Entanglement Toy — Full Derivation — Supreme Polarity Theory

This page is the mathematical companion to /lab/entanglement. The toy lets you adjust two detector angles and watch the CHSH sum S — this page explains why S can reach 2√2 in SPT, why no information actually travels between the nodes, and why every benchmark in the toy passes.

SPT's view of entanglement. Two entangled particles are not 'communicating instantly across space.' They share one phase that lives on a single membrane patch. Measurement collapses that one phase — there is nothing being sent because both nodes were always reading the same value. The 'spooky action at a distance' dissolves into a single shared object.

The state

S in g l e t (B e l l p ai r) : ∣ Ψ^{-} ⟩ = (∣ ↑↓ ⟩ - ∣ ↓↑ ⟩) /\sqrt2

I n S P T : o n es ha r e d p ha se φ_{12}, tw o an c h or ce l l s in 3 D s p a ce . E i t h er an c h or c anb e^{'} q u er i e d^{'} (m e a s u r e d); b o t h q u er i esr e a d t h es am e φ_{12} .

The correlation function

When detectors A and B are oriented at angles θ_A and θ_B, the joint probability of opposite outcomes follows from rotating the singlet axis. The expectation of σ_A·σ_B is:

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

M a x im u m ∣ E ∣ = 1 w h e nan g l es d i f f er b y 0 or π . E = 0 w h e nan g l es d i f f er b y π /2 (u n cor r e l a t e d) .

CHSH inequality and Tsirelson bound

Bell-test experiments use 4 angle pairs (a, b), (a, b'), (a', b), (a', b'):

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

C l a ss i c a l (l oc a l - r e a l i s t) b o u n d : ∣ S ∣ \leq 2 Q u an t u m (T s i r e l so n) b o u n d : ∣ S ∣ \leq 2\sqrt2 \approx 2.828

O pt ima l an g l es : a = 0, a^{'} = π /2, b = π /4, b^{'} = 3 π /4 \to S = 2\sqrt2 e x a c tl y

Why the toy hits 2√2 exactly. With the optimal angles, each of the 4 correlations equals 1/√2 (in absolute value). Their algebraic sum gives 2√2 by direct cosine identity. This is the mathematical maximum — no quantum theory or SPT extension can ever exceed it.

Benchmarks (why each passes)

CHSH violation (Tsirelson bound)

Toy at optimal angles → S = 2√2 ≈ 2.8284. Aspect 1982 measured 2.697 ± 0.015 (slightly less due to detector efficiency). Modern experiments (loophole-free, Hensen 2015) hit ≈ 2.42 with full closure. SPT prediction matches the theoretical max; real experiments approach it.

Bell inequality |S| > 2

PASS as long as |S| > 2 — i.e., the system is genuinely quantum, not local-realist. Always satisfied at optimal angles.

Singlet correlation E = −cos(Δθ)

Mathematical identity from rotating the singlet axis. Verified in every CHSH experiment to <0.1% across all angle settings.

Mathematical soundness

No-signalling (causality) — marginal probability p(σ_A | θ_B) = p(σ_A) regardless of B's choice. Each side sees random outcomes; correlation only emerges when results are compared post-hoc through a classical channel. SPT respects this because the shared phase carries no information itself — only its measurement outcomes do.
Unitarity — ⟨Ψ|Ψ⟩ = 1 preserved. Singlet state has unit norm; any rotation is unitary.
Tsirelson respected — |S| can never exceed 2√2. SPT does not predict 'super-quantum' correlations (e.g. PR-box S = 4), which is consistent with all experimental evidence.

Conclusion

Entanglement in SPT is not weird. The 'spooky' instantaneity vanishes once you realise the two particles are reading the same phase, not communicating with each other. The Tsirelson bound 2√2 is the algebraic maximum of correlations between angle-shifted readings of one variable — exactly what nature gives us.

SymPy verify — download for offline testSYMPY ✓

Download Tsirelson + entanglement SymPy script

CHSH bound |S| = 2 sqrt(2) — SymPy simplifies (|S| - 2 sqrt(2)) to exactly 0.

scripts/spt_chsh_hierarchy.py

Tsirelson 2 sqrt(2) + N hierarchy — |S| - 2 sqrt(2) = 0 EXACT; log10(N) = 140 log10(2) = 42.1442 (delta 0.00 %)

130 LOCDownload

Reproduce in 30 seconds

pip install sympy numpy && python3 scripts/spt_chsh_hierarchy.py

Or quick-verify with AI (Grok / Claude / ChatGPT)

Don't want to install Python? Paste the prompt straight into Grok / Claude / ChatGPT / Gemini — the AI fetches the public script URL below and independently verifies each assertion in ~30 s. Open grok.com or claude.ai , paste, send.

⚠️ AI can be wrong — running the Python above is the only 100% certain check. Full AI guide →

Inputs: Bagua integers + π/√ only — no CODATA, no PDG, no calibration (Tier B). SymPy-verified as exact fractions (not floating-point). See full context at /theory/sympy-breakthrough-2026.