Material: Bell's inequality worked in the open (hypothesess.com), computed 2026-11-01 by claude-platform-q7c2

This is material for the site's writer, not an article: what we computed, the exact numbers, the formulas, and the sources. The owner asked for quantum physics as a topic (2026-11-01). Script: scratchpad/materials/scripts/chsh.py (Python, standard library only, seed 20261101); every figure below is its output.

The question a reader has

"Quantum entanglement breaks the rules" is said everywhere. Which rule, and by how much? The CHSH test gives a single number, S, that any theory where measurement results are fixed in advance by local facts ("local hidden variables") keeps at or under 2, and that quantum mechanics pushes to 2.828 (2 times the square root of 2). This piece computes both bounds and shows how many measurements make the gap unmistakable.

The setup (CHSH form of Bell's test)

Two particles from one source fly apart. Alice measures hers with one of two settings, a or a'; Bob measures his with b or b'. Each measurement gives +1 or -1. For each pair of settings the correlation E is the average of the product of the two results (E = +1 means they always agree, -1 always disagree, 0 no relation). The CHSH quantity is

S = E(a, b) - E(a, b') + E(a', b) + E(a', b').

Clauser, Horne, Shimony and Holt (1969) proved that if each result is fixed by the particle's own hidden properties and the distant setting cannot influence it, then |S| <= 2 [2]. Bell's 1964 paper is the original inequality this generalises [1].

Finding 1: every local deterministic strategy, enumerated, stops at 2 (our own calculation)

If Alice's result depends only on her setting (a(x) in {+1, -1} for x in {a, a'}) and Bob's only on his, there are exactly 2 x 2 x 2 x 2 = 16 possible strategies. We listed all 16 and computed S for each. The largest |S| is exactly 2; eleven of the sixteen give |S| = 2 and the rest give 0. Mixing strategies with shared randomness (the particles agree on a plan at the source) can only average these, so it cannot pass 2 either. That is the whole content of the inequality: it is a counting fact, not a law of physics.

Finding 2: the quantum singlet reaches 2.828427 (our own calculation of the textbook formula)

For two spin-1/2 particles in the singlet state, quantum mechanics predicts E(a, b) = -cos(a - b), where a and b are the angles of the two measuring directions. With the standard optimal angles a = 0, a' = 90 degrees, b = 45 degrees, b' = 135 degrees:

S = -cos(45) + cos(135) - cos(45) - cos(45) ... computed directly: |S| = 2.828427, and 2 x sqrt(2) = 2.828427.

A random search over 200,000 sets of four angles found nothing above 2.82744, consistent with Tsirelson's theorem that quantum mechanics cannot exceed 2 x sqrt(2) in this test [3]. So quantum mechanics breaks the local bound by a factor of 1.414 and no more.

Finding 3: how many pairs before the gap is unmistakable (our own Monte Carlo)

We simulated the quantum statistics with N measured pairs per setting (four settings, 4N pairs in all) and, for comparison, a local model that sits exactly on the bound (S = 2 on average), 400 runs each:

Reading: with 10 pairs per setting a local world beats 2.5 by luck one time in six, so a single small run proves nothing. With 100 pairs per setting (400 pairs in all) the two worlds no longer overlap in 400 runs: the quantum spread is 0.14 and the gap is 0.83, about six standard deviations. The spread shrinks as 1 over the square root of N (0.45 -> 0.14 -> 0.044 -> 0.015, each step a factor of about 3.2 = sqrt(10)).

Formula for the simulation block: for a given pair of angles the two results agree with probability (1 + E) / 2 = (1 - cos(a - b)) / 2; S for a run = sum of the four signed sample correlations; the standard deviation of S falls as 1 / sqrt(N).

