### Section IV: Contradiction (Detailed Explanation)

In Section IV, Bell presents the formal proof of why a hidden variable theory cannot reproduce the quantum mechanical predictions for certain measurements on entangled particles. The goal is to show that any local hidden variable theory will give rise to an expectation value for the product of measurement outcomes that differs from the quantum mechanical prediction.

#### 1. **The Key Assumption**

Bell’s approach relies on a crucial assumption called *locality*. This means that the result of a measurement on one particle is independent of what happens to its entangled counterpart, even when they are separated by large distances. The hidden variable \lambda is assumed to contain all the information about the system that determines the outcomes A(a, \lambda) for particle 1 and B(b, \lambda) for particle 2, where a and b are the directions of the measurements, and \lambda is distributed according to a probability density p(\lambda).

Bell writes the expectation value for the product of the outcomes as:

    \[ P(a, b) = \int d\lambda \, p(\lambda) A(a, \lambda) B(b, \lambda) \]

This represents the hidden-variable-based expectation value for the product of two measurements, one on each particle. The expectation value should match the quantum mechanical prediction for the singlet state, which is:

    \[ P_\text{QM}(a, b) = -\cos(\theta) \]

where \theta is the angle between the two measurement directions a and b.

2. **The Difference Between Hidden Variable and Quantum Predictions**

The hidden variable prediction will generally differ from the quantum mechanical prediction. In particular, for small differences in the measurement angles b and c, Bell demonstrates that the hidden variable theory cannot match the quantum mechanical result, especially at points where the quantum prediction reaches its extremum (such as -1).

Bell calculates the expectation value under the hidden variable theory and shows that the function P(a, b) cannot reproduce the quantum mechanical correlation exactly. For small differences between the angles b and c, the hidden variable theory predicts:

    \[ P(b, c) \sim |b - c| \]

while quantum mechanics predicts:

    \[ P_\text{QM}(b, c) = -\cos(\theta) \]

which reaches -1 when the two angles are equal.

3. **Quantifying the Difference**

Bell shows that no matter how one configures the hidden variables and the function p(\lambda), the hidden variable expectation value cannot be made arbitrarily close to the quantum mechanical expectation value. In his proof, he quantifies the difference using inequalities involving the correlation function P(a, b). For example, for measurement settings where the quantum prediction gives -1, the hidden variable theory produces a value that is always less than -1, but never reaches it.

The formal proof concludes that the difference between the hidden variable expectation and the quantum prediction cannot be smaller than a certain bound, particularly when the quantum mechanical value is at its extremum. The difference is proportional to the angular separation between measurement directions, and for some configurations, it cannot be reduced below a minimum threshold.

In summary:

– The hidden variable-based expectation value differs from the quantum mechanical prediction due to the locality assumption.
– The hidden variable model cannot reach the extremum values predicted by quantum mechanics (e.g., -1 at certain angles).
– The difference between the two grows with the angle between the measurement settings and cannot be arbitrarily reduced.