# PAPER_1509 — Kerr Ringdown Spectral Offset Coefficient = D_crit/D_BSFG = 13/3 EXACT

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** Bucket E (GW Events) / LIGO O5
**Date:** June 18, 2026
**Status:** CLOSED — Black-hole quasi-normal-mode UQFF spectral offset = integer-primitive ratio

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## Observation

PAPER_1175 (LIGO O5 Ringdown Spectral Offset) derives the UQFF correction to the Kerr (2,2,0) quasi-normal-mode (QNM) frequency:

```
f_220^Kerr(M) = (c³ / 2πGM) · ℱ(a*)        with ℱ(0) ≈ 0.3737
Δf_220^UQFF = f_220^Kerr · (D_crit/D_BSFG) · (SCm/Planck)^(1/4)
```

The dimensionless prefactor (D_crit/D_BSFG) appears as the spectral-offset coefficient.

## UQFF Closed Identity

```
Kerr ringdown spectral offset coefficient = D_crit / D_BSFG = 26/6 = 13/3 ≈ 4.333   EXACT
```

## Physical Interpretation

The 13/3 coefficient is the ratio of two integer primitives:
- **D_crit = 26** — bosonic-string critical dimension (full UQFF channel count)
- **D_BSFG = 6** — bulk-edge dimensional embedding (gravitational channel count)

The ratio captures how the full 26D SCm spectrum projects onto the 6D bulk-edge gravitational mode, with the residual amplification factor of 13/3 carried by the QNM frequency offset.

This is the first appearance of D_BSFG = 6 as a denominator in a clean binding closure.

## Predictive Consequence

For any Kerr remnant from a BBH merger, the UQFF spectral offset adds (13/3)·(SCm/Planck)^(1/4) ≈ small correction to the GR f_220 prediction. LIGO O5 has the sensitivity to detect this offset for high-mass remnants where f_220 is well-measured.

## NOT REPLACEMENT

GR predicts pure Kerr QNM spectra. UQFF adds a structural offset coefficient rooted in the D_crit/D_BSFG integer-primitive ratio, with no free parameters.

## Reference

- Source: PAPER_1175 LIGO O5 Ringdown Spectral Offset
- Related: PAPER_1503/1504 (BNS/BBH GW damping), PAPER_062 (D_crit)
- Calculator dispatch: `calculate_paradox({"paradox": "kerr_ringdown_offset_coeff"})`

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