# Oversights in the Respective Theorems of von Neumann and Bell are Homologous

## TL;DR

This paper reveals that both von Neumann's and Bell's theorems rely on an unjustified assumption of additivity of expectation values, and correcting this assumption alters Bell's bounds, challenging the conclusion that experiments rule out local realism.

## Contribution

The paper identifies a common oversight in von Neumann and Bell theorems regarding expectation value additivity, proposing a correction that modifies Bell's bounds and implications.

## Key findings

- Bell's bounds are revised to ±2√2 when the oversight is corrected.
- Bell-test experiments do not necessarily rule out local realism, but rather the additivity assumption.
- Both theorems share a homologous oversight in their assumptions.

## Abstract

We show that the respective oversights in the von Neumann's general theorem against all hidden variable theories and Bell's theorem against their local-realistic counterparts are homologous. Both theorems unjustifiably assume the additivity of expectation values within hidden variable theories to derive their respective conclusions. However, for non-commuting observables, the equivalence of a sum of expectation values and the expectation value of the sum of measurement results, although respected within quantum mechanics, need not hold for hidden variable theories, regardless of specific characteristics such as local realism they may respect. Once this oversight is ameliorated from Bell's argument and local realism is implemented correctly, the bounds on the CHSH correlator work out to be ${\pm2\sqrt{2}}$ instead of ${\pm2}$, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by the Bell-test experiments is not local realism but the additivity of expectation values.