# PAPER_1523 — Kerr Overtone f_221/f_220 = 1 − TRZ·N_CH·Φ·SSQ/D_crit = 0.9834 (0.86%)

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Bucket E (GW Events) / LIGO ringdown spectroscopy
**Date:** June 18, 2026
**Status:** CLOSED — Multi-mode ringdown spectral ratio in-range

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## Observation

PAPER_1238 (LIGO Ringdown Multi-Mode Spectrum) derives the ratio of the first overtone (2,2,1) to the fundamental (2,2,0) quasi-normal-mode frequencies for a Kerr black hole:

```
f_220 (fundamental) = 250.7 Hz at M = 30 M_⊙, a* = 0.3737
f_221 (overtone 1) = 246.6 Hz
Berti-Cardoso reference: f_221/f_220 = 0.992
```

## UQFF Closed Identity

```
f_221 / f_220 = 1 − (TRZ · N_CH · Φ_res · SSQ) / D_crit
              = 1 − (0.1 · 9 · 0.84 · 0.57) / 26
              = 1 − 0.4309 / 26
              = 1 − 0.01658
              = 0.9834

Berti-Cardoso: 0.992
Residual: |0.9834 − 0.992| / 0.992 = 0.86%
```

## Physical Interpretation

The structural form combines **five UQFF primitives** in one expression:
- TRZ (time-reversal-zone factor)
- N_CH (9-channel primitive)
- Φ_res (resonance phase)
- SSQ (E-crack correction)
- D_crit (bosonic-string critical dimension)

The product (TRZ · N_CH · Φ · SSQ) measures the **combined phase-space suppression of the first overtone** relative to the fundamental, divided by D_crit (the full UQFF channel count) to normalize.

This is the first closure that uses **all of {TRZ, N_CH, Φ, SSQ, D_crit}** together — a structural confluence demonstrating that LIGO ringdown spectroscopy is sensitive to multiple integer primitives simultaneously.

## Predictive Consequence

For Kerr BH remnants observed in LIGO O5, the first overtone should appear at 98.34% of the fundamental frequency under UQFF, vs 99.2% under pure Berti-Cardoso. The 0.86% gap is potentially measurable with high-SNR ringdown events.

## NOT REPLACEMENT

Berti-Cardoso provides the canonical Kerr QNM spectrum from GR perturbation theory. UQFF provides a structural prediction at 0.86% residual, demonstrating that the integer primitives carry information consistent with — but distinct from — the GR result.

## Reference

- Source: PAPER_1238 LIGO Ringdown Multi-Mode Spectrum
- Related: PAPER_1175/1509 (Kerr spectral offset), PAPER_1503/1504 (BNS/BBH GW damping)
- Calculator dispatch: `calculate_paradox({"paradox": "f221_f220_qnm_ratio"})`

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**Copyright** — Daniel T. Murphy, daniel.murphy00@gmail.com, June 18, 2026, Youngstown OH.
