \path|SP| & \path|N_tau_k^0x[0,eps_tau](nonempty)| & \path|Xi_0B,0^-1 tau_0B,k,0=tau_0B,k^sp=Xi_0B,B^-1 tau_0B,k,B;S_0B tau_0B,k,0=tau_0B,k,B;full_frame;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G4; count=12}\\*
\path|0B| & \path|G4| & \path|s_j| & \path|SPLIT| & \path|N_s_j^0x[0,eps_spoke](nonempty)| & \path|s_j,u=tau_0B,j,u*sloc_0B,j,u;tildeF_0B(s_j,0)=s_j,B;S_on_tau;L_on_local;all_jets| \\
\path|0B| & \path|G4| & \path|s_k| & \path|SPLIT| & \path|N_s_k^0x[0,eps_spoke](nonempty)| & \path|s_k,u=tau_0B,k,u*sloc_0B,k,u;tildeF_0B(s_k,0)=s_k,B;S_on_tau;L_on_local;all_jets| \\
\path|0B| & \path|G4| & \path|s_i| & \path|SP| & \path|N_s_i^0x[0,eps_spoke](nonempty)| & \path|S_0B(s_i,0)=s_i,B;full_product_germ;all_jets| \\
\path|0B| & \path|G4| & \path|branch_i| & \path|SP| & \path|N_branch_i^0x[0,eps_spoke](nonempty)| & \path|S_0B(branch_i,0)=branch_i,B;eight_ordered_letters;full_product_germ;all_jets| \\
\path|0B| & \path|G4| & \path|f_E_i^-1| & \path|SP| & \path|N_f_E_i^-1^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_f_E_i^-1,0^sp=chi_f_E_i^-1,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0B| & \path|G4| & \path|r_s_i| & \path|SP| & \path|N_r_s_i^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_r_s_i,0^sp=chi_r_s_i,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0B| & \path|G4| & \path|f_i-_1^-1| & \path|SP| & \path|N_f_i-_1^-1^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_f_i-_1^-1,0^sp=chi_f_i-_1^-1,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0B| & \path|G4| & \path|r_c_i| & \path|SP| & \path|N_r_c_i^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_r_c_i,0^sp=chi_r_c_i,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0B| & \path|G4| & \path|f_i+^-1| & \path|SP| & \path|N_f_i+^-1^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_f_i+^-1,0^sp=chi_f_i+^-1,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0B| & \path|G4| & \path|l_c_i^-1| & \path|SP| & \path|N_l_c_i^-1^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_l_c_i^-1,0^sp=chi_l_c_i^-1,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0B| & \path|G4| & \path|f_i-_0^-1| & \path|SP| & \path|N_f_i-_0^-1^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_f_i-_0^-1,0^sp=chi_f_i-_0^-1,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0B| & \path|G4| & \path|l_s_i^-1| & \path|SP| & \path|N_l_s_i^-1^0x[0,eps_spoke](nonempty)| & \path|S_0B chi_l_s_i^-1,0^sp=chi_l_s_i^-1,B^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G5; count=1}\\*
\path|0B| & \path|G5| & \path|detector_deltaE| & \path|SP| & \path|S1x(-delta_det,delta_det)x[0,eps_det](nonempty)| & \path|S_0B delta_0(theta,n,u)=delta_B(theta,n,u);core_b_deltaE_0_to_b_deltaE_B;d(dtheta,dn,du)=(dtheta,dn,du);turn=0;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G6; count=1}\\*
\path|0B| & \path|G6| & \path|A_mod| & \path|SPLIT| & \path|A_mod_openx[0,eps_PC5_work](coordinate_domain;contains_full_closed_A_mod;fixed_nonempty_open_product_germ)| & \path|tildeF_0B q_0^P(zeta_mod(x),u)=q_B^P(zeta_mod(x),u);x_in_A_mod_open;L_0B_restriction=S_0B_restriction_on_nonempty_open_overlap;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G7; count=3}\\*
\path|0B| & \path|G7| & \path|pair_i| & \path|SP| & \path|D_i^0x[0,eps_PC5_pair](nonempty)| & \path|tildeF_0B^i,+ Pi_i,0^-+,(3) kappa_i,0^-,(3)(z,u)=Pi_i,B^-+,(3) tildeF_0B^i,- kappa_i,0^-,(3)(z,u)=kappa_i,B^+,(3)(jmath_iz,u);normal_u_to_u;all_jets| \\
\path|0B| & \path|G7| & \path|pair_j| & \path|LOC| & \path|D_j^0x[0,eps_PC5_pair](nonempty)| & \path|tildeF_0B^j,+ Pi_j,0^-+,(3) kappa_j,0^-,(3)(z,u)=Pi_j,B^-+,(3) tildeF_0B^j,- kappa_j,0^-,(3)(z,u)=kappa_j,B^+,(3)(jmath_jz,u);normal_u_to_u;all_jets| \\
\path|0B| & \path|G7| & \path|pair_k| & \path|LOC| & \path|D_k^0x[0,eps_PC5_pair](nonempty)| & \path|tildeF_0B^k,+ Pi_k,0^-+,(3) kappa_k,0^-,(3)(z,u)=Pi_k,B^-+,(3) tildeF_0B^k,- kappa_k,0^-,(3)(z,u)=kappa_k,B^+,(3)(jmath_kz,u);normal_u_to_u;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G8; count=4}\\*
\path|0B| & \path|G8| & \path|E_cut_trace| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|tildeF_0B eps_E,0^cut=eps_E,B^cut;S_0B_on_K_side;all_jets| \\
\path|0B| & \path|G8| & \path|E_full_bicollar| & \path|GLOBAL-F| & \path|N_E_phys^two-sided(nonempty)| & \path|F_0B_restriction_N_E_phys^two-sided=id;inside_half_from_GLOBAL-E-DESC;outside_half_EXT-ID| \\
\path|0B| & \path|G8| & \path|old_region| & \path|GLOBAL-F| & \path|W_old(nonempty)| & \path|F_0B_restriction_W_old=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\path|0B| & \path|G8| & \path|unused_handles_marking| & \path|GLOBAL-F| & \path|union_unused_t_N_t(nonempty_or_vacuous_only_when_N=3)| & \path|F_0B_restriction_union_unused_t_N_t=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0B; source=0; target=B; roles=j,k;i; group=G9; count=1}\\*
