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this 2011 paper "Born's rule from measurements of classical signals by threshold detectors which are properly calibrated" by Khrennikov investigates the theoretical possibility of Born-like measurement rule/ dynamics for classical systems but appears largely theoretical.

are there any reported classical physics experiments/ measurements that exhibit Born-like measuring rules?

looking for papers etc., esp. in, but not limited to, acoustics.

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The oldest work on this preceeds quantum mechanics by more than 100 years. it was done by Malus in 1809 about experiments with polarized light. See http://arnold-neumaier.at/papers/physpapers.html#CQlightslides

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