The Certification Budget of an Electro-Optical Observation Chain: Diffraction, Sampling, and Noise Limits on Size, Distance, Velocity, and Acceleration Claims

An observation report can state only what its instrument chain can certify. This paper derives, from first principles, the certification budget of a generic electro-optical observation chain: the maximum claim about an object's size, distance, velocity, and acceleration that a given instrument can support. The budget combines the diffraction limit set by aperture and wavelength, the angular sampling set by pixel pitch and focal length, photon and detector noise, platform jitter, exposure-time motion blur, and frame-rate sampling. The central result is an identifiability statement: a single bearing-only sensor measures angles and angle rates, and linear size, transverse velocity, and transverse acceleration are each the product of a measured angle quantity and the range, so without an independent range measurement no unique linear claim is certified, however high the angular signal-to-noise ratio may be. A fully worked numerical example (100 mm aperture, 550 nm wavelength, 1,000 mm focal length, 3.76 micrometre pixel pitch, 30 frames per second, 10 ms exposure) tests a reported-style claim of a 3.00 m object at 10 km travelling at 722 m/s (2,600 km/h) and accelerating at 50.0 g: the angular measurements pass well above the noise floor, but the linear claims fail certification because no range was measured, and the size claim fails independently on motion-blur grounds. Adding a hypothetical independent range measurement with a standard deviation of 25 m at 10 km closes the same budget and the same claims then pass. The paper closes with the report classes that most commonly exceed their budget and a minimum reporting standard.