{
  "kind": "exact proof diagram; no numerical samples",
  "rings": [
    "E",
    "B=E-infinity",
    "C=E-real"
  ],
  "target": "S=colim O(left half-polydisc)/O(full polydisc)",
  "resolution": {
    "F_k": "K_k direct-sum K_(k-1)",
    "d_F(a,b)": "(d_K a+f b,-d_K b)",
    "f(Q)": "Q(theta I-A)",
    "rank_F_k": "q*binomial(d+1,k)",
    "proof": [
      "EE.5",
      "EE.6",
      "EE.7",
      "EE.8",
      "EE.9",
      "EE.10",
      "EE.11",
      "EE.12"
    ]
  },
  "solution_contraction": {
    "operators": [
      "theta I-A",
      "D_x1",
      "...",
      "D_xd"
    ],
    "homotopy_on_alpha_plus_e0_beta": "H_x alpha-e0 wedge H_x beta",
    "last_complex": "S_0^q --theta I-A--> S_0^q",
    "last_arrow": "surjective by EE.17",
    "positive_ext": "zero",
    "hom_dimension": "q"
  },
  "extension": {
    "sequence": "0 -> L -> T -> N -> 0",
    "membership": "finite filtration by arbitrary T_A=B tensor_E N_A",
    "functional_sequence": "0 -> Hom_B(N,S) -> Hom_B(T,S) -> Hom_B(L,S) -> 0",
    "restriction_surjective_reason": "Ext_B^1(N,S)=0; EE.24",
    "separation_steps": [
      "all g(pi v)=0 -> pi v=0 -> v=i(w)",
      "all u(w)=0 -> w=0 -> v=0"
    ]
  },
  "comparison": {
    "arrows": [
      "T -> C tensor_B T",
      "C tensor_B T -> S^r(T)"
    ],
    "formulas": [
      "v -> 1 tensor v",
      "c tensor v -> (c f_j(v))_j"
    ],
    "composite": "injective evaluation; EE.25--EE.27",
    "C_flatness_assumed": false
  },
  "boundary": "No arbitrary holonomic membership or contact theorem is asserted."
}