\path|0B| & \path|G9| & \path|role_exception| & \path|EDGE| & \path|complete_ROLE10_TRANSPORT_product_germ(nonempty)| & \path|j0_to_J_over_k0_k_eps_via_beta0; spectator=i0; J=j0#_{beta0}k_eps| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G1; count=7}\\*
\path|Bstar| & \path|G1| & \path|E| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|S_Bstar eps_E,B^cut=eps_E,star^cut;tildeF_Bstar;all_jets| \\
\path|Bstar| & \path|G1| & \path|i-| & \path|LOC| & \path|D_i^Bx[0,eps_face](nonempty)| & \path|L_Bstar kappa_i,B^-,(3)(z,u)=kappa_i,star^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|Bstar| & \path|G1| & \path|i+| & \path|LOC| & \path|D_i^Bx[0,eps_face](nonempty)| & \path|L_Bstar kappa_i,B^+,(3)(z,u)=kappa_i,star^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|Bstar| & \path|G1| & \path|j-| & \path|LOC| & \path|D_j^Bx[0,eps_face](nonempty)| & \path|L_Bstar kappa_j,B^-,(3)(z,u)=kappa_j,star^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|Bstar| & \path|G1| & \path|j+| & \path|LOC| & \path|D_j^Bx[0,eps_face](nonempty)| & \path|L_Bstar kappa_j,B^+,(3)(z,u)=kappa_j,star^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|Bstar| & \path|G1| & \path|k-| & \path|SP| & \path|D_k^Bx[0,eps_face](nonempty)| & \path|S_Bstar kappa_k,B^-,(3)(z,u)=kappa_k,star^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|Bstar| & \path|G1| & \path|k+| & \path|SP| & \path|D_k^Bx[0,eps_face](nonempty)| & \path|S_Bstar kappa_k,B^+,(3)(z,u)=kappa_k,star^+,(3)(z,u);d_frames_equal;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G2; count=6}\\*
\path|Bstar| & \path|G2| & \path|s_i| & \path|SPLIT| & \path|(-delta_s_i,delta_s_i)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_Bstar=S_Bstar_on_tau_star_L_Bstar_on_local rho_s_i,B^(3)(x,t,u)=rho_s_i,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|Bstar| & \path|G2| & \path|c_i| & \path|LOC| & \path|(-delta_c_i,delta_c_i)x[0,1]x[0,eps_rib](nonempty)| & \path|L_Bstar rho_c_i,B^(3)(x,t,u)=rho_c_i,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|Bstar| & \path|G2| & \path|s_j| & \path|SPLIT| & \path|(-delta_s_j,delta_s_j)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_Bstar=S_Bstar_on_tau_star_L_Bstar_on_local rho_s_j,B^(3)(x,t,u)=rho_s_j,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|Bstar| & \path|G2| & \path|c_j| & \path|LOC| & \path|(-delta_c_j,delta_c_j)x[0,1]x[0,eps_rib](nonempty)| & \path|L_Bstar rho_c_j,B^(3)(x,t,u)=rho_c_j,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|Bstar| & \path|G2| & \path|s_k| & \path|SP| & \path|(-delta_s_k,delta_s_k)x[0,1]x[0,eps_rib](nonempty)| & \path|S_Bstar rho_s_k,B^(3)(x,t,u)=rho_s_k,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|Bstar| & \path|G2| & \path|c_k| & \path|SP| & \path|(-delta_c_k,delta_c_k)x[0,1]x[0,eps_rib](nonempty)| & \path|S_Bstar rho_c_k,B^(3)(x,t,u)=rho_c_k,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G3; count=2}\\*
\path|Bstar| & \path|G3| & \path|tau_i| & \path|SP| & \path|N_tau_i^Bx[0,eps_tau](nonempty)| & \path|Xi_Bstar,B^-1 tau_Bstar,i,B=tau_Bstar,i^sp=Xi_Bstar,star^-1 tau_Bstar,i,star;S_Bstar tau_Bstar,i,B=tau_Bstar,i,star;full_frame;all_jets| \\
\path|Bstar| & \path|G3| & \path|tau_j| & \path|SP| & \path|N_tau_j^Bx[0,eps_tau](nonempty)| & \path|Xi_Bstar,B^-1 tau_Bstar,j,B=tau_Bstar,j^sp=Xi_Bstar,star^-1 tau_Bstar,j,star;S_Bstar tau_Bstar,j,B=tau_Bstar,j,star;full_frame;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G4; count=12}\\*
\path|Bstar| & \path|G4| & \path|s_i| & \path|SPLIT| & \path|N_s_i^Bx[0,eps_spoke](nonempty)| & \path|s_i,u=tau_Bstar,i,u*sloc_Bstar,i,u;tildeF_Bstar(s_i,B)=s_i,star;S_on_tau;L_on_local;all_jets| \\
\path|Bstar| & \path|G4| & \path|s_j| & \path|SPLIT| & \path|N_s_j^Bx[0,eps_spoke](nonempty)| & \path|s_j,u=tau_Bstar,j,u*sloc_Bstar,j,u;tildeF_Bstar(s_j,B)=s_j,star;S_on_tau;L_on_local;all_jets| \\
\path|Bstar| & \path|G4| & \path|s_k| & \path|SP| & \path|N_s_k^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar(s_k,B)=s_k,star;full_product_germ;all_jets| \\
\path|Bstar| & \path|G4| & \path|branch_k| & \path|SP| & \path|N_branch_k^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar(branch_k,B)=branch_k,star;eight_ordered_letters;full_product_germ;all_jets| \\
