{
 "about": {
  "method": "hill-initial-rate",
  "origin": "Original synthetic rates from known parameters with recorded noise vectors (NumPy default_rng seeds named per case); no published dataset",
  "ruling": "CSO memo of 13 October 2026, rulings H1 to H4 (docs/board/memos/2026-10-13-cso-hill-and-bisubstrate.md)",
  "model": "v = Vmax S^h / (S0.5^h + S^h); h in [0.1, 10]",
  "minimumLevels": 6,
  "minimumEachSide": 2,
  "withheld": "fewer than two distinct levels below or above the fitted S0.5 ('not-bracketed'): the whole fit, no constant, the Michaelis-Menten fit offered; any parameter at its box: 'bound'",
  "interval": "95 percent profile extra-sum-of-squares interval on h only; threshold RSS_min (1 + F(0.95; 1, n - 3)/(n - 3))",
  "comparison": "AICc (K = parameters + 1) for Hill and Michaelis-Menten on the same rates, dAicc = AICc_MM - AICc_Hill, and the ordinary F of Michaelis-Menten within Hill (h = 1 interior); withheld at an exact fit",
  "coverage": {
   "range": [
    93.5,
    96.5
   ],
   "runs": [
    {
     "h": 1.0,
     "design": "8 levels 0.25 to 32 uM in duplicate, Vmax 10, S0.5 3, additive noise SD 0.3",
     "seed": 20261015,
     "replicates": 2000,
     "resolved": 2000,
     "notResolved": 0,
     "contained": 1892,
     "percent": 94.6,
     "passes": true
    },
    {
     "h": 2.5,
     "design": "8 levels 0.25 to 32 uM in duplicate, Vmax 10, S0.5 3, additive noise SD 0.3",
     "seed": 20261016,
     "replicates": 2000,
     "resolved": 2000,
     "notResolved": 0,
     "contained": 1910,
     "percent": 95.5,
     "passes": true
    }
   ],
   "passes": true
  },
  "tolerances": {
   "statuses, refusals, reasons, the parameters at the box and the counts below and above S0.5": "exact",
   "vmax, s05 and h": "within 1e-5 relative; hand values within 1e-9 relative",
   "rss": "within 1e-6 relative, or both at the rates' numeric resolution",
   "correlations": "within 1e-4 absolute",
   "h interval bounds": "within 1e-3 in ln h; side states exact",
   "aicc and dAicc": "within 1e-6 absolute",
   "F": "within 1e-5 relative",
   "p": "within 1e-4 relative or 1e-10 absolute"
  },
  "recordedOn": "2026-10-13",
  "script": "pipeline/assay-analyser/oracles/hill_kinetics.py",
  "scriptSha256": "88c18463d39a3dbdc9549f22144155709811db509a2a8a9c25ae332301233c92",
  "helper": {
   "script": "pipeline/assay-analyser/oracles/stage_a_oracle.py",
   "sha256": "492d85ca2fa6750db80b1c64214fd4a0b13cd4c588267244f6262c504c73e3e8"
  },
  "environment": {
   "image": "ghcr.io/biotechmat/drybench/oracle-env:11b8a5e4456df7eb@sha256:7e69891ca1d9f24838660e0a9887bdb654ce7771d2f2ac7e7275e9b1eb3b6a09",
   "r": "R version 4.4.3 (2025-02-28)",
   "packages": {
    "jsonlite": "2.0.0"
   },
   "python": "3.11.15",
   "numpy": "2.0.2",
   "scipy": "1.13.1",
   "recordedOn": "2026-10-13",
   "cpu": "AMD EPYC 9V74 80-Core Processor",
   "cpuClass": "x86-64-v3",
   "simd": [
    "SSSE3",
    "SSE41",
    "POPCNT",
    "SSE42",
    "AVX",
    "F16C",
    "FMA3",
    "AVX2"
   ],
   "openblasCore": "Haswell",
   "threads": 1
  },
  "crossCheck": {
   "tool": "R stats::nls",
   "version": "R version 4.4.3 (2025-02-28)",
   "script": "pipeline/assay-analyser/oracles/r/hill_nls.R",
   "scriptSha256": "cd7abad0d519110ef9f49e4751ec297d627eb13f2b29dff0677efb5d011576db",
   "ran": true,
   "settings": "nls on each estimate from the SciPy values, the same rows, tol 1e-8 then 1e-6, scaleOffset 1",
   "tolerances": {
    "parameters": "1e-5 relative",
    "rss": "1e-6 relative, or both at the rates' numeric resolution"
   },
   "cases": 7,
   "agreeing": 7,
   "scipyOnly": [
    "not-bracketed",
    "h-at-box",
    "refused-five-levels",
    "refused-negative-substrate"
   ],
   "result": [
    {
     "case": "noiseless-h2",
     "converged": true,
     "parameters": {
      "vmax": 10,
      "s05": 3,
      "h": 2
     },
     "relativeDifferences": {
      "vmax": 1.77635683940025e-16,
      "s05": 2.96059473233375e-16,
      "h": 1.1102230246251568e-16
     },
     "rss": 7.14905195356542e-30,
     "agrees": true
    },
    {
     "case": "noiseless-h05",
     "converged": true,
     "parameters": {
      "vmax": 5,
      "s05": 2,
      "h": 0.5
     },
