Bell’s Theorem: Predicted, Not Accommodated

Bell’s theorem (1964) is a mathematical theorem proving that no theory of local hidden variables can reproduce the statistical correlations predicted by quantum mechanics for entangled states.

The Bell inequalities have been violated experimentally (Aspect, 1982; Hensen et al., 2015). This rules out any interpretation where the particles carry pre-determined values that are fixed locally before measurement.

The axioms do not merely accommodate Bell’s result. They predict it — both the qualitative violation and the quantitative value.

Proposition 2 (Bell Prediction). The axioms structurally forbid local hidden variables. Axiom R forbids pre-existing value assignments. The non-spatial pre-state (AP09) dissolves the locality presupposition.

Both pillars of Bell’s theorem — realism and locality — are removed. The Bell inequalities are violated, and the quantitative violation follows from the Born rule (AP09).

Proof. Bell’s theorem requires two assumptions: (1) realism — that each particle carries definite values for all possible measurements before measurement, and (2) locality — that the outcome at A depends only on the hidden variable and the measurement setting at A, not on the distant setting at B.

This yields the factorisation condition: P(a,b|λ) = P(a|λ) · P(b|λ), where λ is the hidden variable. Bell’s inequality is derived from this factorisation. The axioms remove both assumptions.

First, Axiom R: states do not possess definite values prior to the writing of the record. There are no pre-existing values. The hidden variable λ is empty — there is nothing to condition on.

The factorisation condition is undefined because its left-hand side P(a,b|λ) presupposes a λ that does not exist. Second, AP09: the pre-state has no spatial structure.

The factorisation condition P(a|λ) · P(b|λ) presupposes that A and B are spatially separated systems with independent local descriptions. In the pre-state, spatial separation does not exist. The factorisation condition’s presupposition of locality is dissolved.

Bell’s inequality derivation fails at its first step. The inequalities must be violated. ∎

You have just watched the axioms predict one of the most profound experimental results in the history of physics. Not accommodate it after the fact. Predict it from first principles. Axiom R kills realism.

The non-spatial pre-state kills locality. Both pillars of Bell’s theorem removed. The violation is not surprising — it is structurally inevitable.

6.0.1 — Quantitative verification: CHSH

The axioms commit to the Born rule (AP09) and the complex Hilbert space structure. These yield the singlet-state correlations directly.

For the singlet state |AB⟩ = (1/√2)(|↑↓⟩ − |↓↑⟩), if Alice measures spin along axis â and Bob measures along b̂, the quantum-mechanical correlation is E(â, b̂) = −â · b̂ = −cosθ, where θ is the angle between the measurement axes.

This follows from the Born rule applied to the singlet state.

The CHSH inequality (Clauser, Horne, Shimony, Holt, 1969) states that for any local hidden variable theory, S = |E(â, b̂) − E(â, b̂′)| + |E(â′, b̂) + E(â′, b̂′)| ≤ 2. Choosing the standard optimal settings (â = 0°, â′ = 90°, b̂ = 45°, b̂′ = 135°), the Born-rule correlations are E(â, b̂) = −cos 45° = −1/√2, E(â, b̂′) = −cos 135° = +1/√2, E(â′, b̂) = −cos 45° = −1/√2, E(â′, b̂′) = −cos 45° = −1/√2. Then S = |−1/√2 − 1/√2| + |−1/√2 − 1/√2| = 2/√2 + 2/√2 = 2√2 ≈ 2.83. This exceeds 2. The CHSH inequality is violated.

The axioms do not merely predict that Bell inequalities are violated. They predict the exact magnitude of the violation: S = 2√2, the Tsirelson bound. This is the maximum violation permitted by quantum mechanics.

The prediction follows from the Born rule and the Hilbert space structure, both established in AP09.

Note on hidden variables: Bell’s theorem rules out local hidden variables.

The axioms go further: Axiom R structurally forbids all hidden variables, including non-local ones, because states do not possess definite values prior to the writing of the record.

This is an axiomatic consequence, not an experimental result. The experimental confirmation of the Bell inequalities closes the local case. The non-local case is closed by the axiom.

6.1 — Note on decoherence

The standard physics account of why entanglement appears to “collapse” involves decoherence: the interaction of the quantum system with its environment causes the off-diagonal elements of the density matrix to decay, making the system behave as if it has collapsed into a definite state.

Decoherence explains why macroscopic superpositions are not observed and provides the mechanism by which quantum systems become effectively classical.

The axioms’ account is complementary to decoherence, not in conflict with it. Decoherence describes the process by which the pre-state loses coherence through environmental interaction.

The axioms identify when this process constitutes an actualisation: when the interaction creates an irreversible record (Axiom R). Decoherence is the physical mechanism; actualisation is the ontological event.

The two are compatible: decoherence is what actualisation looks like from within the manifold.