Many-to-one projection

In plain terms

If the world is deterministic and you know the quantum state exactly, you can ask why you can't predict the outcome of a measurement. Something must separate what the state tells you from what the world knows, and in CSD that something is a projection, a map from the full arena of world-states down to the space of quantum states, whose important feature is that it forgets.

Many distinct points of the arena map to the same quantum state, as the map is many- to-one by design. Knowing the quantum state therefore fixes only the visible coordinate of the world's position and says nothing about the rest, and outcomes depend on the rest, so exact knowledge of the state is compatible with genuine ignorance of the outcome, even though nothing in the world is random.

This is the oldest move in statistical physics, relocated. The temperature of a gas is a single number many different arrangements of molecules share, and knowing the temperature doesn't tell you where the molecules are. The novelty in CSD isn't the move itself but what's proved about the projection.

In CSD

The projection from the arena to the projective sector, CSD's name for the space of quantum states, is postulated to be measurable and many-to-one. Measurable means that any region of the state space to which a size can be assigned pulls back to a region of the arena to which a size can be assigned, so volumes upstairs and downstairs can be compared, and the many-to-one character is intended and is stated as such in the source.

Everything epistemic, everything about what an observer can know, happens downstream of the projection, and everything ontic, everything about what's actually the case, happens upstream. The Born weight of a region of the state space is the ontic volume of the set of arena points that project into it, so the information the projection forgets is exactly the information typicality averages over.

What keeps this from being a shell game is an anti-circularity discipline. The projection carries no Born equality, no identity between measures and no unitarity, as those are theorems proved about it downstream and not properties packed into the definition.

Mathematically

ProjectiveSector carries the measurable map pi from Sigma to the projective space with no further structure, and the projective law is the pushforward of an ontic measure along pi. On the Kahler instance pi is the product projection CP^(N-1) x T^2 -> CP^(N-1), with every fibre a full torus.

The pushforward identity, Liouville measure to Fubini-Study measure, is proved, and the guard scripts enforce that the projection's descent equation is genuinely consumed downstream rather than decoratively carried.

Related entries

• FibreThe arena of CSD is larger than the space of quantum states by exactly one compact factor, and from three dimensions upward the natural ways of doing without that factor stop working.
• Constraint surfaceThere's one arena, it's deterministic, and it doesn't grow, and everything else in the programme is bookkeeping about how much of it a measurement context lets you see.
• ProjectabilityProjectability is the one genuine postulate at the interface, that the ontic arena projects onto the space quantum mechanics is written on, and downstream of that symmetry does the choosing.
• 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.
• Projective sectorQuantum mechanics is written on a curved projective space, and in CSD that space isn't fundamental, as it's the sector the arena projects onto, with its geometry forced by symmetry.