     "relativeDifferences": {
      "vmax": 1.7763568394002508e-16,
      "s05": 1.1102230246251577e-15,
      "h": 2.2204460492503126e-16
     },
     "rss": 1.03537993810258e-30,
     "agrees": true
    },
    {
     "case": "noisy-duplicate",
     "converged": true,
     "parameters": {
      "vmax": 9.98765478273982,
      "s05": 2.98051428077576,
      "h": 2.00913309170702
     },
     "relativeDifferences": {
      "vmax": 3.557105002207452e-16,
      "s05": 2.9799502234525544e-16,
      "h": 1.9893171364047205e-15
     },
     "rss": 0.203111112800565,
     "agrees": true
    },
    {
     "case": "metamorphic-mM",
     "converged": true,
     "parameters": {
      "vmax": 9.98765478003626,
      "s05": 0.00298051427965871,
      "h": 2.00913309483649
     },
     "relativeDifferences": {
      "vmax": 0.0,
      "s05": 2.910107641181046e-16,
      "h": 6.631057111020369e-16
     },
     "rss": 0.203111112800564,
     "agrees": true
    },
    {
     "case": "metamorphic-rate-unit",
     "converged": true,
     "parameters": {
      "vmax": 9987.65477483995,
      "s05": 2.98051427647636,
      "h": 2.00913310055812
     },
     "relativeDifferences": {
      "vmax": 9.936418260286569e-11,
      "s05": 1.1579416097580513e-10,
      "h": 6.372465754653507e-10
     },
     "rss": 203111.112800564,
     "agrees": true
    },
    {
     "case": "metamorphic-row-order",
     "converged": true,
     "parameters": {
      "vmax": 9.98765477555403,
      "s05": 2.98051427800598,
      "h": 2.00913310133012
     },
     "relativeDifferences": {
      "vmax": 5.335657507150013e-16,
      "s05": 8.939850678665435e-16,
      "h": 1.9893171268765244e-15
     },
     "rss": 0.203111112800564,
     "agrees": true
    },
    {
     "case": "michaelis-menten-rates",
     "converged": true,
     "parameters": {
      "vmax": 9.95563357948649,
      "s05": 2.97625138587588,
      "h": 0.975720260690192
     },
     "relativeDifferences": {
      "vmax": 5.352819060337113e-16,
      "s05": 1.342898287294489e-15,
      "h": 1.1378497191805999e-16
     },
     "rss": 0.493814111135277,
     "agrees": true
    }
   ]
  }
 },
 "oracle": {
  "engine": "scipy.optimize.curve_fit",
  "method": "trf with bounds, log parameters, three starts, lowest RSS; the h profile by curve_fit refits and brentq; Michaelis-Menten by michaelis_menten.py's method; AICc and F by hand, p from scipy.stats.f.sf"
 },
 "cases": [
  {
   "name": "noiseless-h2",
   "about": "Hill rates from Vmax 10, S0.5 3, h 2 at 8 levels 0.25 to 32 uM: the hand values exactly",
   "substrate": [
    0.25,
    0.5,
    1.0,
    2.0,
    4.0,
    8.0,
    16.0,
    32.0
   ],
   "rate": [
    0.06896551724137931,
    0.2702702702702703,
    1.0,
    3.076923076923077,
    6.4,
    8.767123287671234,
    9.660377358490567,
    9.912875121006778
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "estimated",
    "vmax": 10.000000000000002,
    "s05": 3.000000000000001,
    "h": 1.9999999999999998,
    "rss": 3.155636213878486e-30,
    "n": 8,
    "df": 5,
    "levels": 8,
    "below": 4,
    "above": 4,
    "correlation": {
     "order": [
      "vmax",
      "s05",
      "h"
     ],
     "matrix": [
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       0.6593364817529658,
       -0.5930071610664255
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      [
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       1.0,
       -0.39049585938493336
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    },
    "residuals": [
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     0.0,
     0.0,
     0.0,
     1.7763568394002505e-15,
     0.0,
     0.0
    ],
    "hInterval": {
     "status": "not-available",
     "reason": "zero-residual"
    },
    "comparison": {
     "status": "not-available",
     "reason": "zero-residual"
    }
   },
   "hand": {
    "vmax": 10.0,
    "s05": 3.0,
    "h": 2.0
   }
  },
  {
   "name": "noiseless-h05",
   "about": "Hill rates from Vmax 5, S0.5 2, h 0.5 at 8 levels 0.01 to 256 uM: the hand values exactly",
   "substrate": [
    0.01,
    0.05,
    0.2,
    1.0,
    4.0,
    16.0,
    64.0,
    256.0
   ],
   "rate": [
