{
  "protocol": "Isotropic radial continuous homogenized disk; no hydration kinetics, self-contact or original FEM. Ritz restricts in-plane projection rho(r), varies signed height q; stability only within this family. Fixed marked face/boundary normal for sign-sensitive shape target.",
  "source_tables": {
    "positive_c": 0.493961173,
    "negative_c": 0.315198651,
    "positive_function_max_error": 4.117750029841716e-10,
    "negative_function_max_error": 4.5981568552733165e-10,
    "fig6_area_levels_G1_to_G9": [
      0.315198651,
      0.366214841,
      0.403986352,
      0.43195173,
      0.452656815,
      0.467986505,
      0.479336343,
      0.487739569,
      0.493961173
    ],
    "S12_FM_G0_to_G9": [
      [
        0.0,
        0.250911576
      ],
      [
        1.0,
        0.29580172
      ],
      [
        2.0,
        0.384870475
      ],
      [
        3.0,
        0.407110155
      ],
      [
        4.0,
        0.43461573
      ],
      [
        5.0,
        0.44789305
      ],
      [
        6.0,
        0.46190031
      ],
      [
        7.0,
        0.465705778
      ],
      [
        8.0,
        0.495519791
      ],
      [
        9.0,
        0.500996402
      ]
    ],
    "max_cross_table_relative_difference": 0.06153875005004383,
    "note": "Fig6 tabulated design correspondence vs S12 calibration are not assumed identical independent replicate measurements; no invented uncertainty."
  },
  "c": 0.493961173,
  "area_min": 0.315198651,
  "beta_max": 0.25185558601350366,
  "beta": 0.2,
  "profiles": [
    {
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      "beta": 0.2,
      "metric_rms": 0.0,
      "intrinsic_curvature_rms": 1.6845607086287763e-16,
      "axisymmetric_feasible": true,
      "bad_reference_area_fraction": 0.0,
      "min_normalized_radicand": 0.0,
      "shape_rms_over_target_height": 3.966942329316965e-09,
      "height": 0.26192693185051785,
      "allowed": true
    },
    {
      "name": "nine_levels",
      "beta": 0.2,
      "smoothing_radial_width": 0.02,
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      "intrinsic_curvature_rms": 16.36919769343022,
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      "bad_reference_area_fraction": 0.0,
      "min_normalized_radicand": 0.0,
      "shape_rms_over_target_height": 0.11941194540135357,
      "height": 0.18141557379074122
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    {
      "name": "ideal",
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      "axisymmetric_feasible": true,
      "bad_reference_area_fraction": 0.0,
      "min_normalized_radicand": 0.0,
      "shape_rms_over_target_height": 1.1917514228712587e-08,
      "height": 0.34923590608431754,
      "allowed": false
    },
    {
      "name": "range_clipped",
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      "smoothing_radial_width": 0.02,
      "metric_rms": 0.487567374982691,
      "intrinsic_curvature_rms": 1.7107461340217773,
      "axisymmetric_feasible": true,
      "bad_reference_area_fraction": 0.0,
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      "shape_rms_over_target_height": 0.3538813565387528,
      "height": 0.16554835091228107
    }
  ],
  "noise": [
    {
      "frequency": 0,
      "stretch_amplitude": 0.0021213203435596424,
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      "intrinsic_curvature_rms": 0.00422917876989267,
      "axisymmetric_feasible": true,
      "bad_reference_area_fraction": 0.0,
      "min_normalized_radicand": 0.0,
      "shape_rms_over_target_height": 0.00378411539350354,
      "height": 0.26248256277957965
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    {
      "frequency": 1,
      "stretch_amplitude": 0.003,
      "metric_rms": 0.004242644266845853,
      "intrinsic_curvature_rms": 0.1308408505493709,
      "axisymmetric_feasible": true,
      "bad_reference_area_fraction": 0.0,
      "min_normalized_radicand": 0.0,
      "shape_rms_over_target_height": 0.010729367321350157,
      "height": 0.2624408422962238
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    {
      "frequency": 4,
      "stretch_amplitude": 0.003,
      "metric_rms": 0.0042426442668458625,
      "intrinsic_curvature_rms": 2.04958879177603,
      "axisymmetric_feasible": false,
      "bad_reference_area_fraction": 0.007518000000000002,
      "min_normalized_radicand": -0.0009277208096937076,
      "shape_rms_over_target_height": null,
      "height": null
    },
    {
      "frequency": 12,
      "stretch_amplitude": 0.003,
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      "intrinsic_curvature_rms": 18.399789389730202,
      "axisymmetric_feasible": false,
      "bad_reference_area_fraction": 0.0892947200000001,
      "min_normalized_radicand": -0.06579334327887747,
      "shape_rms_over_target_height": null,
      "height": null
    },
    {
      "frequency": 20,
      "stretch_amplitude": 0.003,
      "metric_rms": 0.004242644266845859,
      "intrinsic_curvature_rms": 51.096518143479834,
      "axisymmetric_feasible": false,
      "bad_reference_area_fraction": 0.2772827999999999,
      "min_normalized_radicand": -0.19546931028729228,
      "shape_rms_over_target_height": null,
      "height": null
    },
    {
      "frequency": 40,
      "stretch_amplitude": 0.003,
      "metric_rms": 0.00424264426684586,
      "intrinsic_curvature_rms": 204.35873301096308,
      "axisymmetric_feasible": false,
      "bad_reference_area_fraction": 0.4010669600000003,
      "min_normalized_radicand": -1.0144910150118023,
      "shape_rms_over_target_height": null,
      "height": null
    }
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  "ritz_stretch_coefficient": 0.020175897787429366,
  "mechanics": [
    {
      "h": 0.005,
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        {
          "q": -0.9999456433061431,
          "energy": 3.1165401665521304e-08,
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        },
        {
          "q": 0.0,
          "energy": 0.00010087948893714684,
