Predicting spectral interpolation from classical dynamics: an exact Berry–Robnik→Brody locus, its error budget, and its domain

Abstract

The Brody parameter β is routinely fitted to the level spacings of a mixed quantum system and then read as a measure of its classical chaos. A fitted number cannot test the classical picture — it can only describe the spectrum. This paper asks the question the other way round: given only the classical regular fraction ρ_reg, what should β be, and does the spectrum agree? The answer is a parameter-free locus: the Berry–Robnik superposition with the exact GOE gap probability (not the Wigner surmise) projected onto the Brody family in two canonical coordinates, a Kullback–Leibler projection and a log-variance projection. One consequence is immediate and often overlooked — the Brody parameter of a fully chaotic GOE spectrum is not 1 but 0.954 (KL) and 0.969 (log-variance). The locus was tested pointwise, under scoring rules registered before the spectra were computed, on the quantised standard map, a two-harmonic kicked rotor and the Bunimovich mushroom billiard: 16 of 17 mixed cells fall inside the registered tolerance with nothing fitted to the spectra. The paper is as explicit about where the law stops as where it holds. It is claimed only where the Heisenberg time exceeds the classical transport time, and it exhibits its own failures outside that domain — a localized kicked rotor and a mixed limaçon billiard that misses all six of its cells — together with a registered out-of-class control the pipeline flags rather than fits, and a bounded sub-tolerance residual whose mechanism is left open. Every cell carries a full error budget; the near-Poisson coefficients are derived in closed form, one identity machine-checked in Lean 4. Companion Letter: doi:10.5281/zenodo.23435211. Instruments, canonical data and registered documents: doi:10.5281/zenodo.23435240.