EPR and Bell: Mathematical Foundations, Experimental Evidence, and the Limits of Physical Conclusions

This article presents a systematic account of the EPR argument and Bell tests, focusing on rigorously established mathematical results and experimental evidence published up to the cut-off date of October 28, 2026. The assumptions of local factorization and measurement-setting independence are stated explicitly. The CHSH bound of 2 is proved, and the quantum bound of $2√2$ is derived using operator methods and semidefinite certificates. The joint outcome probabilities for the singlet state are derived, and the no-signalling theorem is proved for completely positive, trace-preserving local operations. Separability is shown to imply Bell locality, and entanglement is carefully distinguished from the violation of a particular Bell inequality. For noisy two-qubit states, spin measurements are explicitly distinguished from binary strategies that allow constant outcomes. The account also includes explicit constructions of local distributions, Fine’s theorem, and self-testing bounds. Experimental evidence is examined together with the statistical assumptions underlying its analysis; historical interpretations and unconfirmed proposals are clearly identified as such. The 1930 photon box is also analysed, with a distinction drawn between a semiclassical reconstruction of the apparatus and an exact relation between energy and time evolution. These results do not, in themselves, constitute particle detections, enable superluminal communication, or single out a unique ontology.