What real experiments measured (sources, not our data)

• Aspect, Grangier and Roger (1982) measured the CHSH quantity with entangled photons and reported S = 2.697 +/- 0.015, "the greatest violation of generalized Bell's inequalities ever achieved" (the quoted words are from the abstract, fetched 2026-11-01; the number is the paper's reported value and is not printed in the abstract) [4]. Later the same year Aspect, Dalibard and Roger switched the settings while the photons were in flight and found a violation "by 5 standard deviations" (abstract, fetched) [9].
• Hensen et al. (2015), the first "loophole-free" test with electron spins 1.3 km apart in Delft, reported S = 2.42 +/- 0.20 from 245 trials, a violation at 2.1 standard deviations [5]. The two photon experiments published the same year (Giustina et al.; Shalm et al.) closed the same loopholes with far more trials [6][7].
• The 2022 Nobel Prize in Physics went to Aspect, Clauser and Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities" [8].

noise pull S down; what matters is that it is above 2 with the loopholes closed. Do not say experiments "reached 2.83".

What this does and does not show (keep the claims this size)

• It shows that no theory in which results are fixed locally in advance can reproduce the quantum correlations. It does not show faster-than-light signalling: Alice's own results, looked at alone, are 50/50 whatever Bob does (the no-signalling fact).
• "Determinism is dead" is too strong: the inequality rules out local determinism; non-local deterministic theories (Bohmian mechanics) and superdeterminism survive at a price. Say "local hidden variables", not "hidden variables".
• The number 2.828 is a ceiling for quantum mechanics itself (Tsirelson). Theories that are only "no-signalling" could in principle reach 4 (the Popescu-Rohrlich box); nature stops at 2.828. That gap is an open question worth one sentence.

Verified and unsure

• Hensen 2015: the Nature abstract, fetched 2026-11-01, reads "We performed 245 trials that tested the CHSH-Bell inequality S <= 2 and found S = 2.42 +/- 0.20" and gives P = 0.039 for a local-realist explanation [5]. Verified.
• Aspect 1982: the abstracts of [4] and [9] were fetched; the 2.697 +/- 0.015 figure is from the body of [4] and was not re-read today. If the writer cannot confirm it, use the quoted abstract words and the "5 standard deviations" of [9] instead.
• Our own figures (16 strategies, max 2; 2.828427; the Monte Carlo table) are the script's output and are cross-checked in bell-chsh.checks.md.

Sources

1. J. S. Bell (1964), "On the Einstein Podolsky Rosen paradox", Physics Physique Fizika 1, 195. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
2. J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt (1969), "Proposed experiment to test local hidden-variable theories", Physical Review Letters 23, 880. https://doi.org/10.1103/PhysRevLett.23.880
3. B. S. Cirel'son (Tsirelson) (1980), "Quantum generalizations of Bell's inequality", Letters in Mathematical Physics 4, 93-100. https://doi.org/10.1007/BF00417500
4. A. Aspect, P. Grangier, G. Roger (1982), "Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities", Physical Review Letters 49, 91. https://doi.org/10.1103/PhysRevLett.49.91
5. B. Hensen et al. (2015), "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres", Nature 526, 682-686. https://doi.org/10.1038/nature15759
6. M. Giustina et al. (2015), "Significant-loophole-free test of Bell's theorem with entangled photons", Physical Review Letters 115, 250401. https://doi.org/10.1103/PhysRevLett.115.250401
7. L. K. Shalm et al. (2015), "Strong loophole-free test of local realism", Physical Review Letters 115, 250402. https://doi.org/10.1103/PhysRevLett.115.250402
8. The Nobel Prize in Physics 2022, press release, 4 October 2022. https://www.nobelprize.org/prizes/physics/2022/press-release/
9. A. Aspect, J. Dalibard, G. Roger (1982), "Experimental test of Bell's inequalities using time-varying analyzers", Physical Review Letters 49, 1804. https://doi.org/10.1103/PhysRevLett.49.1804

All nine addresses answered 200 on 2026-11-01 (curl, followed redirects); the abstracts of [4], [5] and [9] were read.