Bell's theorem, measured

Can the correlations between two entangled particles be explained by any theory in which each particle carries predetermined answers fixed at emission (a 'local hidden variable') — or does nature refuse to be locally real?

Units: dimensionless (CHSH correlator S = 2√2, Tsirelson's bound)

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The finding

Bell's theorem, measured: entangled pairs reach S = 2.828 ≈ 2√2 (Tsirelson's bound), violating the local-realist limit S ≤ 2 on every seed — no local hidden variable can reproduce quantum mechanics

Method

Emit two spin-½ particles in the singlet Bell state |ψ⁻⟩ = (|↑↓⟩ − |↓↑⟩)/√2 toward two distant analyzers. Alice measures spin along an axis at angle a, Bob along b, each getting ±1. Quantum mechanics gives the joint outcome probabilities P(++)=P(−−)=½sin²((a−b)/2), P(+−)=P(−+)=½cos²((a−b)/2), so the correlation is E(a,b)=⟨AB⟩=−cos(a−b). The lab Monte-Carlo-samples the ±1 outcomes from this joint distribution (A drawn fair, B from the conditional — the formula for S never appears in the draw), estimates the four correlators from the counts, and forms the CHSH combination S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′) at the optimal angles a=0°, a′=90°, b=45°, b′=135°, using N = 120000 pairs per setting swept across 24 seeds. Two controls: an explicit local-hidden-variable model (shared emission angle λ ~ U(0,2π), deterministic local responses A=sign(cos(a−λ)), B=sign(cos(b−λ))) scanned over all analyzer settings; and a Werner state of tunable visibility V (singlet mixed with white noise).

The law it recovers

E(a,b) = −cos(a−b); S = E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′); |S| ≤ 2 for local hidden variables (Bell), |S| ≤ 2√2 for quantum mechanics (Tsirelson)

Measurements, controls & cross-checks

Recovered uncertainty

0.0039

Recovered abs offset

0.0001

Worst seed abs error

0.0086

Classical bound

2

Min violation margin

0.8198

Sampler validity

Measured E vs minus cos rms

0.0027

Note

the Monte-Carlo sampler independently reproduces quantum mechanics: the measured correlation E(Δ) matches −cos(Δ) to 0.0027 RMS across a 37-point scan of the relative analyzer angle. So S is pinned to QM, not to any tuning of the CHSH result.

Control local hidden variable

Name

Shared emission angle λ, deterministic local response A=sign(cos(a−λ)), B=sign(cos(b−λ))

Max S over all settings

2.0135

Note

the explicit LOCAL-REALIST model, Monte-Carlo-sampled and scanned over all analyzer settings, saturates at S = 2 (its triangular correlation E=1−2Δ/π gives S≡2 for the optimal-family angles) but can NEVER climb to 2√2. This is the falsification: local realism is capped at 2, quantum mechanics reaches 2.828.

Control werner

Law

S(V) = 2√2·V

Fitted slope

2.8308

Crossing visibility

0.7065

Threshold expected

0.70710678

Note

the CHSH value scales linearly with the source's visibility V, so the Bell violation switches off CONTINUOUSLY: the fitted slope is 2√2 and S(V) crosses the classical bound S=2 exactly at V=1/√2≈0.707. Below that visibility the correlations admit a local-hidden-variable description; above it they cannot. A perturbation on the source purity that moves the answer exactly as predicted — the bound is not hardcoded.

Tsirelson ceiling

Max S over quantum settings

2.8282

Note

scanning the quantum analyzer angles, the measured max |S| never exceeds 2√2: 2√2 is a genuine ceiling (Tsirelson's bound), not a coincidence of one special setting.

What it reduces to

Bell's theorem (J. S. Bell 1964) in its CHSH form (Clauser–Horne–Shimony–Holt 1969), with Tsirelson's quantum ceiling (1980) — a textbook, experimentally-confirmed result (Aspect 1982; loophole-free Hensen et al. 2015; Nobel Prize in Physics 2022 to Aspect, Clauser & Zeilinger). It VALIDATES, not derives: the lab assumes the quantum joint distribution for the singlet state (from which E(a,b)=−cos(a−b) follows) and shows the emergent, measurable CHSH value S = 2.828 ≈ 2√2 violates the local-realist bound |S| ≤ 2 on every one of 24 seeds, with 2√2 recovered from sampled ±1 counts to ±0.01 without ever being plugged in. It does NOT derive the Born rule or −cos(a−b) from a deeper principle (that is quantum mechanics itself); it demonstrates that those correlations are incompatible with ANY local-hidden-variable theory. Two controls make the falsification airtight: an explicit local model saturates at exactly 2 and cannot pass it, and a Werner state's visibility scan crosses the bound continuously at V=1/√2. This is the lab's FIRST world about quantum entanglement and nonlocality — distinct from the early-quantum arc (?world=blackbody … ?world=sterngerlach), which quantizes single-particle observables (energy, charge, spin); here the subject is the correlation between two particles, and the result is that the world is not locally real.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- bell (scripts/bell-derisk.mjs)

Oracle

scripts/oracles/bell.reference.json

Sources

J. S. Bell, 'On the Einstein Podolsky Rosen paradox', Physics 1, 195–200 (1964). J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, Phys. Rev. Lett. 23, 880 (1969) — the CHSH inequality |S| ≤ 2. B. S. Cirel'son (Tsirelson), Lett. Math. Phys. 4, 93 (1980) — the quantum bound 2√2. A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. 49, 1804 (1982); B. Hensen et al., Nature 526, 682 (2015) — loophole-free test. Nobel Prize in Physics 2022. R. F. Werner, Phys. Rev. A 40, 4277 (1989) — Werner states, threshold V = 1/√2.