\path|Bstar| & \path|G4| & \path|f_E_k^-1| & \path|SP| & \path|N_f_E_k^-1^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_f_E_k^-1,B^sp=chi_f_E_k^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|Bstar| & \path|G4| & \path|r_s_k| & \path|SP| & \path|N_r_s_k^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_r_s_k,B^sp=chi_r_s_k,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|Bstar| & \path|G4| & \path|f_k-_1^-1| & \path|SP| & \path|N_f_k-_1^-1^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_f_k-_1^-1,B^sp=chi_f_k-_1^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|Bstar| & \path|G4| & \path|r_c_k| & \path|SP| & \path|N_r_c_k^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_r_c_k,B^sp=chi_r_c_k,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|Bstar| & \path|G4| & \path|f_k+^-1| & \path|SP| & \path|N_f_k+^-1^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_f_k+^-1,B^sp=chi_f_k+^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|Bstar| & \path|G4| & \path|l_c_k^-1| & \path|SP| & \path|N_l_c_k^-1^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_l_c_k^-1,B^sp=chi_l_c_k^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|Bstar| & \path|G4| & \path|f_k-_0^-1| & \path|SP| & \path|N_f_k-_0^-1^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_f_k-_0^-1,B^sp=chi_f_k-_0^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|Bstar| & \path|G4| & \path|l_s_k^-1| & \path|SP| & \path|N_l_s_k^-1^Bx[0,eps_spoke](nonempty)| & \path|S_Bstar chi_l_s_k^-1,B^sp=chi_l_s_k^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G5; count=1}\\*
\path|Bstar| & \path|G5| & \path|detector_deltaE| & \path|SP| & \path|S1x(-delta_det,delta_det)x[0,eps_det](nonempty)| & \path|S_Bstar delta_B(theta,n,u)=delta_star(theta,n,u);core_b_deltaE_B_to_b_deltaE_star;d(dtheta,dn,du)=(dtheta,dn,du);turn=0;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G6; count=1}\\*
\path|Bstar| & \path|G6| & \path|A_mod| & \path|SPLIT| & \path|A_mod_openx[0,eps_PC5_work](coordinate_domain;contains_full_closed_A_mod;fixed_nonempty_open_product_germ)| & \path|tildeF_Bstar q_B^P(zeta_mod(x),u)=q_star^P(zeta_mod(x),u);x_in_A_mod_open;L_Bstar_restriction=S_Bstar_restriction_on_nonempty_open_overlap;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G7; count=3}\\*
\path|Bstar| & \path|G7| & \path|pair_i| & \path|LOC| & \path|D_i^Bx[0,eps_PC5_pair](nonempty)| & \path|tildeF_Bstar^i,+ Pi_i,B^-+,(3) kappa_i,B^-,(3)(z,u)=Pi_i,star^-+,(3) tildeF_Bstar^i,- kappa_i,B^-,(3)(z,u)=kappa_i,star^+,(3)(jmath_iz,u);normal_u_to_u;all_jets| \\
\path|Bstar| & \path|G7| & \path|pair_j| & \path|LOC| & \path|D_j^Bx[0,eps_PC5_pair](nonempty)| & \path|tildeF_Bstar^j,+ Pi_j,B^-+,(3) kappa_j,B^-,(3)(z,u)=Pi_j,star^-+,(3) tildeF_Bstar^j,- kappa_j,B^-,(3)(z,u)=kappa_j,star^+,(3)(jmath_jz,u);normal_u_to_u;all_jets| \\
\path|Bstar| & \path|G7| & \path|pair_k| & \path|SP| & \path|D_k^Bx[0,eps_PC5_pair](nonempty)| & \path|tildeF_Bstar^k,+ Pi_k,B^-+,(3) kappa_k,B^-,(3)(z,u)=Pi_k,star^-+,(3) tildeF_Bstar^k,- kappa_k,B^-,(3)(z,u)=kappa_k,star^+,(3)(jmath_kz,u);normal_u_to_u;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G8; count=4}\\*
\path|Bstar| & \path|G8| & \path|E_cut_trace| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|tildeF_Bstar eps_E,B^cut=eps_E,star^cut;S_Bstar_on_K_side;all_jets| \\
\path|Bstar| & \path|G8| & \path|E_full_bicollar| & \path|GLOBAL-F| & \path|N_E_phys^two-sided(nonempty)| & \path|F_Bstar_restriction_N_E_phys^two-sided=id;inside_half_from_GLOBAL-E-DESC;outside_half_EXT-ID| \\
\path|Bstar| & \path|G8| & \path|old_region| & \path|GLOBAL-F| & \path|W_old(nonempty)| & \path|F_Bstar_restriction_W_old=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\path|Bstar| & \path|G8| & \path|unused_handles_marking| & \path|GLOBAL-F| & \path|union_unused_t_N_t(nonempty_or_vacuous_only_when_N=3)| & \path|F_Bstar_restriction_union_unused_t_N_t=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=Bstar; source=B; target=star; roles=i,j;k; group=G9; count=1}\\*
\path|Bstar| & \path|G9| & \path|role_exception| & \path|EDGE| & \path|complete_ROLE10_TRANSPORT_product_germ(nonempty)| & \path|i0_to_Istar_over_complete_J_Jplus_via_alpha; spectator=k0; Jplus=j_eps#_{beta1}k_2eps; alpha_endpoint_on_j_eps_lobe| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G1; count=7}\\*
\path|0A| & \path|G1| & \path|E| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|S_0A eps_E,0^cut=eps_E,A^cut;tildeF_0A;all_jets| \\
\path|0A| & \path|G1| & \path|i-| & \path|LOC| & \path|D_i^0x[0,eps_face](nonempty)| & \path|L_0A kappa_i,0^-,(3)(z,u)=kappa_i,A^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|0A| & \path|G1| & \path|i+| & \path|LOC| & \path|D_i^0x[0,eps_face](nonempty)| & \path|L_0A kappa_i,0^+,(3)(z,u)=kappa_i,A^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|0A| & \path|G1| & \path|j-| & \path|LOC| & \path|D_j^0x[0,eps_face](nonempty)| & \path|L_0A kappa_j,0^-,(3)(z,u)=kappa_j,A^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|0A| & \path|G1| & \path|j+| & \path|LOC| & \path|D_j^0x[0,eps_face](nonempty)| & \path|L_0A kappa_j,0^+,(3)(z,u)=kappa_j,A^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|0A| & \path|G1| & \path|k-| & \path|SP| & \path|D_k^0x[0,eps_face](nonempty)| & \path|S_0A kappa_k,0^-,(3)(z,u)=kappa_k,A^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|0A| & \path|G1| & \path|k+| & \path|SP| & \path|D_k^0x[0,eps_face](nonempty)| & \path|S_0A kappa_k,0^+,(3)(z,u)=kappa_k,A^+,(3)(z,u);d_frames_equal;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G2; count=6}\\*
