GHZ state

In plain terms

Here is the problem with Bell's argument, if you want to call it a problem: it is statistical. No single run of a Bell experiment is impossible on the common-sense picture, just collectively improbable. You need thousands of runs and error bars, and a determined skeptic can always mutter about statistics.

Greenberger, Horne and Zeilinger found a way to remove the error bars. With three particles instead of two, there is a set of measurements where the common-sense picture predicts one definite answer and quantum mechanics predicts the opposite definite answer. Not different averages. Opposite certainties.

One run settles it, in principle. The conflict stops being about probability and becomes arithmetic: multiply out the answers the common-sense picture demands, and the product must be plus one; quantum mechanics says minus one. Something has to give, and experiment says it is the answer sheet.

In CSD

The corpus carries the three-party obstruction alongside the two-party one for a specific defensive reason: they have different logical shapes, and a substrate account has to survive both. You could imagine a mechanism that reproduces statistical Bell violations through some averaging trick but chokes on a deterministic contradiction. Carrying GHZ closes that escape.

The proof quantifies over arbitrary state spaces rather than poking at one model. That is what makes it a constraint on the whole class of product-form hidden-variable accounts, this programme's naive rewrites included -- which is the point.

Mathematically

No product partition of a shared hidden-state space reproduces the GHZ correlations. Proved over an arbitrary measurable state space.

The contradiction is algebraic. The four predicted outcome-sign products cannot be consistently assigned: multiplying the local predictions forces plus one where the quantum state forces minus one. No inequality is involved and no bound is approached -- the two predictions simply differ, which is why a single ideal run distinguishes them.

The name

Daniel Greenberger of City College New York, Michael Horne (1943-2019) of Stonehill College -- who as Shimony's graduate student at Boston University had already co- authored the CHSH inequality -- and Anton Zeilinger of Vienna and Innsbruck introduced the argument in 1989.

The publication story is odd: the result appeared in the proceedings of a conference at Erice, not a journal, and spread mostly by word of mouth until Mermin wrote it up and made it famous. Zeilinger shared the 2022 Nobel Prize for entanglement experiments; Horne had died three years before.