There is exactly one way to measure the space of quantum states without secretly favouring a direction. The Born rule is what that measure says.
fubiniStudyMeasure_uniqueEvery state a quantum system can be in is a point in a curved space, much as every place on Earth is a point on a globe. To ask how likely an outcome is, you need to know how large a region of that space is. But size needs a ruler, and a carelessly chosen ruler would quietly build the answer you wanted into the question. The Fubini-Study measure is the one ruler that treats every direction alike. Because it is the only such ruler, the sizes it reports are not anybody's choice.
CSD does not assume the Born rule. It derives outcome probabilities as ratios of volumes, and a ratio of volumes means nothing until the ruler is fixed. This is the ruler, and the point is that it is forced rather than picked. The unitary group acts transitively on projective state space, and a space with a transitive symmetry admits exactly one invariant probability measure. So the measure falls out of the state space and its symmetry instead of going in as an assumption, which is the whole reason Born weights arrive here as theorems rather than postulates. Helland reaches the same uniqueness from group representation theory over accessible variables. CSD reaches it from geometry.
CP^(N-1) is the compact homogeneous space U(N)/(U(1) x U(N-1)). The Fubini-Study measure is the unique U(N)-invariant Borel probability measure on it: the normalised Riemannian volume of the Fubini-Study metric, equivalently the top exterior power of the associated Kahler form over n factorial. Uniqueness is Haar uniqueness. An invariant measure on a homogeneous space is unique up to scale when the action is transitive, and normalisation fixes the scale. Proved here at general N with no imported axioms.
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