{
  "schema": "MH043-original-exact-diagram-geometry/v1",
  "scope": "Exact local coordinates and actual cylinder parameter plots, not numerical proofs or global embeddings",
  "figure1": {
    "q": "-u^2+v^2",
    "lambda": 1,
    "nu": 1,
    "block": [
      [
        -2,
        2
      ],
      [
        -1,
        1
      ]
    ],
    "lower_level": -1,
    "upper_level": 2,
    "max_q_on_block": 1,
    "max_q_on_u_exit": -3,
    "lower_boundary": "|u|=sqrt(1+v^2)",
    "field": [
      "du/dt=2u",
      "dv/dt=-2v"
    ],
    "initial_point": [
      0.5,
      1.5
    ],
    "first_block_entry": [
      0.75,
      1
    ],
    "q_at_entry": "7/16",
    "T_B": "log(3/2)/2",
    "hypothetical_T_A": "log(2+sqrt(13))/4",
    "radial_point": [
      "sqrt(5)/2",
      "1/2"
    ],
    "radial_exit": [
      2,
      0.5
    ],
    "q_at_radial_exit": "-15/4",
    "interpretation": "rescaled local Morse chart; relative core is not asserted to be an absolute loop",
    "proof": [
      "MH9",
      "MH10",
      "MH11",
      "MH12",
      "MH13",
      "MH14",
      "MH15",
      "MH19",
      "MH20"
    ]
  },
  "figure2": {
    "standard_pairs": [
      [
        "D0",
        "empty"
      ],
      [
        "D1",
        "S0"
      ],
      [
        "D2",
        "S1"
      ]
    ],
    "relative_nonzero_degrees": [
      0,
      1,
      2
    ],
    "coefficient": "arbitrary unital ring k",
    "index_zero_based_quotient": "original point disjoint added point",
    "interval_boundary": "right minus left",
    "index_two_quotient": "D2/boundaryD2 = S2, not planar embedding",
    "proof": [
      "MH4",
      "MH16",
      "MH17",
      "MH18",
      "MH19",
      "MH20"
    ]
  },
  "figure3": {
    "manifold": "S1 x R",
    "function": "t^2-cos(theta)",
    "parameter_identification": "theta=-pi and theta=pi are the same edge; markers at (\u00b1pi,0) are one point",
    "critical_points": [
      {
        "theta": 0,
        "t": 0,
        "value": -1,
        "index": 0
      },
      {
        "theta": "pi modulo 2pi",
        "t": 0,
        "value": 1,
        "index": 1
      }
    ],
    "closed_plot_levels": [
      0,
      2
    ],
    "bounds": "|t| <= sqrt(level+cos(theta)) where level+cos(theta)>=0",
    "open_stage_heights": [
      -2,
      0,
      2,
      3
    ],
    "relative_group_ledger": [
      "k at degree0",
      "k at degree1",
      "zero"
    ],
    "actual_absolute_homology": {
      "H0": "k",
      "H1": "k",
      "Hj_j>=2": 0
    },
    "proof": [
      "MH8",
      "MH21",
      "MH22",
      "learner Example 3"
    ]
  },
  "source_credit": "Original diagrams from the written MH043 proof; Morse-coordinate programme method AN04 Lemma4.1, inverse/ODE DGCHAR Lemma3.1, finite chains DGCHAR Thom \u00a7\u00a71\u20133; classical Morse and Hatcher-style singular excision methods.",
  "blender_assessment": "Planar exact block, trajectory, relative disc pairs and cylinder parameter plots explain these objects directly; a 3D scene would not improve the finite-chain or quotient argument."
}