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    1.2012653667602107,
    2.0710678118654755,
    2.928932188134525,
    3.6939806251812928,
    4.248894475888326,
    4.593948484291938
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "estimated",
    "vmax": 4.999999999999999,
    "s05": 1.9999999999999978,
    "h": 0.5000000000000001,
    "rss": 1.0384614260135976e-30,
    "n": 8,
    "df": 5,
    "levels": 8,
    "below": 4,
    "above": 4,
    "correlation": {
     "order": [
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     "matrix": [
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     0.0,
     8.881784197001252e-16,
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    "hInterval": {
     "status": "not-available",
     "reason": "zero-residual"
    },
    "comparison": {
     "status": "not-available",
     "reason": "zero-residual"
    }
   },
   "hand": {
    "vmax": 5.0,
    "s05": 2.0,
    "h": 0.5
   }
  },
  {
   "name": "noisy-duplicate",
   "about": "The h = 2 model in duplicate with additive noise SD 0.15 (seed 20261013)",
   "substrate": [
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    0.25,
    0.5,
    0.5,
    1.0,
    1.0,
    2.0,
    2.0,
    4.0,
    4.0,
    8.0,
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    16.0,
    16.0,
    32.0,
    32.0
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   "rate": [
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    3.1195194634923995,
    2.909263065858298,
    6.569230301416336,
    6.5234288107121525,
    8.671849124124668,
    8.67288758959861,
    9.797581705346257,
    9.46135050064174,
    9.90400637398006,
    10.041556565576027
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   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
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    "vmax": 9.987654782739824,
    "s05": 2.980514280775759,
    "h": 2.009133091707016,
    "rss": 0.20311111280056382,
    "n": 16,
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    "levels": 8,
    "below": 4,
    "above": 4,
    "correlation": {
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     "matrix": [
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    "hInterval": {
     "status": "computed",
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     "low": {
      "status": "crossing",
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     "michaelisMenten": {
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     "aiccHill": -58.22908945525506,
     "aiccMichaelisMenten": 0.5569399163627988,
     "dAicc": 58.78602937161786,
     "F": 630.1313115774427,
     "p": 2.1239657975287405e-12,
     "df1": 1,
     "df2": 13
    }
   }
  },
  {
   "name": "metamorphic-mM",
   "about": "The noisy rates with substrate in mM (every S x 1e-3): S0.5 x 1e-3, Vmax, h and RSS unchanged",
   "substrate": [
    0.00025,
    0.00025,
    0.0005,
    0.0005,
    0.001,
    0.001,
    0.002,
    0.002,
    0.004,
    0.004,
    0.008,
    0.008,
    0.016,
    0.016,
    0.032,
    0.032
   ],
   "rate": [
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    0.220665772274722,
    0.3934078981547179,
    0.9767077787404874,
    1.0222427330783015,
    3.1195194634923995,
    2.909263065858298,
    6.569230301416336,
    6.5234288107121525,
    8.671849124124668,
    8.67288758959861,
    9.797581705346257,
    9.46135050064174,
    9.90400637398006,
    10.041556565576027
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   "substrateUnit": "mM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "estimated",
    "vmax": 9.98765478003626,
    "s05": 0.0029805142796587108,
    "h": 2.0091330948364914,
    "rss": 0.20311111280056374,
    "n": 16,
    "df": 13,
    "levels": 8,
    "below": 4,
    "above": 4,
    "correlation": {
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     "matrix": [
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     "michaelisMenten": {
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     "aiccHill": -58.22908945525506,
     "aiccMichaelisMenten": 0.5569399163628042,
     "dAicc": 58.78602937161786,
     "F": 630.1313115774432,
     "p": 2.123965797528731e-12,
     "df1": 1,
     "df2": 13
    }
