A symmetry-blind chaoticity coordinate for spectral statistics: experimental validation and legacy-data reanalysis

A reported Brody parameter is not comparable across symmetry classes. The same fully chaotic dynamics returns q ≈ 0.95 in an orthogonal system, 1.52 in a unitary one, and 2.34 in a symplectic one — three incommensurable numbers for one physical state — and none of them has been tied to the classical chaotic fraction f. This paper turns the Brody parameter into a coordinate that fixes both problems. Building on the factorization of the Berry–Robnik Brody parameter into a symmetry factor and a dynamical factor (companion theory paper, below), we define a symmetry-blind chaoticity coordinate g = q / qpure(βD) ≈ If(2, 3/4) ∈ 0, 1. Dividing any reported Brody parameter by its class-specific chaotic-limit value collapses fully chaotic orthogonal, unitary, and symplectic systems — whose raw parameters span 0.95 to 2.34 — onto the single point g = 1, and renders systems of different symmetry directly comparable for the first time. Because the coordinate is invertible, f = I−1(g) reads the classical chaotic fraction straight off any fitted Brody parameter. Two results give the coordinate physical teeth. First experimental test. Using a coupled superconducting microwave billiard for which both a Berry–Robnik fit (parameter μ) and a plain Brody fit (parameter ω) were performed on the same measured spacing distribution, the parameter-free projection ω = qpure Iμ(2, 3/4) reproduces the independently fitted Brody parameter to 2.9% across three coupling strengths (RMSE 0.011) — the first direct experimental confirmation that the plain Brody parameter of a mixed system is the Berry–Robnik projection. Legacy missing-level diagnostic. A Brody parameter below the fully chaotic value is ambiguous: genuinely intermediate dynamics, or a complete chaotic spectrum with missing levels. The inverse coordinate converts the ambiguity into a number. Applied to the 238U s-wave neutron resonances, it attributes the sub-chaotic Brody value to incompleteness rather than to intermediate dynamics, consistent with independent level-count and Porter–Thomas width estimates of 3–7% missing levels. Extended to a seven-nuclide panel of even-even targets from a single evaluated library, the concordance pattern across three independent observables — spacings, level counts, and neutron widths — separates complete (232Th), mildly incomplete (238U), and width-depleted (166,168Er, 152Sm) resonance sequences, and a same-window census against the primary measurements identifies the mechanism as weak-resonance culling at the compilation stage. The width route is blind-validated on 152Sm, where it infers 26 ± 8% missing against ~24% from the evaluation's own level-density anchor. The paper is deliberately honest about scope: the Brody route carries lower Fisher information than a direct GOE fit and is offered as a one-step consistency check for legacy spectra rather than a frontline estimator; its shortfall is a calibrated lower bound on the deletion fraction; and the experimental test, with three coupling values, places the projection on the right curve rather than measuring it to sub-percent precision. 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 — and the projection paper derives the q ↔ f law, this paper is the series' experimental and legacy-data readout: it takes that derived coordinate, tests it against real laboratory and nuclear spectra, and inverts it to convert decades of published Brody fits into physical chaotic fractions and missing-level verdicts. 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 derivation: q ≈ 0.944·βD2/3·If(2,¾); a=2; P(0)-sensitive repulsion/projection diagnostic 10.5281/zenodo.20584079 A symmetry-blind chaoticity coordinate (this deposit) Experimental + legacy readout: cross-class collapse; first experimental projection test; inverts Brody fits to f; seven-nuclide missing-level diagnostic this