Fundamental physical and statistical limits (Abbe’s diffraction limit, Cramér-Rao bound, Fisher information) mean that no amount of sophisticated analysis or AI can extract more spatial or parametric information than the underlying measurement actually contains. 

AI models can produce confident numerical outputs even when the experiment is weakly informative, ambiguous or nearly non-identifiable, because confidence scores primarily reflect the model’s internal assumptions rather than the information content of the data. 

Scientific AI systems should therefore incorporate structured, fail-closed abstention mechanisms that refuse to issue a qualified estimate when the measurement geometry, noise level or sensitivity is insufficient, and should report both error and coverage so that limits on the claim are treated as valuable scientific information rather than failure.

AI can produce a precise estimate even when an experiment no longer contains enough information to identify the target. Science should treat abstention as a valid result, not a failure.

In 1873, Ernst Abbe formalized a hard truth about microscopes: optical resolution is constrained by the wavelength of light and the aperture of the instrument. Below that limit, the problem is not that the lens needs a cleverer analyst. The measurement itself does not carry arbitrarily fine spatial information.

Seven decades later, C. R. Rao and Harald Cramér expressed the same idea in statistical language. Under the assumptions of the Cramér-Rao framework, Fisher information places a lower bound on the variance attainable by an unbiased estimator. The floor is set by the measurement model, its sensitivity to the quantity being estimated, and the noise. Better mathematics can use available information more efficiently. It cannot manufacture information that the experiment never recorded.

Nature keeps accounts.

Artificial intelligence is now becoming part of the analytical machinery of science. Learned models solve inverse problems, infer parameters from sparse observations, separate overlapping signals, classify images, and extract structure from data too large for manual inspection. Physics-informed neural networks, for example, have been used for forward and inverse problems in differential equations, while later work has explicitly confronted the identifiability of........

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