Does the PBR theorem rule out CSD?

The short answer

The Pusey-Barrett-Rudolph theorem is often summarised as "the quantum state is real", which makes any programme calling the state epistemic, a summary of knowledge rather than a complete description of the world, sound refuted from the start. The summary is too coarse, and the coarseness is exactly where the quick answer goes wrong.

What PBR actually rules out is a specific class of models. In these models two distinct pure quantum states can both be knowledge about the same underlying world- state, so the sets of world-states compatible with each overlap. Add an independence assumption about preparing systems side by side, and the theorem shows that such overlap contradicts quantum predictions, and it says nothing against models in which distinct states describe disjoint sets of world-states.

CSD is in the second camp, and the corpus proves it rather than arranging it. Two distinct exactly prepared pure states correspond to distributions over the underlying states with no overlap at all, so CSD doesn't evade PBR and PBR isn't inapplicable to it, as CSD lands on the side of the disjointness conclusion PBR argues for. The state still functions epistemically, as it summarises what a preparation fixes and doesn't determine the underlying microstate, but that's a different sense of the word, and the distinction between the two senses is what the whole question turns on.

The longer answer

Three claims have to be kept apart, and running them together is what produces the wrong answer to this question.

The first is CSD's epistemicity, that the quantum state is a many-to-one, incomplete coordinate on the arena, the underlying state space. The second is the classification, due to Harrigan and Spekkens, in which PBR is stated, where an account is called psi-ontic when distinct exact pure-state preparations have mutually singular ontic measures, meaning the distributions over underlying states they induce live on disjoint sets. On that classification CSD's exact sharp interface, its account of preparations that fix a pure state exactly, is psi-ontic, and that's a machine-checked theorem. The third is finite-resolution overlap, as preparations that fix the state only to within a region of positive volume around it, rather than exactly, provably do overlap for distinct states whose regions overlap. All three are true, and they're about different objects.

Two tempting framings are wrong, and earlier versions of this answer used both, so both are withdrawn. Disjointness is a theorem and not something that holds by construction, and "the PBR premises are simply not met" isn't the right description. And region-preparation overlap isn't a second way in which CSD sits outside PBR, as it concerns a different class of preparations, so it neither establishes nor softens the classification of exact states.

There's an honest boundary. PBR also assumes preparation independence, a compositional premise about systems prepared side by side, and the corpus neither proves nor refutes it, and the fact that the composite ontology doesn't factorise globally doesn't settle it either way. Reading the geometry of composite systems, the Segre layer, as a contradiction with PBR was a superseded interpretation and is recorded as such.

What is actually proved

The exact half is proved and not definitional. epistemicMeasure_projectiveLaw says the exact sharp preparation delta_p (x) Haar has projective pushforward exactly delta_p, so the corpus's own witness demonstrably meets the classification's hypothesis. sharp_preparations_mutuallySingular says any two ontic measures whose projective laws are Diracs at distinct points are mutually singular, with no Preparation structure, no region and no finiteness assumed. epistemicMeasure_mutuallySingular is the concrete corollary, routed through the general theorem, and pbr_sharp_preparation_capstone bundles both for citation. The theorem landed on 2026-08-25, and before that the entry described the disjointness as holding by construction.

The finite-resolution half is also proved, about different objects. The theorem kMuL_fibre_null shows that a single exact projective fibre is Liouville-null, so an exact sharp preparation isn't obtainable by conditioning on a positive-volume region. The class separation is stated at the level of probability laws by exact_sharp_ne_region_conditional (no positive-volume region-conditioned law equals an exact sharp law, however the region is chosen), with no_region_preparation_exact_fibre the weaker set-level companion, conditional_not_mutuallySingular and kahler_preparations_overlap (region preparations on overlapping neighbourhoods have non-separable conditional laws), and the rho_ep density theorems.

The superseded reading of the Segre-layer geometry as a PBR contradiction is recorded in specs/c2-support-plan.md, and PBR preparation independence isn't proved either way.

Related entries

• Psi-epistemicThe quantum state is what you know and not what exists, and the hard part was always saying what it's knowledge of, so CSD's whole architecture is an answer to that.
• Is the wavefunction real in CSD?The wavefunction is a genuine but incomplete coordinate of a real underlying world, so it can be read off the world's actual configuration, yet it leaves out the part of that configuration which decides what a measurement will show.
• Does the excess-baggage theorem count against CSD?No. The excess-baggage theorems say any model with real underlying states beneath quantum mechanics needs a continuum of them, and CSD's constraint surface is a continuum from the outset, so the requirement is met before it's raised.
• Ontic vs epistemicWhat exists is one question and what a measurement context lets you say is another, and most quantum paradoxes are category errors between the two.
• Epistemic measureCondition on what a preparation actually fixes, and the ignorance left over is a measure, and that measure is a theorem rather than a stipulation, as the arena's own volume disintegrates into exactly it.