Outcome region

In plain terms

If outcome probabilities are the sizes of regions, you can ask who draws the regions. There's a wrong answer available, and early versions of any volume picture flirt with it, which is to let the prepared state draw its own regions, sized to the probabilities you already knew. That's bookkeeping wearing the costume of an explanation.

The honest requirement is that the measuring device fixes the partition before the state shows up, the way a roulette wheel's pockets exist before the ball drops, and it's the same wheel for every ball. Then the fact that the frequencies match the pocket sizes is a discovery about the dynamics and not a definition.

CSD's outcome regions meet that requirement. They're built from the apparatus's own data, and no preparation appears anywhere in their definition.

In CSD

The outcome regions are the epistemic half of the two-level structure, the half that concerns descriptions rather than what exists. A measurement context fixes a partition of the sector, the space of quantum states, and Paper C demands that the partition depend on the context alone. The corpus discharges this with basins built from a context's rate field, the outcome weights the apparatus assigns, read at the ontic point, so the preparation plays no part, and it then proves that their epistemic weights are exactly the Born weights, the probabilities quantum mechanics assigns.

The dimension story is a genuine discovery. For the qubit, the simplest quantum system, the regions can live on the base of the sector, the projected state space itself, and at higher dimension the natural way of doing so is squeezed very hard. Covariance, the requirement that the construction respect the symmetry, together with positivity, forces a chain of machine-checked conditions on any base density that depends on the preparation alone and only through distance from it, namely disjoint supports and vanishing below a threshold that rises with dimension, with the qubit exempted for the same structural reason in three independent ways. The scope matters, because it's easy to overstate and an earlier version of this page did overstate it, so that's a tight constraint and not a no-go. The closing step needs machinery that doesn't exist yet, so base-only at higher dimension is open in both directions, neither proved impossible nor exhibited, and is parked as such. The fibre, the part of the arena above each state, is where the corpus builds, by a judgement that either answer would leave unchanged. The regions stay epistemic throughout, and what the trajectory selects among them, the record, is ontic.

Mathematically

Paper C's A7 is the requirement that the partition depend on the context M alone. A ContextField assigns each base point a measurable probability vector over outcomes, and the basin for outcome i is the set of arena points whose fibre coordinate falls in the i-th arc of the field's cumulative weights.

Basins are measurable, pairwise disjoint, almost-everywhere exhaustive, and their epistemic probability at preparation psi is the Born weight, the headline globalBasin_born, from a partition that never mentions psi. The higher-dimensional constraint is on radial, preparation-only base densities, where covariance plus positivity force disjoint supports and vanishing below a dimension-dependent threshold.

Related entries

• Born ruleQuantum probabilities aren't a law of nature in CSD, as they're the sizes of regions in the space of states, measured with a ruler nobody was free to choose.
• Typicality volumeAlmost every way the experiment could have started gives the quantum statistics, and that isn't an assumption about probability, as it's a measured volume.
• Preparation regionRepeating the same experiment is a physical claim, and this is what it refers to, a region of the arena and not a single point, that the preparation procedure lands the world in.
• CarvingCarving is the programme's own word for the move it refuses to count as a derivation, which is cutting a region to have exactly the size you wanted to explain.
• Borel setBefore any region can have a probability, someone has to say which regions are allowed to have a size at all, and the Borel sets are the answer everyone uses.