Universality of the Berry–Robnik–Brody projection exponent: 𝑎 = 2 for all symmetry classes

Fit a Brody distribution to a mixed regular–chaotic spectrum and ask how the fitted parameter grows as chaos turns on. The textbook expectation is that the growth exponent tracks the level-repulsion index of the chaotic component, β ~ f^(βD+1) — quadratic for orthogonal, cubic for unitary, quintic for symplectic systems. This paper shows that expectation is wrong everywhere except the one case in which it had ever been checked. The small-f growth law is universal: β(f) ~ κf² with exponent exactly a = 2 and coefficient κ = 1/ζ(2) = 6/π², independent of the symmetry class of the chaotic component. The orthogonal class is special only because there βD + 1 = 2; the intuitive law is a coincidence. The repulsion index enters at subleading order f^(βD+2), with exact first-splitting coefficients −1/2, −4/3, and −256π²/75 for the orthogonal, unitary, and symplectic classes, derived in closed form from the population Kullback–Leibler projection of the Berry–Robnik gap distribution onto the Brody family. Three confirmations give the law its teeth. Bias-free population projection. Solving the projection's score equation on the analytic Berry–Robnik density — no sampling, no estimator bias — gives fitted small-f slopes of 1.95–2.01 for every Dyson index including non-integer values, and the coefficient extrapolates to 6/π² within 0.7% for every class. Falsification at scale. A production Monte-Carlo sweep over seventeen values of continuous βD ∈ [0.5, 5] (~4.8×10⁵ spacings per point, two independent bias corrections) leaves the corrected small-f slope flat — d(slope)/dβD = 0.11 against the skeleton's required 1.00 — with the two corrections bracketing the population a = 2 prediction (mean deviations 0.24 and 0.21, versus 1.25 to the skeleton). Physical realization. In the quantum kicked top, a real time-periodic spin system, the finite-ℏ statistics rise toward the a = 2 Berry–Robnik prediction from below as ℏ = 1/j → 0, with a common upward trend at 2.7σ across two independent chaotic fractions, error-barred by a parametric ensemble of independent spectra. The coefficient traces to a single number: ζ(2) = π²/6 is the Fisher information of the Brody family at its Poisson point, and κ is its reciprocal; ζ(3) enters one cumulant higher through the exact projection-feedback coefficient c₄ = κ² + κ³ζ(3). That one sine-kernel constant controls the universal onset of level repulsion in every symmetry class is the compact content of the result. The paper is deliberately honest about scope: the exponent, not the coefficient, is the directly measurable quantity — κ is a population-level statement masked at small f by the Brody-MLE floor and at finite ℏ by the slow semiclassical approach; the kicked-top evidence is a directional trend, not an attained limit; and the symplectic splitting coefficient is obtained analytically, with the direct residual verified for βD ≤ 2. Part of the ICP series on the information geometry of chaos. Where the geometric papers build the Brody manifold from the inside, the projection paper derives the full q ↔ f law with its class-dependent anchor, and the coordinate paper inverts that law against laboratory and nuclear spectra, this Letter isolates what all symmetry classes share: the universal germ of the projection — the exponent a = 2 and the coefficient 6/π² — and proves the familiar class-tracking exponent to be an orthogonal-class accident. 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·βD^(2/3)·If(2,¾); a=2; P(0)-sensitive repulsion/projection diagnostic 10.5281/zenodo.20584079 A symmetry-blind chaoticity coordinate Experimental + legacy readout: cross-class collapse; first experimental projection test; seven-nuclide missing-level diagnostic 10.5281/zenodo.20681797 Universality of the Berry–Robnik–Brody projection exponent (this deposit) Universality Letter: a=2 for all symmetry classes; parameter-free κ=6/π²; exact class-splitting coefficients −1/2, −4/3, −256π²/75 10.5281/zenodo.21360207this