Born volume ratio

In plain terms

Here is the problem. Showing "probability equals volume" for one contrived example proves little. The claim earns its keep only at full strength: every dimension, every state, every outcome coordinate, no fine print.

This theorem is the full-strength version. Take a quantum state in any finite dimension. Its Born weight for outcome i -- the number the textbook computes as a squared amplitude -- equals the Fubini-Study volume of an explicitly described region, divided by the total. Not approximately, not generically: exactly, and for states with vanishing amplitudes too.

The regions are barycentric cells over the probability simplex, and nobody carved them to fit. The sizes come out of the geometry.

In CSD

This is the derived side of the carving line, at general N: the theorem that turned the volume story from realisation into derivation. The route runs through the moment map and the Duistermaat-Heckman law -- the pushforward of the symmetric measure to the flat simplex law -- so the volume of a cell is an honest simplex integral, computed once and equal to the Born weight.

Downstream, the frequency companion turns it empirical: i.i.d. draws from the Fubini-Study measure give frequencies converging jointly to the full Born vector.

Mathematically

fs_born_volume_ratio_N and its apex variant: the Born weight of each of the N coordinates equals the Fubini-Study volume ratio of the corresponding barycentric region, unconditional in the preparation. The proof composes the Gaussian representation of the Fubini-Study measure, the product-index bridge to exponential laws, and the joint Dirichlet pushforward; the earlier positivity hypothesis was retired by a per-cell dichotomy on closed simplex faces.