Entanglement reaches Tsirelson boundS = 2√2 — Bell-violated, no-signalling, unitarity preserved

Entanglement — Bell / CHSH

Two DANodes share one phase; measuring one side instantly reflects on the other.

SPT Model

Two DANodes share one membrane patch — their phase φ₁₂ is fixed (locked), not two separate phases. Measurement on either side collapses the SAME phase; the other side reflects this instantly because there was never anything to send.

Singlet state (Bell pair)

∣ Ψ^{-} ⟩ = \frac{1}{2} (∣ ↑↓ ⟩ - ∣ ↓↑ ⟩)

Two nodes locked in opposite-polarity superposition. SPT: one shared phase, two anchor cells.

Quantum correlation

E (θ_{A}, θ_{B}) = - cos (θ_{A} - θ_{B})

Expectation of joint spin measurement when detectors are at angles θ_A, θ_B.

CHSH inequality (classical bound)

∣ S ∣ = ∣ E (a, b) - E (a, b^{'}) + E (a^{'}, b) + E (a^{'}, b^{'}) ∣ \leq 2

Any local-realist theory must obey |S| ≤ 2. Quantum mechanics violates this.

Tsirelson bound (quantum max)

∣ S ∣_{m a x} = 22 \approx 2.828

Reached at a=0, a'=π/2, b=π/4, b'=3π/4. SPT recovers this exactly.

No-signalling (causality)

p_{A} (σ_{A} ∣ θ_{B}) = p_{A} (σ_{A})

Marginal probability on A is independent of B's measurement choice — no FTL communication.

Live Derivation

θ_{A}

= 0.0°

θ_{B}

= 45.0°

E (θ_{A}, θ_{B})

= - 0.7071

- cos (θ_{A} - θ_{B})

S

= - 2.8284

E (a, b) - E (a, b^{'}) + E (a^{'}, b) + E (a^{'}, b^{'})

Bell inequality VIOLATED — quantum non-locality

∣ S ∣/∣ S ∣_{m a x}

= 100.00%

Physics Recovery

CHSH violation (Tsirelson bound)

Aspect 1982; CHSH 1969

PASS

Predicted (SPT)

2.828

Measured

2.828

Δ = 0.0000%

∣ S ∣_{m a x} = 22

Bell-inequality violation (>2)

Bell 1964 (violation = quantum non-locality)

PASS

Predicted (SPT)

2.828

Measured

2.690

Δ = 5.15%

∣ S ∣ > 2

Singlet correlation E = -cos(Δθ)

QM textbook; verified to <0.1% in all Bell tests

PASS

Predicted (SPT)

-0.7071

Measured

-0.7071

Δ = 0.0000%

E = - cos (θ_{A} - θ_{B})

Soundness

Entanglement controls

θ_A — detector A angle0°

θ_B — detector B angle45°

Use CHSH-optimal angles

a=0, a'=π/2, b=π/4, b'=3π/4 → S=2√2