A loophole of all `loophole-free' Bell-type theorems

TL;DR

This paper identifies a fundamental loophole in all existing 'loophole-free' Bell-type theorems, showing that certain assumptions about measurement representations lead to contradictions at a basic level.

Contribution

It reveals a critical flaw in Bell-type theorems related to the mathematical representation of measurement results, challenging their claimed loophole-free status.

Findings

Contradictions arise from mixing different arithmetics in measurement models.

Bell's theorem cannot be proved under certain conditions involving non-identical domains.

A new perspective on the EPR argument highlights limitations in current Bell theorem proofs.

Abstract

Bell's theorem cannot be proved if complementary measurements have to be represented by random variables which cannot be added or multiplied. One such case occurs if their domains are not identical. The case more directly related to the Einstein-Rosen-Podolsky argument occurs if there exists an `element of reality' but nevertheless addition of complementary results is impossible because they are represented by elements from different arithmetics. A naive mixing of arithmetics leads to contradictions at a much more elementary level than the Clauser-Horne-Shimony-Holt inequality.