Berry–Robnik to Brody Projection: A Universal Incomplete Beta Function and a 𝑃 (0)-Sensitive Diagnostic Separating Repulsion from Projection

What does the Brody parameter actually measure? For fifty years it has gauged "chaoticity" in quantum spectra with no link to the classical dynamics. This paper supplies the dictionary: the Brody MLE of a mixed Berry–Robnik spectrum obeys q ≈ 0.944 · βD2/3 · If(2, 3/4), an invertible law that reads the classical chaotic fraction f straight off a fitted Brody parameter. The exponent a = 2 is derived and directly measured (slope 2.13 ± 0.02), and a new P(0)-sensitive fidelity-susceptibility diagnostic separates genuine level repulsion from the projection defect. The universal shape If(2, 3/4) matches Berry–Robnik simulations (N = 3000, 200 realizations, single fixed protocol) at RMSE 3.8% (GOE), 1.5% (GUE), and 4.8% (GSE); per-class (a, b) calibration drives residuals to ≈1%. The law is validated on the Csórdás billiard (r = 0.9993, RMSE 1.4%), reconstructed across four lemon-billiard geometries, and issued as seven parameter-free mushroom-billiard predictions held as a standing, falsifiable test. As an application, the inverse f = I−1(q/qpure) is invertible by construction, turning any reported Brody parameter into a classical chaotic fraction. Its development into a legacy missing-level diagnostic — and its validation against laboratory spectra (a coupled microwave billiard) and nuclear data (238U and a seven-nuclide resonance panel) — is the subject of the companion paper [10.5281/zenodo.20681798]. Part of the ICP series on the information geometry of chaos. Where the geometric papers build the Brody manifold from the inside — its metric, connections, and symmetries — this paper closes the loop back to measurement: it projects that geometry onto a single observable and inverts it, turning any reported Brody parameter into a classical chaotic fraction. Its coordinate is then taken up as a measurement convention, tested against real laboratory and nuclear spectra, and applied to legacy missing-level diagnostics in the companion paper [10.5281/zenodo.20681798] — the series' experimental and legacy readout. Paper Role DOI The Instability Compression Principle ICP empirical foundation: β → compression scaling across 30 chaotic systems 10.5281/zenodo.18099118 The Compressibility of Chaos (Ordo ab Chao) ICP theoretical derivation: scaling coefficient α from information geometry 10.5281/zenodo.18834609 Variance Excess ε(β) formula; one-point/two-point divide at βc = π 10.5281/zenodo.18650473 Information Geometry of the Brody Distribution Riemannian foundation: exact Fisher metric, deff = 2/β + 1, spectral duality theorem 10.5281/zenodo.18879754 The α-Connection Structure of the Brody Manifold Amari–Chentsov tensor, orbit-universal connection 10.5281/zenodo.19151206 Dual Symmetries of the Brody Statistical Manifold Z₂×Z₂ symmetry group, GOE=GUE orbit-equivalence 10.5281/zenodo.19239285 Berry–Robnik → Brody Projection Physical readout: q ≈ 0.944·βD2/3·If(2,¾); inverts Brody fits to chaotic fraction f; P(0)-sensitive repulsion/projection diagnostic this A symmetry-blind chaoticity coordinate Experimental + legacy readout: cross-class collapse; first experimental projection test; inverts Brody fits to f; seven-nuclide missing-level diagnostic 10.5281/zenodo.20681798