{
  "title": "Relative commutants: four exact mechanisms",
  "mathematical_status": "Exact formulas and coordinates; the three shear iterates and four finite subsets are illustrative finite samples.",
  "shear": {
    "definition": "Tg(r,q)=exp(r)g(r,q-r)",
    "bump": "rho(t)=exp(-1/(1-t^2)) for |t|<1, zero otherwise",
    "test": "g(r,q)=rho(2r+3)rho(q)",
    "iteration": "T^n g(r,q)=exp(nr)g(r,q-nr)",
    "closed_support": "-2<=r<=-1 and |q-nr|<=1",
    "samples": [
      {"n": 0, "vertices_r_q": [[-2,-1],[-1,-1],[-1,1],[-2,1]], "center_r_q": [-1.5,0], "center_value_exact": "exp(-2)"},
      {"n": 2, "vertices_r_q": [[-2,-5],[-1,-3],[-1,-1],[-2,-3]], "center_r_q": [-1.5,-3], "center_value_exact": "exp(-5)"},
      {"n": 4, "vertices_r_q": [[-2,-9],[-1,-5],[-1,-3],[-2,-7]], "center_r_q": [-1.5,-6], "center_value_exact": "exp(-8)"}
    ],
    "bound": "||T^n g||_1+||d_r^2 T^n g||_1 <= C_g (1+n)^2 exp(-n) -> 0",
    "locator": "RCC7.x-RCC7.aa; RCC8.f-RCC8.h"
  },
  "finite_net": {
    "index_set": "All finite subsets of an uncountable I, ordered by inclusion",
    "sample_nodes": [
      {"id":"A","set":["alpha"],"xy":[0.5,0.25]},
      {"id":"B","set":["alpha","beta"],"xy":[0.2,0.53]},
      {"id":"C","set":["alpha","gamma"],"xy":[0.8,0.53]},
      {"id":"D","set":["alpha","beta","gamma"],"xy":[0.5,0.81]}
    ],
    "sample_edges": [["A","B"],["A","C"],["B","D"],["C","D"]],
    "limit": "p_F -> 1 strongly as F ranges over all finite subsets; no sequence of finite subsets suffices",
    "assembly": "A_F=sum_(i in F) p_i tensor M_f_i -> direct_sum_i M_f_i strongly",
    "locator": "RCC7.j-RCC7.o; RCC8.b"
  },
  "matrix": {
    "density_diagonal": [1,4],
    "time_exact": "pi/(2 log(4))",
    "lambda_coefficient_phase_exact": "-i",
    "compensating_phase_exact": "i",
    "central_group": "z_t=pi_phi(h^(-it)) lambda(t)",
    "coordinate_unitary": "F xi(r)=h^(ir) xi(r)",
    "fourier_sign": "negative",
    "images": ["z_t", "I tensor T_t", "I tensor M_exp(-itq)"],
    "locator": "RCC7.e-RCC7.i; RCC8.c"
  },
  "translation": {
    "trace": "tau(f)=integral_R exp(-q)f(q)dq",
    "action": "theta_s(f)(q)=f(q-s)",
    "scaling": "tau(theta_s(f))=exp(-s)tau(f)",
    "unitary": "v_b(q)=exp(ibq)",
    "coboundary": "v_b(q)conjugate(v_b(q-s))=exp(ibs)",
    "coordinate_change": {"forward": ["y=r+q","z=q"], "inverse": ["r=y-z","q=z"], "jacobian": 1},
    "full_crossed_product": "B(L2(R_y)) tensor I_z",
    "relative_commutant": "pi(N)' intersect P=pi(N)",
    "center": "Z(P)=C1",
    "locator": "RCC7.p-RCC7.w; RCC8.d-RCC8.e"
  }
}