\path|0A| & \path|G2| & \path|s_i| & \path|SPLIT| & \path|(-delta_s_i,delta_s_i)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_0A=S_0A_on_tau_star_L_0A_on_local rho_s_i,0^(3)(x,t,u)=rho_s_i,A^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0A| & \path|G2| & \path|c_i| & \path|LOC| & \path|(-delta_c_i,delta_c_i)x[0,1]x[0,eps_rib](nonempty)| & \path|L_0A rho_c_i,0^(3)(x,t,u)=rho_c_i,A^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0A| & \path|G2| & \path|s_j| & \path|SPLIT| & \path|(-delta_s_j,delta_s_j)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_0A=S_0A_on_tau_star_L_0A_on_local rho_s_j,0^(3)(x,t,u)=rho_s_j,A^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0A| & \path|G2| & \path|c_j| & \path|LOC| & \path|(-delta_c_j,delta_c_j)x[0,1]x[0,eps_rib](nonempty)| & \path|L_0A rho_c_j,0^(3)(x,t,u)=rho_c_j,A^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0A| & \path|G2| & \path|s_k| & \path|SP| & \path|(-delta_s_k,delta_s_k)x[0,1]x[0,eps_rib](nonempty)| & \path|S_0A rho_s_k,0^(3)(x,t,u)=rho_s_k,A^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|0A| & \path|G2| & \path|c_k| & \path|SP| & \path|(-delta_c_k,delta_c_k)x[0,1]x[0,eps_rib](nonempty)| & \path|S_0A rho_c_k,0^(3)(x,t,u)=rho_c_k,A^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G3; count=2}\\*
\path|0A| & \path|G3| & \path|tau_i| & \path|SP| & \path|N_tau_i^0x[0,eps_tau](nonempty)| & \path|Xi_0A,0^-1 tau_0A,i,0=tau_0A,i^sp=Xi_0A,A^-1 tau_0A,i,A;S_0A tau_0A,i,0=tau_0A,i,A;full_frame;all_jets| \\
\path|0A| & \path|G3| & \path|tau_j| & \path|SP| & \path|N_tau_j^0x[0,eps_tau](nonempty)| & \path|Xi_0A,0^-1 tau_0A,j,0=tau_0A,j^sp=Xi_0A,A^-1 tau_0A,j,A;S_0A tau_0A,j,0=tau_0A,j,A;full_frame;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G4; count=12}\\*
\path|0A| & \path|G4| & \path|s_i| & \path|SPLIT| & \path|N_s_i^0x[0,eps_spoke](nonempty)| & \path|s_i,u=tau_0A,i,u*sloc_0A,i,u;tildeF_0A(s_i,0)=s_i,A;S_on_tau;L_on_local;all_jets| \\
\path|0A| & \path|G4| & \path|s_j| & \path|SPLIT| & \path|N_s_j^0x[0,eps_spoke](nonempty)| & \path|s_j,u=tau_0A,j,u*sloc_0A,j,u;tildeF_0A(s_j,0)=s_j,A;S_on_tau;L_on_local;all_jets| \\
\path|0A| & \path|G4| & \path|s_k| & \path|SP| & \path|N_s_k^0x[0,eps_spoke](nonempty)| & \path|S_0A(s_k,0)=s_k,A;full_product_germ;all_jets| \\
\path|0A| & \path|G4| & \path|branch_k| & \path|SP| & \path|N_branch_k^0x[0,eps_spoke](nonempty)| & \path|S_0A(branch_k,0)=branch_k,A;eight_ordered_letters;full_product_germ;all_jets| \\
\path|0A| & \path|G4| & \path|f_E_k^-1| & \path|SP| & \path|N_f_E_k^-1^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_f_E_k^-1,0^sp=chi_f_E_k^-1,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0A| & \path|G4| & \path|r_s_k| & \path|SP| & \path|N_r_s_k^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_r_s_k,0^sp=chi_r_s_k,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0A| & \path|G4| & \path|f_k-_1^-1| & \path|SP| & \path|N_f_k-_1^-1^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_f_k-_1^-1,0^sp=chi_f_k-_1^-1,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0A| & \path|G4| & \path|r_c_k| & \path|SP| & \path|N_r_c_k^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_r_c_k,0^sp=chi_r_c_k,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0A| & \path|G4| & \path|f_k+^-1| & \path|SP| & \path|N_f_k+^-1^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_f_k+^-1,0^sp=chi_f_k+^-1,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0A| & \path|G4| & \path|l_c_k^-1| & \path|SP| & \path|N_l_c_k^-1^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_l_c_k^-1,0^sp=chi_l_c_k^-1,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0A| & \path|G4| & \path|f_k-_0^-1| & \path|SP| & \path|N_f_k-_0^-1^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_f_k-_0^-1,0^sp=chi_f_k-_0^-1,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|0A| & \path|G4| & \path|l_s_k^-1| & \path|SP| & \path|N_l_s_k^-1^0x[0,eps_spoke](nonempty)| & \path|S_0A chi_l_s_k^-1,0^sp=chi_l_s_k^-1,A^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G5; count=1}\\*
\path|0A| & \path|G5| & \path|detector_deltaE| & \path|SP| & \path|S1x(-delta_det,delta_det)x[0,eps_det](nonempty)| & \path|S_0A delta_0(theta,n,u)=delta_A(theta,n,u);core_b_deltaE_0_to_b_deltaE_A;d(dtheta,dn,du)=(dtheta,dn,du);turn=0;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G6; count=1}\\*