   }
  },
  {
   "name": "metamorphic-rate-unit",
   "about": "The noisy rates in nM/min (every v x 1e3): Vmax x 1e3, RSS x 1e6, S0.5 and h unchanged",
   "substrate": [
    0.25,
    0.25,
    0.5,
    0.5,
    1.0,
    1.0,
    2.0,
    2.0,
    4.0,
    4.0,
    8.0,
    8.0,
    16.0,
    16.0,
    32.0,
    32.0
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   "rate": [
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    2909.263065858298,
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    8671.849124124668,
    8672.88758959861,
    9797.581705346258,
    9461.350500641742,
    9904.00637398006,
    10041.556565576027
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   "substrateUnit": "µM",
   "rateUnit": "nM/min",
   "expected": {
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    "vmax": 9987.654773847535,
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    "h": 2.009133101838433,
    "rss": 203111.11280056412,
    "n": 16,
    "df": 13,
    "levels": 8,
    "below": 4,
    "above": 4,
    "correlation": {
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     "matrix": [
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     "aiccHill": 162.81907947217334,
     "aiccMichaelisMenten": 221.6051088437912,
     "dAicc": 58.78602937161784,
     "F": 630.1313115774418,
     "p": 2.1239657975287594e-12,
     "df1": 1,
     "df2": 13
    }
   }
  },
  {
   "name": "metamorphic-row-order",
   "about": "The noisy rates with the rows permuted (seed 20261013): everything unchanged",
   "substrate": [
    32.0,
    8.0,
    2.0,
    8.0,
    32.0,
    4.0,
    16.0,
    0.25,
    0.5,
    16.0,
    1.0,
    2.0,
    0.5,
    1.0,
    4.0,
    0.25
   ],
   "rate": [
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    3.1195194634923995,
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    9.797581705346257,
    0.1536973147118371,
    0.3934078981547179,
    9.46135050064174,
    1.0222427330783015,
    2.909263065858298,
    0.220665772274722,
    0.9767077787404874,
    6.569230301416336,
    0.18317388365821274
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "estimated",
    "vmax": 9.987654775554025,
    "s05": 2.9805142780059772,
    "h": 2.009133101330124,
    "rss": 0.2031111128005639,
    "n": 16,
    "df": 13,
    "levels": 8,
    "below": 4,
    "above": 4,
    "correlation": {
     "order": [
      "vmax",
      "s05",
      "h"
     ],
     "matrix": [
      [
       1.0000000000000002,
       0.6562778641836876,
       -0.5903641437818112
      ],
      [
       0.6562778641836875,
       1.0000000000000002,
       -0.3869674475569794
      ],
      [
       -0.5903641437818112,
       -0.3869674475569794,
       1.0
      ]
     ]
    },
    "residuals": [
     0.13797541699884341,
     -0.10805825227428478,
     0.026382928248460225,
     -0.1070197868003433,
     0.0004252254028767055,
     0.09523622963206702,
     0.13995234187438932,
     0.08547070106981615,
     0.12432930300101963,
     -0.19627886283012685,
     0.02072452038529371,
     -0.1838734693856412,
     -0.04841282287897625,
     -0.02481043395252036,
     0.14103772033625006,
     0.1149472700161918
    ],
    "hInterval": {
     "status": "computed",
     "level": 0.95,
     "df": 13,
     "threshold": 0.2760310135248758,
     "low": {
      "status": "crossing",
      "value": 1.8889222061666011
     },
     "high": {
      "status": "crossing",
      "value": 2.139204495158941
     }
    },
    "comparison": {
     "status": "computed",
     "michaelisMenten": {
      "status": "estimated",
      "vmax": 12.288152173967084,
      "km": 4.732925912097924,
      "rss": 10.04823972087543
     },
     "aiccHill": -58.22908945525504,
     "aiccMichaelisMenten": 0.5569399163628077,
     "dAicc": 58.78602937161785,
     "F": 630.1313115774427,
     "p": 2.1239657975287405e-12,
     "df1": 1,
     "df2": 13
    }
   }
  },
  {
   "name": "michaelis-menten-rates",
   "about": "Michaelis-Menten rates (Vmax 10, Km 3) in duplicate with noise SD 0.15 (seed 20261014): the interval on h includes 1 and the F is reported",
   "substrate": [
    0.25,
    0.25,
    0.5,
    0.5,
    1.0,
    1.0,
    2.0,
    2.0,
