{
  "title": "Hypothetical calibration and bounded transition readout",
  "status": "Analytical teaching models; no measured thermometer or material",
  "rows": {
    "calibration": 201,
    "transition": 401
  },
  "parameters": {
    "linear_intercept_V": 0.2,
    "linear_slope_V_per_K": 0.01,
    "linear_temperature_C_range": [
      0,
      100
    ],
    "linear_readout_standard_uncertainty_V": 0.001,
    "transition_midpoint_K": 4,
    "transition_width_parameter_K": 0.02,
    "transition_temperature_K_range": [
      3.8,
      4.2
    ],
    "transition_normalized_readout_standard_uncertainty": 0.001,
    "uncertainty_signal_domain": [
      0.05,
      0.95
    ]
  },
  "equations": {
    "linear": "S=a+b*T_C; inferred T_C=(S-a)/b",
    "linear_readout_uncertainty": "uT=uS/abs(b), a and b treated exact",
    "transition": "s=1/(1+exp(-(T_K-T0)/w))",
    "inverse": "T_K=T0+w*ln(s/(1-s)), strictly 0<s<1",
    "sensitivity": "ds/dT=s*(1-s)/w",
    "local_uncertainty": "uT=us/(ds/dT), shown only for 0.05<=s<=0.95"
  },
  "checks": {
    "all_pass": true,
    "linear_inverse_max_error_K": 1.4210854715202004e-14,
    "transition_inverse_max_error_K": 3.8191672047105385e-14,
    "derivative_finite_difference_max_relative_error": 4.363698842543524e-08,
    "linearization_symmetric_span_max_relative_difference": 0.00011620288209246432,
    "invalid_inverse_inputs_rejected": 6,
    "outside_uncertainty_view_returns_null": true
  },
  "limitations": [
    "Assigned mathematical responses, not properties of a named crystal or superconducting specimen.",
    "The logistic response is phenomenological and has no exact zero-resistance plateau. It is not a microscopic superconductivity model.",
    "Only readout standard uncertainty is propagated; calibration coefficients and midpoint/width are treated exact. No total accuracy claim follows.",
    "The local uncertainty approximation is deliberately restricted away from saturation; no finite inverse exists at normalized signal 0 or 1.",
    "No thermal lag, contact error, self-heating, magnetic-field dependence, hysteresis or electrical feedback is modeled.",
    "A binary threshold indicates a side of a transition; an analog calibrated transition response can resolve temperature locally.",
    "Linear example uses Celsius values relative to its chosen zero; temperature differences in C and K have equal numerical size."
  ],
  "sources": [
    {
      "title": "Swetz et al. (2012), Applied Physics Letters 101, 242603 \u2014 Current distribution and transition width in superconducting transition-edge sensors",
      "url": "https://doi.org/10.1063/1.4771984"
    },
    {
      "title": "NIST publication record (journal metadata conflicts with linked article PDF)",
      "url": "https://www.nist.gov/publications/current-distribution-and-transition-width-superconducting-transition-edge-sensors",
      "note": "NIST record title page says Physical Review Letters; its linked PDF and DOI identify Applied Physics Letters 101, 242603 (2012). Do not use the landing page's journal field as authoritative."
    },
    {
      "title": "NIST Industrial Thermometer Calibrations",
      "url": "https://www.nist.gov/pml/sensor-science/thermodynamic-metrology/industrial-thermometer-calibrations"
    }
  ]
}