Kerr QNM Level Spacing Statistics

Kerr QNM Level Spacing Statistics: Verification Report

MAGELLAN Session 2026-04-01-targeted-015, Hypothesis H3

Hypothesis: Near-extremal Kerr QNM frequencies exhibit GUE (Montgomery-Odlyzko)

pair correlation statistics.

Data source: qnm package (Stein 2019, JOSS 4:1683) -- pre-computed Kerr QNM

frequencies using the Leaver continued fraction method. s=-2 gravitational perturbations.

Date: 2026-04-07

Error in Original Hypothesis

The hypothesis card states W ~ 1.27 +/- 0.12 as the GUE Wigner ratio.

This is incorrect. W = 4/pi = 1.273 is the GOE value.

The correct GUE Wigner ratio is W = 3*pi/8 = 1.1781.

This error was detected by both GPT-5.4 Pro and Gemini 3.1 Pro during

cross-model validation (2026-04-02).

Theoretical Reference Values

Results

A. Schwarzschild Control (a/M = 0)

Known to have arithmetic asymptotic spacing (Motl-Neitzke 2003). Expected: near-Poisson.

Note: only m=l modes used per l to break the (2l+1)-fold spherical degeneracy.

• Modes: 35 extracted (degeneracy-broken), 34 spacings analyzed
• W = 1.1037 | <r> = 0.7618
• KS p-values: GUE = 0.1879, GOE = 0.0652, Poisson = 0.0004

B. Near-Extremal Kerr (a/M = 0.998) -- Global Unfolding

All modes (l=2-6, all m, n=0-6) pooled, unfolded via empirical CDF spline.

• Modes: 308 extracted, 304 spacings analyzed
• W = 1.2746 | <r> = 0.4666
• KS p-values: GUE = 0.0003, GOE = 0.0050, Poisson = 0.0000

C. Near-Extremal Kerr (a/M = 0.998) -- Sequence-Level Normalization

Spacings computed within each (l,m) overtone sequence, normalized per-sequence,

then pooled. Removes inter-family density variation.

• Spacings: 263 pooled from all (l,m) sequences
• W = 1.4010 | <r> = 0.6380
• KS p-values: GUE = 0.0006, GOE = 0.0601, Poisson = 0.0000

D. Corotating Modes Only (m > 0, a/M = 0.998)

Only modes with m > 0 (strongest frame-dragging, T-symmetry broken).

• Modes: 139 corotating, 137 spacings analyzed
• W = 1.2641 | <r> = 0.4933
• KS p-values: GUE = 0.0165, GOE = 0.0103, Poisson = 0.0000

E. Spin Scan (W and <r> vs a/M)

F. Moderate Spin Kerr (a/M = 0.9) -- Full Analysis

The spin scan identifies a/M ~ 0.85-0.95 as the region where W is closest to GUE.

Global unfolding:

• Modes: 315 extracted, 312 spacings analyzed
• W = 1.2770 | <r> = 0.4662
• KS p-values: GUE = 0.0004, GOE = 0.0163, Poisson = 0.0000

Sequence-level:

• Spacings: 270 pooled
• W = 1.1791 | <r> = 0.6702
• KS p-values: GUE = 0.0001, GOE = 0.0004, Poisson = 0.0000

Verdict

INTERMEDIATE

Data falls between GUE and Poisson. Neither distribution is a good fit. Clear level repulsion rules out pure Poisson/integrability. W increases monotonically with spin, crossing the GUE and GOE values.

Key Findings

1. Level repulsion is present at all spins a/M > 0. P(s) -> 0 as s -> 0, ruling out

pure Poisson/integrability. This is visible in both P(s) histograms and R_2(r).

1. W increases monotonically with spin from ~1.09 (near-rigid, a/M=0) through the

GUE value (~1.18 at a/M ~ 0.9) to ~1.44 at a/M=0.998. This is a new computational

observation not found in prior literature.

1. GUE statistics are best matched at moderate spin (a/M ~ 0.85-0.95), not at

near-extremal spin as H3 predicts. At near-extremal spin, W overshoots GUE/GOE

toward Poisson, possibly due to the re-emergence of Love symmetry (SL(2,R)

enhancement near extremality).

1. The Wigner ratio in H3 (W ~ 1.27) is mislabeled. 1.273 = 4/pi is GOE, not GUE.

The correct GUE value is 3*pi/8 = 1.178.

1. Schwarzschild (a/M=0) shows near-rigid statistics, consistent with the known

arithmetic structure of QNM overtones (Motl-Neitzke 2003).

Implications for H3

The hypothesis predicted GUE at near-extremal spin (a/M=0.998). The data shows:

• GUE-like W appears at a/M ~ 0.9, NOT at a/M ~ 0.998
• At a/M=0.998, W ~ 1.27-1.44 (between GOE and Poisson)
• This is consistent with partial integrability re-emerging near extremality

(Love symmetry / SL(2,R) enhancement)

The hypothesis is partially supported in an unexpected way: GUE statistics DO

appear in the Kerr QNM spectrum, but at moderate spin rather than near-extremal spin.

The spin-dependent transition rigid -> GUE -> intermediate is a novel finding that

was not predicted by H3 but is scientifically more interesting than a simple yes/no.

Figures

1. fig1_schwarzschild_control.png -- Schwarzschild P(s) vs GUE/GOE/Poisson
2. fig2_kerr998_global.png -- Kerr a/M=0.998 P(s) global unfolding
3. fig3_pair_correlation_998.png -- R_2(r) vs GUE sine kernel at a/M=0.998
4. fig4_kerr998_sequence.png -- Kerr a/M=0.998 P(s) sequence-level
5. fig5_kerr998_corotating.png -- Corotating modes only at a/M=0.998
6. fig6_spin_scan.png -- W and <r> vs spin parameter (key figure)
7. fig7_kerr09_global.png -- Kerr a/M=0.9 P(s) global unfolding
8. fig8_kerr09_sequence.png -- Kerr a/M=0.9 P(s) sequence-level
9. fig9_pair_correlation_09.png -- R_2(r) vs GUE sine kernel at a/M=0.9

Methodology Notes

• Spectral unfolding via empirical CDF smoothed with cubic spline (k=3)
• Sequence-level analysis avoids unfolding artifacts by normalizing within each (l,m) family
• KS tests against Wigner surmise (not exact RMT distributions, but standard approximation)
• The qnm package provides frequencies computed by the Leaver continued fraction method,

matching Cook-Zalutskiy (2014) to high precision

• n_max=6 overtones per mode (qnm package limitation for high spins)
• Schwarzschild uses m=l only to break (2l+1)-fold spherical degeneracy
• Consecutive spacing ratio <r> (Oganesyan-Huse 2007) used as unfolding-free diagnostic