Tsirelson Bound

The Tsirelson bound constrains correlations in Bell-type experiments:

This bound is a mathematical consequence of Hilbert space structure—and in Logic Realism Theory, Hilbert space structure is itself derived from Determinate Identity.

The Correlation Hierarchy

Bell-type experiments measure correlations between distant measurements. The bounds form a hierarchy:

Nature obeys the quantum bound. But why $2\sqrt{2}$ and not $4$?

Derivation from Hilbert Space

The Tsirelson bound follows from the inner product structure of Hilbert space. For observables A, B with eigenvalues $\pm 1$:

The Cauchy-Schwarz inequality on the Hilbert space inner product yields:

This is a mathematical fact about Hilbert space—not an empirical observation that could have been otherwise.

The LRT Derivation Chain

The complete derivation chain is:

L₃ (specifically Determinate Identity)
    ↓ [Theorem: Anti-holism]
Local Tomography
    ↓ [Masanes-Müller reconstruction]
Complex Hilbert Space
    ↓ [Cauchy-Schwarz inequality]
Tsirelson Bound: S ≤ 2√2

Each step is either:

• A theorem from Id (our derivations)
• An established mathematical result (reconstruction theorems, Cauchy-Schwarz)

What PR-Boxes Would Violate

Popescu-Rohrlich (PR) boxes are hypothetical systems saturating the algebraic maximum $S = 4$. They would satisfy no-signaling but violate the Tsirelson bound.

In LRT terms, PR-boxes would violate local tomography:

• Two PR-box states could have identical local statistics
• But differ in their global correlation structure
• This means parts would not determine wholes

PR-boxes exhibit exactly the global holism that Determinate Identity rules out. If the whole has properties not grounded in parts, identity becomes extrinsic—violating Id.

What This Means

The Tsirelson bound is not a separate postulate or unexplained fact. It is a downstream consequence of Determinate Identity applied to composite systems.

Correlations cannot exceed $2\sqrt{2}$ because:

1. Id forces local tomography (parts ground wholes)
2. Local tomography forces complex Hilbert space
3. Hilbert space structure mathematically caps correlations at $2\sqrt{2}$

Prediction: Any observation of correlations exceeding the Tsirelson bound would falsify the $L_3$ constraint framework.

Comparison with Information Causality

Pawlowski et al. (2009) derive the Tsirelson bound from Information Causality—the principle that Bob’s accessible information cannot exceed the bits physically sent to him.

These approaches are complementary:

• Information Causality shows what is ruled out (super-quantum correlations)
• LRT shows why nature has the arena it does (complex Hilbert space satisfies Id)

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