Bell Inequality Violation

1. Setting

Bell violation is a stress test for any claimed substrate theory. RS must produce non-classical correlations without enabling superluminal signaling. The Bell page therefore sits at the intersection of recognition nonlocality and ledger conservation. The relevant output is the Tsirelson bound: the CHSH parameter can exceed 2 but cannot exceed 2 sqrt(2).

2. Equations

(E1)

$$ S_{\mathrm{CHSH}}\le 2\sqrt{2} $$

Tsirelson bound.

(E2)

$$ S_{\mathrm{classical}}\le 2 $$

Classical local hidden-variable bound.

3. Prediction or structural target

• Max CHSH value: predicted 2 sqrt(2) (dimensionless); empirical Observed violations above 2 and below 2 sqrt(2). Source: Aspect 1982; Hensen 2015

Experiments violate the classical bound but stay below Tsirelson, matching standard quantum mechanics and the RS no-signaling constraint.

4. Formal anchor

The primary anchor is Quantum.BellInequality..

5. What is inside the Lean module

Key theorems:

• quantum_correlation_bounded
• perfect_anticorrelation
• classical_chsh_bound
• tsirelson_bound_value
• cos_three_pi_div_four
• optimal_chsh_value
• quantum_violation
• bell_from_shared_ledger
• no_signaling
• nobel_prize_2022
• max_entanglement_entropy
• entanglement_monogamy

Key definitions:

• Outcome
• BellPair
• singlet
• quantumCorrelation
• chshCombination
• tsirelsonBound
• optimalAngles
• optimalCHSH

6. Derivation chain

• Quantum.BellInequality - Bell inequality

7. Falsifier

A loophole-free experiment producing a CHSH value above 2 sqrt(2), or allowing controllable superluminal signaling from entanglement, refutes the RS quantum-recognition account.

8. Where this derivation stops

Below this page the chain reduces to the RS forcing sequence: J-cost uniqueness, phi forcing, the eight-tick cycle, and the D=3 recognition substrate. If any upstream theorem changes, this page must be versioned rather than patched silently. The published URL is stable, but the version field is the contract.

9. Reading note

The minimal way to audit this page is to open the first Lean anchor and then walk the supporting declarations listed above. If the primary theorem is a module-level anchor, the key theorems section names the internal declarations that carry the mathematical load. This keeps the public derivation readable without severing it from the proof object.

10. Audit path

To audit bell-inequality-violation, start with the primary Lean anchor Quantum.BellInequality. Then inspect the theorem names listed in the module-content section. The page is intentionally built so the public explanation is not a substitute for the proof object; it is a map into it. The mathematical dependency is the same in every case: reciprocal cost fixes J, J fixes the phi-ladder, the eight-tick cycle fixes the recognition clock, and the domain theorem listed above supplies the last step. If that last step is empirical, the falsifier section names what observation would break it. If that last step is formal, a Lean-checkable counterexample is the relevant failure mode.

11. Why this belongs in the derivations corpus

The corpus is organized around load-bearing consequences, not around file names. This entry is included because Quantum.BellInequality contributes a reusable theorem or definitional bridge that other pages can cite. Keeping the page public gives readers a stable URL, a JSON record, and a direct path into the Lean theorem page. If the entry becomes redundant with a stronger derivation later, the current slug should be retired rather than silently rewritten; the replacement should absorb its anchors and preserve the audit history.