\path|0A| & \path|G6| & \path|A_mod| & \path|SPLIT| & \path|A_mod_openx[0,eps_PC5_work](coordinate_domain;contains_full_closed_A_mod;fixed_nonempty_open_product_germ)| & \path|tildeF_0A q_0^P(zeta_mod(x),u)=q_A^P(zeta_mod(x),u);x_in_A_mod_open;L_0A_restriction=S_0A_restriction_on_nonempty_open_overlap;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G7; count=3}\\*
\path|0A| & \path|G7| & \path|pair_i| & \path|LOC| & \path|D_i^0x[0,eps_PC5_pair](nonempty)| & \path|tildeF_0A^i,+ Pi_i,0^-+,(3) kappa_i,0^-,(3)(z,u)=Pi_i,A^-+,(3) tildeF_0A^i,- kappa_i,0^-,(3)(z,u)=kappa_i,A^+,(3)(jmath_iz,u);normal_u_to_u;all_jets| \\
\path|0A| & \path|G7| & \path|pair_j| & \path|LOC| & \path|D_j^0x[0,eps_PC5_pair](nonempty)| & \path|tildeF_0A^j,+ Pi_j,0^-+,(3) kappa_j,0^-,(3)(z,u)=Pi_j,A^-+,(3) tildeF_0A^j,- kappa_j,0^-,(3)(z,u)=kappa_j,A^+,(3)(jmath_jz,u);normal_u_to_u;all_jets| \\
\path|0A| & \path|G7| & \path|pair_k| & \path|SP| & \path|D_k^0x[0,eps_PC5_pair](nonempty)| & \path|tildeF_0A^k,+ Pi_k,0^-+,(3) kappa_k,0^-,(3)(z,u)=Pi_k,A^-+,(3) tildeF_0A^k,- kappa_k,0^-,(3)(z,u)=kappa_k,A^+,(3)(jmath_kz,u);normal_u_to_u;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G8; count=4}\\*
\path|0A| & \path|G8| & \path|E_cut_trace| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|tildeF_0A eps_E,0^cut=eps_E,A^cut;S_0A_on_K_side;all_jets| \\
\path|0A| & \path|G8| & \path|E_full_bicollar| & \path|GLOBAL-F| & \path|N_E_phys^two-sided(nonempty)| & \path|F_0A_restriction_N_E_phys^two-sided=id;inside_half_from_GLOBAL-E-DESC;outside_half_EXT-ID| \\
\path|0A| & \path|G8| & \path|old_region| & \path|GLOBAL-F| & \path|W_old(nonempty)| & \path|F_0A_restriction_W_old=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\path|0A| & \path|G8| & \path|unused_handles_marking| & \path|GLOBAL-F| & \path|union_unused_t_N_t(nonempty_or_vacuous_only_when_N=3)| & \path|F_0A_restriction_union_unused_t_N_t=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=0A; source=0; target=A; roles=i,j;k; group=G9; count=1}\\*
\path|0A| & \path|G9| & \path|role_exception| & \path|EDGE| & \path|complete_ROLE10_TRANSPORT_product_germ(nonempty)| & \path|i0_to_IA_over_j0_j_eps_via_alpha; spectator=k0; IA=i0#_{alpha}j_eps| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G1; count=7}\\*
\path|AAC| & \path|G1| & \path|E| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|S_AAC eps_E,A^cut=eps_E,AC^cut;tildeF_AAC;all_jets| \\
\path|AAC| & \path|G1| & \path|i-| & \path|LOC| & \path|D_i^Ax[0,eps_face](nonempty)| & \path|L_AAC kappa_i,A^-,(3)(z,u)=kappa_i,AC^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|AAC| & \path|G1| & \path|i+| & \path|LOC| & \path|D_i^Ax[0,eps_face](nonempty)| & \path|L_AAC kappa_i,A^+,(3)(z,u)=kappa_i,AC^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|AAC| & \path|G1| & \path|j-| & \path|SP| & \path|D_j^Ax[0,eps_face](nonempty)| & \path|S_AAC kappa_j,A^-,(3)(z,u)=kappa_j,AC^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|AAC| & \path|G1| & \path|j+| & \path|SP| & \path|D_j^Ax[0,eps_face](nonempty)| & \path|S_AAC kappa_j,A^+,(3)(z,u)=kappa_j,AC^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|AAC| & \path|G1| & \path|k-| & \path|LOC| & \path|D_k^Ax[0,eps_face](nonempty)| & \path|L_AAC kappa_k,A^-,(3)(z,u)=kappa_k,AC^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|AAC| & \path|G1| & \path|k+| & \path|LOC| & \path|D_k^Ax[0,eps_face](nonempty)| & \path|L_AAC kappa_k,A^+,(3)(z,u)=kappa_k,AC^+,(3)(z,u);d_frames_equal;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G2; count=6}\\*
\path|AAC| & \path|G2| & \path|s_i| & \path|SPLIT| & \path|(-delta_s_i,delta_s_i)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_AAC=S_AAC_on_tau_star_L_AAC_on_local rho_s_i,A^(3)(x,t,u)=rho_s_i,AC^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|AAC| & \path|G2| & \path|c_i| & \path|LOC| & \path|(-delta_c_i,delta_c_i)x[0,1]x[0,eps_rib](nonempty)| & \path|L_AAC rho_c_i,A^(3)(x,t,u)=rho_c_i,AC^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|AAC| & \path|G2| & \path|s_j| & \path|SP| & \path|(-delta_s_j,delta_s_j)x[0,1]x[0,eps_rib](nonempty)| & \path|S_AAC rho_s_j,A^(3)(x,t,u)=rho_s_j,AC^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|AAC| & \path|G2| & \path|c_j| & \path|SP| & \path|(-delta_c_j,delta_c_j)x[0,1]x[0,eps_rib](nonempty)| & \path|S_AAC rho_c_j,A^(3)(x,t,u)=rho_c_j,AC^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|AAC| & \path|G2| & \path|s_k| & \path|SPLIT| & \path|(-delta_s_k,delta_s_k)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_AAC=S_AAC_on_tau_star_L_AAC_on_local rho_s_k,A^(3)(x,t,u)=rho_s_k,AC^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|AAC| & \path|G2| & \path|c_k| & \path|LOC| & \path|(-delta_c_k,delta_c_k)x[0,1]x[0,eps_rib](nonempty)| & \path|L_AAC rho_c_k,A^(3)(x,t,u)=rho_c_k,AC^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G3; count=2}\\*
