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            -0.460557001677674,
            0.6403963891678337
          ],
          "first_projection_max_m": -2.241243313572648,
          "second_projection_min_m": -1.2134987926303928,
          "gap_m": 1.0277445209422553,
          "plane_offset_m": -1.7273710531015205,
          "method": "minimum/maximum projections of every actual component vertex; all triangles and their enclosed convex solids remain in those halfspaces"
        },
        "classification": "separated_at_required_clearance"
      },
      {
        "parts": [
          "same-chair-east/back_rail",
          "same-stool/brace_2_3"
        ],
        "passed": true,
        "required_clearance_m": 0.0055,
        "certified_clearance_lower_bound_m": 1.0302325135606034,
        "broadphase_aabb_distance_lower_bound_m": 0.9765167407236256,
        "separating_plane": {
          "unit_normal_world": [
            -0.6058618331754627,
            -0.4616009538822383,
            0.6479629607286789
          ],
          "first_projection_max_m": -2.2455546816854364,
          "second_projection_min_m": -1.215322168124833,
          "gap_m": 1.0302325135606034,
          "plane_offset_m": -1.7304384249051346,
          "method": "minimum/maximum projections of every actual component vertex; all triangles and their enclosed convex solids remain in those halfspaces"
        },
        "classification": "separated_at_required_clearance"
      },
      {
        "parts": [
          "same-chair-east/back_rail",
          "same-stool/brace_3_1"
        ],
        "passed": true,
        "required_clearance_m": 0.0055,
        "certified_clearance_lower_bound_m": 1.2878882659748185,
        "broadphase_aabb_distance_lower_bound_m": 1.099137194561698,
        "separating_plane": {
          "unit_normal_world": [
            -0.7027084584917013,
            -0.4101546133008187,
            0.5813553264160162
          ],
          "first_projection_max_m": -2.202973450237537,
          "second_projection_min_m": -0.9150851842627186,
          "gap_m": 1.2878882659748185,
          "plane_offset_m": -1.5590293172501277,
          "method": "minimum/maximum projections of every actual component vertex; all triangles and their enclosed convex solids remain in those halfspaces"
        },
        "classification": "separated_at_required_clearance"
      }
    ],
    "same_object_contacts": "excluded from inter-furniture checks; independently accepted structural contacts remain intact"
  },
  "translation_search": [
    {
      "uuid": "same-chair-west",
      "outward_unit_direction_world": [
        -1.0,
        -0.0,
        -6.12323399574e-17
      ],
      "selected_chair_local_axis": 2,
      "minimum_shift_bracket_m": [
        0.11182479858398438,
        0.1118255615234375
      ],
      "applied_shift_m": 0.1118755615234375,
      "minimum_shift_certificate": {
        "parts": [
          "same-table/leg_1",
          "same-chair-west/seat"
        ],
        "blocked_shift_interval_m": [
          0.0,
          0.11182479858398438
        ],
        "endpoint_witnesses": [
          {
            "shift_m": 0.0,
            "first_point_m": [
              -0.6795057463794054,
              0.45376137690119445,
              -2.3548912771157897
            ],
            "second_point_m": [
              -0.6795057463794058,
              0.4537613769011948,
              -2.3548912771157906
            ],
            "distance_upper_bound_m": 1.0053497077208614e-15,
            "maximum_actual_triangle_halfspace_error_m": 0.0
          },
          {
            "shift_m": 0.11182479858398438,
            "first_point_m": [
              -0.7013088035633919,
              0.41487768299524913,
              -2.376136983253868
            ],
            "second_point_m": [
              -0.7068079658222867,
              0.414912048,
              -2.3761322326131338
            ],
            "distance_upper_bound_m": 0.005499271685577266,
            "maximum_actual_triangle_halfspace_error_m": 4.891557159325544e-10
          }
        ],
        "fixed_object_translations_world_m": {
          "same-table": [
            0.0,
            0.0,
            0.0
          ],
          "same-chair-east": [
            0.0,
            0.0,
            0.0
          ],
          "same-stool": [
            0.0,
            0.0,
            0.0
          ]
        },
        "numerical_distance_headroom_m": 5e-08,
        "proof": "For two convex solids under linear translation, interpolate each pair of contained endpoint witness points. Every intermediate pair remains in the translated solids and has distance at most the larger endpoint distance, strictly below the required clearance."
      },
      "authoring_precision_margin_m": 5e-05,
      "minimality_scope": "smallest feasible outward translation along the existing chair axis, with other furniture fixed; bisection bracket <=1micrometre"
    },
    {
      "uuid": "same-chair-east",
      "outward_unit_direction_world": [
        0.9902680687415877,
        -0.0,
        -0.13917310095994204
      ],
      "selected_chair_local_axis": 2,
      "minimum_shift_bracket_m": [
        0.10590667724609376,
        0.10590744018554688
      ],
      "applied_shift_m": 0.10595744018554687,
      "minimum_shift_certificate": {
        "parts": [
          "same-table/leg_4",
          "same-chair-east/seat"
        ],
        "blocked_shift_interval_m": [
          0.0,
          0.10590667724609376
        ],
        "endpoint_witnesses": [
          {
            "shift_m": 0.0,
            "first_point_m": [
              0.7290436251824141,
              0.47079679920950773,
              -1.8468998096485798
            ],
            "second_point_m": [
              0.7290436251824143,
              0.4707967992095075,
              -1.8468998096485794
            ],
            "distance_upper_bound_m": 5.438959822042073e-16,
            "maximum_actual_triangle_halfspace_error_m": 0.0
          },
          {
            "shift_m": 0.10590667724609376,
            "first_point_m": [
              0.7561899459497043,
              0.5202217718875581,
              -1.831001177890488
            ],
            "second_point_m": [
              0.7616304728936821,
              0.519973571,
              -1.8317671450910118
            ],
            "distance_upper_bound_m": 0.005499785710462835,
            "maximum_actual_triangle_halfspace_error_m": 2.449751512756393e-11
          }
        ],
        "fixed_object_translations_world_m": {
          "same-table": [
            0.0,
            0.0,
            0.0
          ],
          "same-chair-west": [
            -0.1118755615234375,
            -0.0,
            -6.850402416128144e-18
          ],
          "same-stool": [
            0.0,
            0.0,
            0.0
          ]
        },
        "numerical_distance_headroom_m": 5e-08,
        "proof": "For two convex solids under linear translation, interpolate each pair of contained endpoint witness points. Every intermediate pair remains in the translated solids and has distance at most the larger endpoint distance, strictly below the required clearance."
      },
      "authoring_precision_margin_m": 5e-05,
      "minimality_scope": "smallest feasible outward translation along the existing chair axis, with other furniture fixed; bisection bracket <=1micrometre"
    }
  ],
  "source_frame_policy": "all original observations and fitted component PLYs retain their original coordinates and bytes; authored root transforms are recorded separately",
  "limitations": [
    "The separation proof applies to the supplied closed convex component triangles and these recorded root translations.",
    "The minimum search is restricted to outward translation along each existing chair axis, with other furniture fixed; it is not an unrestricted layout optimisation.",
    "Positive common interior ball radius is a certified penetration lower bound, not the maximum penetration depth.",
    "This layout uses no evaluation reference geometry or furniture construction parameters."
  ]
}