Contextuality

In plain terms

It would be natural to believe that a measurement reveals a property the system already had, and that the property doesn't care what else happened to be measured alongside it. Call this the answer-sheet picture, in which every question has a pre- written answer and the companion questions are irrelevant.

Quantum mechanics refutes the picture outright. Two measurements are called compatible when they can be made together without either disturbing the other's result, and there are collections of questions for which no answer sheet exists, not hidden or hard to find but provably nonexistent, once you demand that each answer stay the same across the different sets of compatible companions it could appear with. And this isn't a matter of disturbance, as the companion measurements are compatible and needn't touch the system at all.

The conclusion has a name, which is that outcomes are contextual. What you get depends on the company the question keeps.

In CSD

CSD doesn't grudgingly accommodate contextuality, as it's built from it, and the distinction matters because contextuality is usually presented as a tax that hidden- variable theories reluctantly pay. Here it's the architecture.

The mechanism is this. The substrate, the underlying space of definite states, holds a definite microstate, and a measurement context fixes which partition of the epistemic space the outcome is read against, that is, which way the space of quantum states is divided into outcome regions. Change the context and you change the partition, so the same microstate can honestly answer differently, and nothing is hidden, nothing is disturbed, and no answer sheet is ever demanded.

The corpus proves both halves of the story. The obstruction is that no single global assignment of answers reproduces the correlations of the singlet, the entangled pair of particles used in the standard tests, and the existence is that an explicit contextual model does reproduce them. Impossibility plus a working alternative is a two-sided claim, and either half alone would be much weaker.

Mathematically

Over one shared measurable state space, no jointly-defined outcome assignment for the four CHSH settings reproduces the singlet correlations. A separate result rules out a different class, as no setting-local (product) pair of response functions reproduces the singlet at every setting pair. The corpus keeps these apart deliberately, since neither subsumes the other, and it proves the GHZ analogue for three parties.

One conflation is flagged in the corpus itself, which is that its fibre analysis isn't a Kochen-Specker or Gleason consequence. Those constrain non-contextual assignments, which this account never makes. The fibre argument comes from covariance plus non-negativity, a different argument with a different shape, and, unlike the Bell obstructions above, it's a constraint chain rather than a no-go, so whether contextual content can stay on the projective base at N of three and above is open in both directions and not settled.

Related entries

• Bell and CHSHNo pre-written answer sheet for all four measurement settings can match what a singlet pair of particles does, and the corpus proves this as a theorem about every possible answer sheet.
• Kochen-Specker theoremThere's a finite list of directions in space that admits no consistent yes-or-no labelling, so contextuality follows by pure counting, with no experiment, no statistics and no separated particles.
• Singlet contextual modelOne machine-checked theorem settles both halves of the Bell question, as an explicit model in which outcomes depend on the measurement context reproduces the singlet pair's statistics in full, and no assignment that ignores the context can.
• Hidden variablesThe no-go theorems don't kill hidden variables, as they price them, and CSD reads the price tag and pays it.
• Measurement contextA measurement isn't a question put to the quantum state, as it's a piece of apparatus, and the apparatus brings its own way of dividing the world into outcomes.