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        "exact_space_boundary_right_derivative_decimal": "-0.74535420818028405918242345507887943784194992689642428123093113514",
        "representation_note": "The exact-space maximum has a flat triangle. Rounded triangle sums can change the endpoint singularity; the lifted Clausen model has its own nearby stationary reference. Neither is an interval certificate for machine arithmetic."
      },
      "invariance_diagnostics": {
        "parameter_dimension": 1,
        "parameter_minimum": [
          -0.0499526960107977
        ],
        "parameter_maximum": [
          0.02434055890665182
        ],
        "maximum_triangle_sum_change": 8.881784197001252e-16,
        "maximum_recorded_angle_extrema_residual": 0.0
      },
      "checks": {
        "solver_converged": {
          "passed": true
        },
        "result_is_not_a_rigorous_certificate": {
          "passed": true
        },
        "positive_trace": {
          "passed": true
        },
        "triangle_sums": {
          "residual": 4.440892098500626e-16,
          "tolerance": 1e-12,
          "passed": true
        },
        "final_affine_reconstruction": {
          "residual": 0.0,
          "tolerance": 1e-14,
          "passed": true
        },
        "clausen_endpoint_comparison": {
          "residual": 1.887379141862766e-15,
          "tolerance": 1e-11,
          "passed": true
        },
        "bloch_wigner_endpoint_comparison": {
          "residual": 1.1102230246251565e-16,
          "tolerance": 1e-11,
          "passed": true
        },
        "independent_final_decrement": {
          "residual": 3.573052173144048e-12,
          "tolerance": 1e-08,
          "passed": true
        },
        "independent_initial_decrement": {
          "residual": 1.1102230246251565e-16,
          "tolerance": 1e-10,
          "passed": true
        },
        "final_centering": {
          "residual": 0.0003411538864913114,
          "tolerance": 0.25,
          "passed": true
        },
        "scheduled_gap_target": {
          "residual": 0.0001,
          "tolerance": 0.00010000000000010001,
          "passed": true
        },
        "exact_space_gap_below_estimate": {
          "gap": 3.5270098608987925e-05,
          "estimate": 7.059807123005772e-05,
          "allowance": 1e-11,
          "passed": true
        },
        "lifted_input_gap_below_estimate": {
          "gap": 3.527009860349972e-05,
          "estimate": 7.059807123005772e-05,
          "allowance": 1e-11,
          "passed": true
        },
        "reference_representation_difference": {
          "residual": 7.144541053897869e-15,
          "tolerance": 1e-11,
          "passed": true
        },
        "path_centering_invariants": {
          "passed": true
        },
        "path_parameter_schedule": {
          "passed": true
        },
        "work_counts": {
          "passed": true
        },
        "full_trace_affine_invariants": {
          "passed": true
        }
      }
    }
  ],
  "exercises_initial_centering": true,
  "all_checks_passed": true,
  "scope": {
    "coordinates": "alpha=base+C*u, positive triangle angles; fixed affine incidence supplied explicitly",
    "arithmetic": "binary64 dense solver; 80-digit independent references",
    "gap": "finite-iterate theorem formula evaluated numerically, not an interval certificate",
    "centering": "strict feasible inputs are supplied; starting-point centering and path steps are counted separately",
    "not_claimed": [
      "nearly-linear solver",
      "certified floating-point volume gap",
      "automatic geometric realization",
      "full three-manifold tangent basis",
      "scaling benchmark"
    ]
  }
}
