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 sound refuted from the start. The summary is too coarse, and the coarseness is exactly where the default machine-generated answer goes wrong.

What PBR actually rules out is a specific class of models: those where two distinct pure states can be knowledge about overlapping sets of underlying world-states, plus an independence assumption about preparing systems side by side. The theorem shows such overlap contradicts quantum predictions. It says nothing against models where 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 exact pure preparations correspond to ontic measures with no overlap at all. So CSD does not evade PBR and PBR is not inapplicable to it: CSD lands on the side of the disjointness conclusion that PBR argues for. The state still functions epistemically -- it summarises what a preparation pins down, and it does not determine the microstate -- but that is a different sense of the word, which is the distinction the whole question turns on.

The longer answer

Three claims have to be kept apart here, and running them together is what generated the old answer to this question.

First, CSD epistemicity: the projective state is a many-to-one, incomplete coordinate on the arena. Second, the Harrigan-Spekkens classification that PBR is stated in: an interface is psi-ontic when distinct exact pure-state preparations have mutually singular ontic measures. On that classification CSD's exact sharp interface is psi-ONTIC, and since 2026-08-25 that is a machine-checked theorem -- sharp_preparations_mutuallySingular, with the concrete witness epistemicMeasure_mutuallySingular. Third, finite-resolution overlap: positive-volume REGION preparations around distinct states provably do overlap. All three are true, of different objects.

What changed. This entry previously said the disjointness held "by construction" and that "the PBR premises are simply not met", and it offered region-preparation overlap as a second way CSD sits outside PBR. Both framings are withdrawn. Disjointness is a theorem, not a construction; and region overlap concerns a different preparation class, so it neither establishes nor softens the exact-state classification.

The honest boundary. PBR also assumes preparation independence, a compositional premise about systems prepared side by side. The corpus neither proves nor refutes it, and -- this matters -- global non-factorisation of the composite ontology does NOT settle it. Reading the Segre-layer geometry as a PBR contradiction was a superseded interpretation, recorded as such in specs/c2-support-plan.md.

What is actually proved

The exact half is now proved, not definitional. epistemicMeasure_projectiveLaw: 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: ANY two ontic measures whose projective laws are Diracs at distinct points are mutually singular -- no Preparation structure, no region, no finiteness assumed. epistemicMeasure_ mutuallySingular is the concrete corollary, routed through the general theorem. pbr_sharp_preparation_capstone bundles both for citation.

The finite-resolution half is also proved, about different objects: kMuL_fibre_null (a single exact projective fibre is Liouville-null, so an exact sharp preparation is not 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.

Not proved, either way: PBR preparation independence.