The claim that quantum mechanics holds even for infinitesimally small probabilities is based on the empirical success and experimental verification of quantum mechanics over the past century. Quantum mechanics has been extensively tested and has consistently provided accurate predictions for a wide range of phenomena, from the behavior of subatomic particles to the properties of materials and the interactions of light.
The probabilistic nature of quantum mechanics arises from the fundamental principles of the theory, such as wave-particle duality, superposition, and the uncertainty principle. These principles have been experimentally confirmed in numerous experiments, including famous ones like the double-slit experiment and Bell's theorem tests.
Quantum mechanics has been applied to a vast array of practical technologies, including transistors, lasers, and atomic clocks, with remarkable precision and success. It is the foundation of modern physics and has been validated by countless experiments and observations.
Regarding the claim's testability, it is important to note that science operates by making predictions based on theories and then testing those predictions through experiments and observations. Quantum mechanics makes predictions about the behavior of particles and systems at all scales, including infinitesimally small probabilities.
While it may not be possible to perform direct experiments in certain extreme regimes, the consistency of quantum mechanics with a wide range of empirical observations provides strong evidence for its validity across various scales. Furthermore, the mathematical framework of quantum mechanics allows for rigorous calculations and predictions, which have been confirmed experimentally in many cases.
It is worth mentioning that quantum mechanics is still an active area of research, and physicists continue to explore its foundations and potential limits. Alternative theories and interpretations have been proposed, but to date, quantum mechanics remains the most successful and well-tested framework for understanding the behavior of particles and systems at the microscopic scale.