\path|AAC| & \path|G3| & \path|tau_i| & \path|SP| & \path|N_tau_i^Ax[0,eps_tau](nonempty)| & \path|Xi_AAC,A^-1 tau_AAC,i,A=tau_AAC,i^sp=Xi_AAC,AC^-1 tau_AAC,i,AC;S_AAC tau_AAC,i,A=tau_AAC,i,AC;full_frame;all_jets| \\
\path|AAC| & \path|G3| & \path|tau_k| & \path|SP| & \path|N_tau_k^Ax[0,eps_tau](nonempty)| & \path|Xi_AAC,A^-1 tau_AAC,k,A=tau_AAC,k^sp=Xi_AAC,AC^-1 tau_AAC,k,AC;S_AAC tau_AAC,k,A=tau_AAC,k,AC;full_frame;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G4; count=12}\\*
\path|AAC| & \path|G4| & \path|s_i| & \path|SPLIT| & \path|N_s_i^Ax[0,eps_spoke](nonempty)| & \path|s_i,u=tau_AAC,i,u*sloc_AAC,i,u;tildeF_AAC(s_i,A)=s_i,AC;S_on_tau;L_on_local;all_jets| \\
\path|AAC| & \path|G4| & \path|s_k| & \path|SPLIT| & \path|N_s_k^Ax[0,eps_spoke](nonempty)| & \path|s_k,u=tau_AAC,k,u*sloc_AAC,k,u;tildeF_AAC(s_k,A)=s_k,AC;S_on_tau;L_on_local;all_jets| \\
\path|AAC| & \path|G4| & \path|s_j| & \path|SP| & \path|N_s_j^Ax[0,eps_spoke](nonempty)| & \path|S_AAC(s_j,A)=s_j,AC;full_product_germ;all_jets| \\
\path|AAC| & \path|G4| & \path|branch_j| & \path|SP| & \path|N_branch_j^Ax[0,eps_spoke](nonempty)| & \path|S_AAC(branch_j,A)=branch_j,AC;eight_ordered_letters;full_product_germ;all_jets| \\
\path|AAC| & \path|G4| & \path|f_E_j^-1| & \path|SP| & \path|N_f_E_j^-1^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_f_E_j^-1,A^sp=chi_f_E_j^-1,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|AAC| & \path|G4| & \path|r_s_j| & \path|SP| & \path|N_r_s_j^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_r_s_j,A^sp=chi_r_s_j,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|AAC| & \path|G4| & \path|f_j-_1^-1| & \path|SP| & \path|N_f_j-_1^-1^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_f_j-_1^-1,A^sp=chi_f_j-_1^-1,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|AAC| & \path|G4| & \path|r_c_j| & \path|SP| & \path|N_r_c_j^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_r_c_j,A^sp=chi_r_c_j,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|AAC| & \path|G4| & \path|f_j+^-1| & \path|SP| & \path|N_f_j+^-1^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_f_j+^-1,A^sp=chi_f_j+^-1,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|AAC| & \path|G4| & \path|l_c_j^-1| & \path|SP| & \path|N_l_c_j^-1^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_l_c_j^-1,A^sp=chi_l_c_j^-1,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|AAC| & \path|G4| & \path|f_j-_0^-1| & \path|SP| & \path|N_f_j-_0^-1^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_f_j-_0^-1,A^sp=chi_f_j-_0^-1,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|AAC| & \path|G4| & \path|l_s_j^-1| & \path|SP| & \path|N_l_s_j^-1^Ax[0,eps_spoke](nonempty)| & \path|S_AAC chi_l_s_j^-1,A^sp=chi_l_s_j^-1,AC^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G5; count=1}\\*
\path|AAC| & \path|G5| & \path|detector_deltaE| & \path|SP| & \path|S1x(-delta_det,delta_det)x[0,eps_det](nonempty)| & \path|S_AAC delta_A(theta,n,u)=delta_AC(theta,n,u);core_b_deltaE_A_to_b_deltaE_AC;d(dtheta,dn,du)=(dtheta,dn,du);turn=0;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G6; count=1}\\*
\path|AAC| & \path|G6| & \path|A_mod| & \path|SPLIT| & \path|A_mod_openx[0,eps_PC5_work](coordinate_domain;contains_full_closed_A_mod;fixed_nonempty_open_product_germ)| & \path|tildeF_AAC q_A^P(zeta_mod(x),u)=q_AC^P(zeta_mod(x),u);x_in_A_mod_open;L_AAC_restriction=S_AAC_restriction_on_nonempty_open_overlap;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G7; count=3}\\*
\path|AAC| & \path|G7| & \path|pair_i| & \path|LOC| & \path|D_i^Ax[0,eps_PC5_pair](nonempty)| & \path|tildeF_AAC^i,+ Pi_i,A^-+,(3) kappa_i,A^-,(3)(z,u)=Pi_i,AC^-+,(3) tildeF_AAC^i,- kappa_i,A^-,(3)(z,u)=kappa_i,AC^+,(3)(jmath_iz,u);normal_u_to_u;all_jets| \\
\path|AAC| & \path|G7| & \path|pair_j| & \path|SP| & \path|D_j^Ax[0,eps_PC5_pair](nonempty)| & \path|tildeF_AAC^j,+ Pi_j,A^-+,(3) kappa_j,A^-,(3)(z,u)=Pi_j,AC^-+,(3) tildeF_AAC^j,- kappa_j,A^-,(3)(z,u)=kappa_j,AC^+,(3)(jmath_jz,u);normal_u_to_u;all_jets| \\
\path|AAC| & \path|G7| & \path|pair_k| & \path|LOC| & \path|D_k^Ax[0,eps_PC5_pair](nonempty)| & \path|tildeF_AAC^k,+ Pi_k,A^-+,(3) kappa_k,A^-,(3)(z,u)=Pi_k,AC^-+,(3) tildeF_AAC^k,- kappa_k,A^-,(3)(z,u)=kappa_k,AC^+,(3)(jmath_kz,u);normal_u_to_u;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G8; count=4}\\*
\path|AAC| & \path|G8| & \path|E_cut_trace| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|tildeF_AAC eps_E,A^cut=eps_E,AC^cut;S_AAC_on_K_side;all_jets| \\
\path|AAC| & \path|G8| & \path|E_full_bicollar| & \path|GLOBAL-F| & \path|N_E_phys^two-sided(nonempty)| & \path|F_AAC_restriction_N_E_phys^two-sided=id;inside_half_from_GLOBAL-E-DESC;outside_half_EXT-ID| \\
