# Compatible quantities and compatible times

Run `python generate.py` with matplotlib to regenerate 1,987 analytical rows, metadata, validation and two original PNG/SVG/PDF figures. No random measurements, sensor performance or classifier accuracy is claimed.

Two estimates of the same scalar have equal noise standard deviation 1 unit and correlation rho from −1 to +1 in steps of 0.01. Their equally weighted average has variance (1+rho)/2. A deterministic shared bias b of 0, 0.5 or 1 unit adds b² to mean squared error, giving 603 rows. Independence gives variance 0.5, perfect positive correlation variance 1, and perfect anticorrelation variance 0. Shared bias persists in every case. The zero-variance endpoint is an ideal algebraic limit, not real certainty; no singular covariance inverse is attempted.

A separate clock example uses constant fractional rate error 10 parts per million, zero starting offset, and no resynchronisation over three hours. Its 181 minute-spaced rows reach 0.108 second (108 ms) offset. At assigned constant speed 2 m/s this corresponds to 0.216 m position mismatch. Rate error, offset, network delay and random jitter are different quantities; the last two are not modeled.

For vertical motion define upward positive and gravity downward with magnitude 9.81 m/s². Ideal specific force is f_z=a_z+g. A stationary supported sensor reads +g and ideal freefall reads zero. With assigned a(t)=sin(2*pi*t) m/s², subtracting delayed acceleration produces g_est=9.81+a(t)−a(t−delta). Delays 0, 0.01 and 0.1 seconds over 0–2 seconds in 0.005 second steps give 1203 rows. The sinusoid exists before time zero, avoiding artificial initial padding. This scenario is separate from the accumulated clock drift example.

Checks independently expand the covariance of the average, verify nonnegative variances and shared-bias MSE, confirm the clock/time units and endpoint, and use the trigonometric identity a(t)−a(t−delta)=2*cos(2*pi*t−pi*delta)*sin(pi*delta). Zero delay cancels the motion exactly. The resulting residual is a timing artifact in this ideal model, not a real gravity anomaly or altitude measurement.

Excluded effects include orientation errors, calibration drift, vibrations, Earth rotation, lever arms, gravity-field variation, packet loss and unknown-class recognition. Reducing the variance of one compatible scalar is not a validation of fusing unlike sensor modalities. See the chapter source review for NIST covariance and IEEE1588 context; no measured uncertainty or timing guarantee is borrowed from those references.