Interactive Simulation

Same Predictions. Different Physical Explanation. That Is the Point.

What this simulator shows - for every reader

In 1964, John Bell proved that if two particles are completely independent from the moment they separate, the correlations between measurements on them cannot exceed a specific limit - the Bell inequality, or more precisely the CHSH bound of |S| = 2. Experiments by Clauser, Aspect, and Zeilinger (Nobel Prize 2022) showed that real quantum particles violate this bound, reaching up to S = 2√2 ≈ 2.828. This is called Bell violation.

Here is the critical point most explanations miss: This model and standard quantum mechanics make identical predictions about the measurement outcomes. Both predict the cosine correlation E(θ) = cos(θ) for a maximally coherent pair, and both predict Bell violation. They agree on every number. What they disagree on is the physical explanation.

Standard quantum mechanics says: the particles are entangled - measuring one instantly affects the other nonlocally. This model says: the particles were formed from a single shared substrate configuration. Their correlations were built in at formation. No signal passes between them. Bell's assumption of independent local hidden variables never applies, because this shared substrate is not independent.

The contrast table below makes this explicit. The classical column shows what an independent hidden-variable theory predicts - it cannot reach Bell violation. The substrate column shows the substrate prediction - it matches quantum mechanics exactly, for a completely different physical reason.

Vijay Shankar Sharma · ORCID 0009-0001-9622-6121

ρ_s = 5.9×10⁻²⁷ kg/m³ Substrate equilibrium density - the single physical constant from which all results are derived

Detector B angle b22°

Primary angle of detector B. CHSH test also uses b' = b + 45°. For maximum Bell violation set b = a + 22.5°.

Environmental noise ε0.08

Random substrate disturbance during propagation. Reduces correlation. Real experiments minimise this with cryogenic isolation.

Trials N800

Number of paired measurement events for the outcome distribution and table.

Model position: The pair originates from one shared substrate configuration. Correlations exist at formation - not generated at measurement. Bell's independence assumption never applies. Same outcomes as QM. Different physical cause.

CHSH test - proper four-angle computation

CHSH score |S|

Classical limit = 2.000

E(a,b) correlation

C·I·cos(b−a)·(1−ε)

Physical substrate

EXISTS or ABSENT

Classical max |S|

Independent HV limit

The key contrast - same predictions, different physical explanation

What is "HV" (Hidden Variable)? A hidden variable theory assumes each particle carries a pre-set, self-contained instruction - a "hidden variable" - that determines how it will respond to any measurement. The particles are assumed to be completely independent of each other after separation. Bell proved mathematically that any such theory is bounded: the CHSH score can never exceed 2. Experiments consistently exceed 2. This model is not a hidden variable theory - the shared substrate is a physically extended, globally constrained structure, not an independent local instruction set.

Question	Classical independent HV (particles independent at separation)	Standard QM (nonlocal entanglement)	Substrate Model (shared substrate)

CHSH result interpretation

Press RUN TEST to compute.

Correlation E(θ) vs angle - three predictions compared

CHSH score |S| vs substrate coherence C - classical limit vs substrate/QM prediction

Outcome distribution across N trials

Trial table - first 20 events

Trial	A outcome	B outcome	Substrate C·I	Substrate account	Standard QM label