\path|AAC| & \path|G8| & \path|old_region| & \path|GLOBAL-F| & \path|W_old(nonempty)| & \path|F_AAC_restriction_W_old=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\path|AAC| & \path|G8| & \path|unused_handles_marking| & \path|GLOBAL-F| & \path|union_unused_t_N_t(nonempty_or_vacuous_only_when_N=3)| & \path|F_AAC_restriction_union_unused_t_N_t=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=AAC; source=A; target=AC; roles=i,k;j; group=G9; count=1}\\*
\path|AAC| & \path|G9| & \path|role_exception| & \path|EDGE| & \path|complete_ROLE10_TRANSPORT_product_germ(nonempty)| & \path|whole_IA_including_Omega_j_j_eps_lobe_and_marking_to_Istar_over_k0_k_2eps_via_beta1; spectator=j0| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G1; count=7}\\*
\path|ACstar| & \path|G1| & \path|E| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|S_ACstar eps_E,AC^cut=eps_E,star^cut;tildeF_ACstar;all_jets| \\
\path|ACstar| & \path|G1| & \path|i-| & \path|SP| & \path|D_i^ACx[0,eps_face](nonempty)| & \path|S_ACstar kappa_i,AC^-,(3)(z,u)=kappa_i,star^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|ACstar| & \path|G1| & \path|i+| & \path|SP| & \path|D_i^ACx[0,eps_face](nonempty)| & \path|S_ACstar kappa_i,AC^+,(3)(z,u)=kappa_i,star^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|ACstar| & \path|G1| & \path|j-| & \path|LOC| & \path|D_j^ACx[0,eps_face](nonempty)| & \path|L_ACstar kappa_j,AC^-,(3)(z,u)=kappa_j,star^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|ACstar| & \path|G1| & \path|j+| & \path|LOC| & \path|D_j^ACx[0,eps_face](nonempty)| & \path|L_ACstar kappa_j,AC^+,(3)(z,u)=kappa_j,star^+,(3)(z,u);d_frames_equal;all_jets| \\
\path|ACstar| & \path|G1| & \path|k-| & \path|LOC| & \path|D_k^ACx[0,eps_face](nonempty)| & \path|L_ACstar kappa_k,AC^-,(3)(z,u)=kappa_k,star^-,(3)(z,u);d_frames_equal;all_jets| \\
\path|ACstar| & \path|G1| & \path|k+| & \path|LOC| & \path|D_k^ACx[0,eps_face](nonempty)| & \path|L_ACstar kappa_k,AC^+,(3)(z,u)=kappa_k,star^+,(3)(z,u);d_frames_equal;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G2; count=6}\\*
\path|ACstar| & \path|G2| & \path|s_i| & \path|SP| & \path|(-delta_s_i,delta_s_i)x[0,1]x[0,eps_rib](nonempty)| & \path|S_ACstar rho_s_i,AC^(3)(x,t,u)=rho_s_i,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|ACstar| & \path|G2| & \path|c_i| & \path|SP| & \path|(-delta_c_i,delta_c_i)x[0,1]x[0,eps_rib](nonempty)| & \path|S_ACstar rho_c_i,AC^(3)(x,t,u)=rho_c_i,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|ACstar| & \path|G2| & \path|s_j| & \path|SPLIT| & \path|(-delta_s_j,delta_s_j)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_ACstar=S_ACstar_on_tau_star_L_ACstar_on_local rho_s_j,AC^(3)(x,t,u)=rho_s_j,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|ACstar| & \path|G2| & \path|c_j| & \path|LOC| & \path|(-delta_c_j,delta_c_j)x[0,1]x[0,eps_rib](nonempty)| & \path|L_ACstar rho_c_j,AC^(3)(x,t,u)=rho_c_j,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|ACstar| & \path|G2| & \path|s_k| & \path|SPLIT| & \path|(-delta_s_k,delta_s_k)x[0,1]x[0,eps_rib](nonempty)| & \path|tildeF_ACstar=S_ACstar_on_tau_star_L_ACstar_on_local rho_s_k,AC^(3)(x,t,u)=rho_s_k,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets| \\
\path|ACstar| & \path|G2| & \path|c_k| & \path|LOC| & \path|(-delta_c_k,delta_c_k)x[0,1]x[0,eps_rib](nonempty)| & \path|L_ACstar rho_c_k,AC^(3)(x,t,u)=rho_c_k,star^(3)(x,t,u);d(dx,dt,du)=(dx,dt,du);endpoints;all_jets;contains_fixed_product_normal_H1_ear| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G3; count=2}\\*
\path|ACstar| & \path|G3| & \path|tau_j| & \path|SP| & \path|N_tau_j^ACx[0,eps_tau](nonempty)| & \path|Xi_ACstar,AC^-1 tau_ACstar,j,AC=tau_ACstar,j^sp=Xi_ACstar,star^-1 tau_ACstar,j,star;S_ACstar tau_ACstar,j,AC=tau_ACstar,j,star;full_frame;all_jets| \\
\path|ACstar| & \path|G3| & \path|tau_k| & \path|SP| & \path|N_tau_k^ACx[0,eps_tau](nonempty)| & \path|Xi_ACstar,AC^-1 tau_ACstar,k,AC=tau_ACstar,k^sp=Xi_ACstar,star^-1 tau_ACstar,k,star;S_ACstar tau_ACstar,k,AC=tau_ACstar,k,star;full_frame;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G4; count=12}\\*
\path|ACstar| & \path|G4| & \path|s_j| & \path|SPLIT| & \path|N_s_j^ACx[0,eps_spoke](nonempty)| & \path|s_j,u=tau_ACstar,j,u*sloc_ACstar,j,u;tildeF_ACstar(s_j,AC)=s_j,star;S_on_tau;L_on_local;all_jets| \\
\path|ACstar| & \path|G4| & \path|s_k| & \path|SPLIT| & \path|N_s_k^ACx[0,eps_spoke](nonempty)| & \path|s_k,u=tau_ACstar,k,u*sloc_ACstar,k,u;tildeF_ACstar(s_k,AC)=s_k,star;S_on_tau;L_on_local;all_jets| \\
\path|ACstar| & \path|G4| & \path|s_i| & \path|SP| & \path|N_s_i^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar(s_i,AC)=s_i,star;full_product_germ;all_jets| \\
