{
  "title": "Correlated averages and synchronised motion correction",
  "status": "Deterministic analytical teaching study; no sensor measurements or accuracy claims",
  "rows": {
    "fusion": 603,
    "timing": 181,
    "motion": 1203
  },
  "parameters": {
    "noise_sd_units": 1,
    "correlation_range": [
      -1,
      1
    ],
    "correlation_step": 0.01,
    "shared_bias_units": [
      0,
      0.5,
      1
    ],
    "clock_rate_error_ppm": 10,
    "clock_duration_hours": 3,
    "constant_velocity_m_s": 2,
    "gravity_m_s2": 9.81,
    "acceleration_amplitude_m_s2": 1,
    "acceleration_frequency_Hz": 1,
    "delays_s": [
      0,
      0.01,
      0.1
    ]
  },
  "equations": {
    "average_variance": "sigma\u00b2*(1+rho)/2",
    "MSE": "average_variance+bias\u00b2",
    "clock_offset": "fractional_rate_error*elapsed_time",
    "position_mismatch": "constant_velocity*clock_offset",
    "specific_force_up": "a(t)+g",
    "gravity_estimate": "g+a(t)-a(t-delay)"
  },
  "checks": {
    "all_pass": true,
    "covariance_expansion": true,
    "shared_bias_MSE_identity": true,
    "delay_trig_identity_max_error_m_s2": 1.887379141862766e-15,
    "zero_delay": true,
    "clock_108ms_at3h": true,
    "zero_clock_offset": true,
    "stationary_and_freefall_convention": true
  },
  "limits": [
    "Only equal-weight estimates of the same scalar in compatible units are averaged. Noise covariance must be physically meaningful; no independence is inferred from sensor count.",
    "MSE includes an assigned deterministic common bias. A variance reduction is not a guarantee of improved real accuracy.",
    "Perfect correlation endpoints are singular ideal covariance cases; no covariance inversion is performed.",
    "Clock example omits initial offset, jitter, network delay, synchronisation correction and packet loss. Constant-velocity mismatch is a separate kinematic illustration.",
    "Vertical specific force is expressed in a known upward inertial axis, with downward gravity. No body-axis orientation error, Earth rotation, vibration, lever arm, drift or gravity field model.",
    "The sinusoid is defined before t=0 so delayed samples need no invented zero padding.",
    "No classifier, unknown-class detection, altitude estimate, physiological validation or physical sensor performance is established."
  ]
}