          "hessian": -0.000403424581067735,
          "stable": false
        },
        {
          "q": 0.9999456433061431,
          "energy": 3.1165401665521304e-08,
          "hessian": 0.0008069026206347025,
          "stable": true
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      ],
      "global_minimum_in_family": {
        "q": -0.9999456433061431,
        "energy": 3.1165401665521304e-08,
        "hessian": 0.0008069026206347025,
        "stable": true
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    },
    {
      "h": 0.02,
      "stationary": [
        {
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          "energy": 1.9934402360346645e-06,
          "hessian": 0.003219607964598475,
          "stable": true
        },
        {
          "q": 0.0,
          "energy": 0.00040351795574858736,
          "hessian": -0.0016080886622305,
          "stable": false
        },
        {
          "q": 0.9991292003400554,
          "energy": 1.9934402360346654e-06,
          "hessian": 0.0032196079434226514,
          "stable": true
        }
      ],
      "global_minimum_in_family": {
        "q": -0.99912920034005,
        "energy": 1.9934402360346645e-06,
        "hessian": 0.003219607964598475,
        "stable": true
      }
    },
    {
      "h": 0.05,
      "stationary": [
        {
          "q": -0.9945187293795887,
          "energy": 3.104670910586625e-05,
          "hessian": 0.007936538592664443,
          "stable": true
        },
        {
          "q": 0.0,
          "energy": 0.0010087948893714683,
          "hessian": -0.00394167771339271,
          "stable": false
        },
        {
          "q": 0.9945187293796707,
          "energy": 3.1046709105866244e-05,
          "hessian": 0.007936538728189715,
          "stable": true
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      ],
      "global_minimum_in_family": {
        "q": 0.9945187293796707,
        "energy": 3.1046709105866244e-05,
        "hessian": 0.007936538728189715,
        "stable": true
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    },
    {
      "h": 0.1,
      "stationary": [
        {
          "q": -0.9774910300522518,
          "energy": 0.0002454285994822806,
          "hessian": 0.015056820807518577,
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        },
        {
          "q": 0.0,
          "energy": 0.0020175897787429366,
          "hessian": -0.007322363201889281,
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        {
          "q": 0.9774910300522518,
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          "hessian": 0.015056820807518577,
          "stable": true
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      ],
      "global_minimum_in_family": {
        "q": -0.9774910300522518,
        "energy": 0.0002454285994822806,
        "hessian": 0.015056820807518577,
        "stable": true
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    },
    {
      "h": 0.2,
      "stationary": [
        {
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        {
          "q": 0.0,
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        {
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          "hessian": 0.02310973678046335,
          "stable": true
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      ],
      "global_minimum_in_family": {
        "q": 0.8980993702342845,
        "energy": 0.0018594720327995567,
        "hessian": 0.02310973678046335,
        "stable": true
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    },
    {
      "h": 0.4,
      "stationary": [
        {
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          "energy": 0.008070359114971747,
          "hessian": 0.015590445601176837,
          "stable": true
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      ],
      "global_minimum_in_family": {
        "q": 0.0,
        "energy": 0.008070359114971747,
        "hessian": 0.015590445601176837,
        "stable": true
      }
    }
  ],
  "biased_h02_eta02": [
    {
      "q": -0.9966130696446964,
      "energy": 1.0046631471181506e-05,
      "hessian": 0.0031952988601661055,
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    },
    {
      "q": -0.005018732235706661,
      "energy": 0.00040353820697942286,
      "hessian": -0.0016079650631828366,
      "stable": false
    },
    {
      "q": 1.0016264518274056,
      "energy": -6.079980641414995e-06,
      "hessian": 0.0032437952995743474,
      "stable": true
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  ],
  "curvature_projection": [
    {
      "target": "sphere",
      "theta": 0.0,
      "coefficient": 0.0,
      "residual": 1.4142135623730951,
      "C": [
        [
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        [
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        ]
      ]
    },
    {
      "target": "sphere",
      "theta": 0.7853981633974483,
      "coefficient": 0.0,
      "residual": 1.4142135623730951,
      "C": [
        [
          -1.0,
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        ],
        [
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          1.0
        ]
      ]
    },
    {
      "target": "saddle",
      "theta": 0.0,
      "coefficient": 0.0,
      "residual": 1.4142135623730951,
      "C": [
        [
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        ],
        [
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      ]
    },
    {
      "target": "saddle",
      "theta": 0.7853981633974483,
      "coefficient": -1.0,
      "residual": 2.684153782866527e-16,
      "C": [
        [
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        ],
        [
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          1.0
        ]
      ]
    }
  ],
  "noise_note": "Same stretch RMS to first order: uniform epsilon/sqrt2 vs cosine epsilon .003. Intrinsic curvature exists even when axisymmetric immersion fails; do not clip negative radicands into fake witnesses."
}