Is the Born rule derived or assumed in CSD?

The short answer

The Born rule is the textbook's recipe for quantum probabilities. A quantum state assigns each possible outcome a number called an amplitude, and the rule says the probability of that outcome is the square of the amplitude's size. In the textbook this is an axiom, postulated and not explained, and in CSD it's a computed quantity. The probability of an outcome is the size of a region of the arena, the underlying state space, measured with the unique measure the space's symmetry allows, and a convergence theorem turns those sizes into the frequencies observed over many runs.

Two things make this more than a restatement. The measure is forced, as its uniqueness is a theorem, so nobody tuned the ruler to get the right sizes. And the regions, in the mature form of the results, aren't carved to fit either, as their sizes fall out of the geometry through the moment map, the function that reads off from each point of the state space the quantities its symmetry conserves.

What isn't claimed is derivation from nothing. The projection from the arena to the space of quantum states is a postulate, and reading probability as typicality, the fraction of the compatible microstates that lead to an outcome across repeated preparations, is a stance the theorems make precise rather than prove. The programme's habit is to say exactly this out loud.

The longer answer

The chain runs in three steps. The Fubini-Study measure, the natural volume on the space of quantum states, is the unique measure invariant under the symmetries of that space, meaning unchanged by every symmetry transformation, and this is the argument of Paper B, proved at general dimension. The Born weight equals a ratio of volumes in that measure, unconditionally and for every coordinate. And empirical frequencies converge to those volumes almost surely under repeated preparation. The historically carved constructions, in which regions were cut to reproduce the right sizes, are retained and labelled as such, using the corpus's own word carving, and the derived route stands independent of them.

The remaining posits are structural and named. Projectability, the sector postulate, says the arena projects onto the space of quantum states. The typicality reading interprets the conditioned measure as probability. The cell law says a measurement context's outcome rates generate its pointer torus, the family of rotations of the state space associated with the apparatus, and this is what ties the outcome regions to the moment map. A separate route to the Born rule of the Gleason type exists on the operational stratum, the layer that describes measurements in the language of laboratory operations, and it's kept distinct so nobody can mistake which layer a citation leans on.

What is actually proved

The route is fsMeasure_unique, fs_born_volume_ratio_N with the apex variant, born_frequency_convergence_N for the joint almost-sure statement, and the moment-map identity momentMap_mk_eq_inner_sq with the Duistermaat-Heckman pushforward behind it.

The volume route is Gleason-free end to end, and the effect-Gleason representation is proved separately on the operational stratum. The remaining posits (projectability, the typicality reading, the cell law) are registered in specs/POSITS.md.

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.
• TypicalityTypicality is probability without randomness, as the world is deterministic, you don't know which state you have, and the odds are the sizes of the possibilities.
• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• If CSD is deterministic, where do probabilities come from?They come from ignorance of the exact starting point, as the dynamics never rolls dice, and the probabilities are the sizes of the sets of microstates a preparation can't tell apart, measured with a ruler nobody was free to choose.
• LF1 main theoremIn a deterministic world with repeated preparations, the frequencies you observe converge to volume ratios with probability one, and the theorem is machine-checked with no randomness anywhere in the dynamics.