\path|ACstar| & \path|G4| & \path|branch_i| & \path|SP| & \path|N_branch_i^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar(branch_i,AC)=branch_i,star;eight_ordered_letters;full_product_germ;all_jets| \\
\path|ACstar| & \path|G4| & \path|f_E_i^-1| & \path|SP| & \path|N_f_E_i^-1^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_f_E_i^-1,AC^sp=chi_f_E_i^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|ACstar| & \path|G4| & \path|r_s_i| & \path|SP| & \path|N_r_s_i^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_r_s_i,AC^sp=chi_r_s_i,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|ACstar| & \path|G4| & \path|f_i-_1^-1| & \path|SP| & \path|N_f_i-_1^-1^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_f_i-_1^-1,AC^sp=chi_f_i-_1^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|ACstar| & \path|G4| & \path|r_c_i| & \path|SP| & \path|N_r_c_i^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_r_c_i,AC^sp=chi_r_c_i,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|ACstar| & \path|G4| & \path|f_i+^-1| & \path|SP| & \path|N_f_i+^-1^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_f_i+^-1,AC^sp=chi_f_i+^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|ACstar| & \path|G4| & \path|l_c_i^-1| & \path|SP| & \path|N_l_c_i^-1^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_l_c_i^-1,AC^sp=chi_l_c_i^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|ACstar| & \path|G4| & \path|f_i-_0^-1| & \path|SP| & \path|N_f_i-_0^-1^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_f_i-_0^-1,AC^sp=chi_f_i-_0^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\path|ACstar| & \path|G4| & \path|l_s_i^-1| & \path|SP| & \path|N_l_s_i^-1^ACx[0,eps_spoke](nonempty)| & \path|S_ACstar chi_l_s_i^-1,AC^sp=chi_l_s_i^-1,star^sp;parameter_and_full_product_germ_preserved;STAB-CORR_position_preserved| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G5; count=1}\\*
\path|ACstar| & \path|G5| & \path|detector_deltaE| & \path|SP| & \path|S1x(-delta_det,delta_det)x[0,eps_det](nonempty)| & \path|S_ACstar delta_AC(theta,n,u)=delta_star(theta,n,u);core_b_deltaE_AC_to_b_deltaE_star;d(dtheta,dn,du)=(dtheta,dn,du);turn=0;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G6; count=1}\\*
\path|ACstar| & \path|G6| & \path|A_mod| & \path|SPLIT| & \path|A_mod_openx[0,eps_PC5_work](coordinate_domain;contains_full_closed_A_mod;fixed_nonempty_open_product_germ)| & \path|tildeF_ACstar q_AC^P(zeta_mod(x),u)=q_star^P(zeta_mod(x),u);x_in_A_mod_open;L_ACstar_restriction=S_ACstar_restriction_on_nonempty_open_overlap;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G7; count=3}\\*
\path|ACstar| & \path|G7| & \path|pair_i| & \path|SP| & \path|D_i^ACx[0,eps_PC5_pair](nonempty)| & \path|tildeF_ACstar^i,+ Pi_i,AC^-+,(3) kappa_i,AC^-,(3)(z,u)=Pi_i,star^-+,(3) tildeF_ACstar^i,- kappa_i,AC^-,(3)(z,u)=kappa_i,star^+,(3)(jmath_iz,u);normal_u_to_u;all_jets| \\
\path|ACstar| & \path|G7| & \path|pair_j| & \path|LOC| & \path|D_j^ACx[0,eps_PC5_pair](nonempty)| & \path|tildeF_ACstar^j,+ Pi_j,AC^-+,(3) kappa_j,AC^-,(3)(z,u)=Pi_j,star^-+,(3) tildeF_ACstar^j,- kappa_j,AC^-,(3)(z,u)=kappa_j,star^+,(3)(jmath_jz,u);normal_u_to_u;all_jets| \\
\path|ACstar| & \path|G7| & \path|pair_k| & \path|LOC| & \path|D_k^ACx[0,eps_PC5_pair](nonempty)| & \path|tildeF_ACstar^k,+ Pi_k,AC^-+,(3) kappa_k,AC^-,(3)(z,u)=Pi_k,star^-+,(3) tildeF_ACstar^k,- kappa_k,AC^-,(3)(z,u)=kappa_k,star^+,(3)(jmath_kz,u);normal_u_to_u;all_jets| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G8; count=4}\\*
\path|ACstar| & \path|G8| & \path|E_cut_trace| & \path|SP-CUT| & \path|N_E_cut^3(nonempty)| & \path|tildeF_ACstar eps_E,AC^cut=eps_E,star^cut;S_ACstar_on_K_side;all_jets| \\
\path|ACstar| & \path|G8| & \path|E_full_bicollar| & \path|GLOBAL-F| & \path|N_E_phys^two-sided(nonempty)| & \path|F_ACstar_restriction_N_E_phys^two-sided=id;inside_half_from_GLOBAL-E-DESC;outside_half_EXT-ID| \\
\path|ACstar| & \path|G8| & \path|old_region| & \path|GLOBAL-F| & \path|W_old(nonempty)| & \path|F_ACstar_restriction_W_old=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\path|ACstar| & \path|G8| & \path|unused_handles_marking| & \path|GLOBAL-F| & \path|union_unused_t_N_t(nonempty_or_vacuous_only_when_N=3)| & \path|F_ACstar_restriction_union_unused_t_N_t=id;after_GLOBAL-E-DESC_and_EXT-ID;no_tildeF_or_Xi_claim| \\
\multicolumn{6}{@{}l@{}}{\bfseries\ttfamily edge=ACstar; source=AC; target=star; roles=j,k;i; group=G9; count=1}\\*
\path|ACstar| & \path|G9| & \path|role_exception| & \path|EDGE| & \path|complete_ROLE10_TRANSPORT_product_germ(nonempty)| & \path|j0_to_J_over_k0_k_eps_via_beta0; actual_spectator=Istar; H1:Kki_natural(-Lji)^plus_to_Kkj; qstar_pstar_unchanged| \\
\end{longtable}
\endgroup