    4.0,
    4.0,
    8.0,
    8.0,
    16.0,
    16.0,
    32.0,
    32.0
   ],
   "rate": [
    0.6985911122340909,
    0.8385051276918859,
    1.307046874630719,
    1.7077153095561248,
    2.280754215999607,
    2.4615185717885923,
    4.156423521451021,
    4.240457597966199,
    5.736883902246124,
    5.92242205129898,
    7.080544874391006,
    6.9757259282775514,
    8.064146410264849,
    8.391798972875849,
    9.268621262903753,
    9.124787159191513
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "estimated",
    "vmax": 9.955633579486495,
    "s05": 2.976251385875876,
    "h": 0.9757202606901919,
    "rss": 0.49381411113527635,
    "n": 16,
    "df": 13,
    "levels": 8,
    "below": 4,
    "above": 4,
    "correlation": {
     "order": [
      "vmax",
      "s05",
      "h"
     ],
     "matrix": [
      [
       1.0,
       0.9262692883887191,
       -0.8294097575615327
      ],
      [
       0.9262692883887192,
       1.0,
       -0.7693777336988109
      ],
      [
       -0.8294097575615326,
       -0.7693777336988107,
       1.0
      ]
     ]
    },
    "residuals": [
     -0.11676651813314287,
     0.023147497324652155,
     -0.17882420731386195,
     0.221844227611544,
     -0.2729732007147754,
     -0.09220884492579007,
     0.13204606250396989,
     0.21608013901914802,
     0.046073450992816234,
     0.23161160004567272,
     -0.12808811165553813,
     -0.23290705776899312,
     -0.27551713724262683,
     0.052135425368373234,
     0.2059252970340495,
     0.0620911933218089
    ],
    "hInterval": {
     "status": "computed",
     "level": 0.95,
     "df": 13,
     "threshold": 0.6711006980863607,
     "low": {
      "status": "crossing",
      "value": 0.8745006726936801
     },
     "high": {
      "status": "crossing",
      "value": 1.0851824287057648
     }
    },
    "comparison": {
     "status": "computed",
     "michaelisMenten": {
      "status": "estimated",
      "vmax": 9.848320855492585,
      "km": 2.8886483827360503,
      "rss": 0.5031164631255491
     },
     "aiccHill": -44.014593933212275,
     "aiccMichaelisMenten": -47.352357134301045,
     "dAicc": -3.3377632010887694,
     "F": 0.24489088737364656,
     "p": 0.6289538946615074,
     "df1": 1,
     "df2": 13
    }
   }
  },
  {
   "name": "not-bracketed",
   "about": "Hill rates (S0.5 1.8, h 2) at 8 levels 0.1 to 2 uM: one level above S0.5, so the fit is withheld",
   "substrate": [
    0.1,
    0.2,
    0.4,
    0.6,
    0.8,
    1.0,
    1.5,
    2.0
   ],
   "rate": [
    0.03076923076923077,
    0.12195121951219512,
    0.4705882352941177,
    1.0,
    1.6494845360824744,
    2.358490566037736,
    4.098360655737705,
    5.524861878453039
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "withheld",
    "reason": "not-bracketed",
    "n": 8,
    "levels": 8,
    "below": 7,
    "above": 1,
    "michaelisMenten": {
     "status": "bound"
    }
   }
  },
  {
   "name": "h-at-box",
   "about": "A step from 0 to 10 between 3 and 4 uM: h runs to its upper box (bound)",
   "substrate": [
    0.25,
    0.5,
    1.0,
    2.0,
    4.0,
    8.0,
    16.0,
    32.0
   ],
   "rate": [
    0.0,
    0.0,
    0.0,
    0.0,
    10.0,
    10.0,
    10.0,
    10.0
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "bound",
    "parameters": [
     "h"
    ],
    "n": 8,
    "levels": 8
   }
  },
  {
   "name": "refused-five-levels",
   "about": "Five substrate levels: fewer than six",
   "substrate": [
    1.0,
    2.0,
    4.0,
    8.0,
    16.0
   ],
   "rate": [
    1.0,
    3.076923076923077,
    6.4,
    8.767123287671234,
    9.660377358490567
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "refused",
    "reason": "substrate-levels"
   }
  },
  {
   "name": "refused-negative-substrate",
   "about": "A negative substrate concentration",
   "substrate": [
    -1.0,
    0.5,
    1.0,
    2.0,
    4.0,
    8.0,
    16.0,
    32.0
   ],
   "rate": [
    0.06896551724137931,
    0.2702702702702703,
    1.0,
    3.076923076923077,
    6.4,
    8.767123287671234,
    9.660377358490567,
    9.912875121006778
   ],
   "substrateUnit": "µM",
   "rateUnit": "µM/min",
   "expected": {
    "status": "refused",
    "reason": "invalid"
   }
  }
 ]
}