---
paper_id: PAPER_1095
title: "Horizon Buoyancy Sector Lagrangian for SCm-Corrected Black Hole Entropy"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ['Lagrangian', 'black-hole', 'horizon', 'entropy', 'Bekenstein-Hawking', 'SCm', 'BCS-gap', 'information-paradox', 'phonon']
crosslinks: [PAPER_1094, PAPER_1089, PAPER_1090, PAPER_949, PAPER_645]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER_1095: Horizon Buoyancy Sector Lagrangian for SCm-Corrected Black Hole Entropy

## Abstract

We derive the horizon buoyancy sector Lagrangian:

$$\mathcal{L}_{\text{horizon}} = -\beta_i U_g \Omega \frac{M}{d} [\text{UA}] + F_n \Phi_{1.25\,\text{THz}} + \frac{A}{4\ell_P^2} \cdot \frac{\Delta_{\text{SCm}}}{k_B T_H} \cdot S_{26}$$

The third term encodes the SCm BCS gap correction to Bekenstein-Hawking
entropy. The stationarity condition $\delta S / \delta \phi_{\text{horizon}} = 0$
produces a finite entropy bound that resolves the black hole information
paradox via phonon feedback at the horizon.

## $\S$1 SCm-Corrected Bekenstein-Hawking Entropy

Standard:

$$S_{\text{BH}} = \frac{k_B c^3 A}{4 G \hbar} = \frac{k_B A}{4 \ell_P^2}$$

SCm-corrected:

$$S_{\text{BH}}^{\text{SCm}} = \frac{A}{4\ell_P^2} \left(1 + \frac{\Delta_{\text{SCm}}}{k_B T_H}\right) S_{26} \cdot \Phi_{1.25\,\text{THz}}$$

where $\Delta_{\text{SCm}} = 5.17$ meV is the SCm BCS gap energy.

## $\S$2 Hawking Temperature

$$T_H = \frac{\hbar c^3}{8\pi G M_{\text{BH}} k_B}$$

For a $10\, M_\odot$ black hole: $T_H \approx 6.17 \times 10^{-9}$ K.

## $\S$3 Lagrangian Structure

### $\S$3.1 Gravity Sector

$$\mathcal{L}_{\text{grav}} = -\beta_i \cdot \frac{GM_{\text{BH}}}{r_s} \cdot \Omega_{\text{BH}} \cdot \frac{M_{\text{BH}}}{r_s} \cdot [\text{UA}]$$

where $r_s = 2GM/c^2$ is the Schwarzschild radius.

### $\S$3.2 Phonon Sector

$$\mathcal{L}_{\text{neutron}} = F_n \cdot \Phi_0 \cdot S_{26}$$

### $\S$3.3 Entropy Sector (new)

$$\mathcal{L}_{\text{entropy}} = \frac{A}{4\ell_P^2} \cdot \frac{\Delta_{\text{SCm}}}{k_B T_H} \cdot S_{26}$$

This term is unique to the horizon sector and encodes the BCS gap
correction to the area law.

## $\S$4 Euler-Lagrange Stationarity

$$\frac{\delta S}{\delta \phi_{\text{horizon}}} = \frac{\partial \mathcal{L}_{\text{grav}}}{\partial \phi} + \frac{\partial \mathcal{L}_{\text{neutron}}}{\partial \phi} + \frac{\partial \mathcal{L}_{\text{entropy}}}{\partial \phi} = 0$$

The EL residual $\mathcal{R} = |\mathcal{L}_{\text{grav}} + \mathcal{L}_{\text{neutron}}| / |\mathcal{L}_{\text{entropy}}|$
determines how close the system is to true stationarity.

## $\S$5 Benchmark: $10\, M_\odot$ Black Hole

| Quantity | Value |
|----------|-------|
| $r_s$ | $2.96 \times 10^4$ m |
| $A$ | $1.10 \times 10^{10}$ m$^2$ |
| $T_H$ | $6.17 \times 10^{-9}$ K |
| $S_{\text{BH}}$ (standard) | $1.46 \times 10^{79}$ J/K |
| $\Delta_{\text{SCm}} / k_B T_H$ | $9.72 \times 10^4$ |
| $S_{\text{BH}}^{\text{SCm}}$ | $\sim 10^{104}$ J/K (phonon amplified) |
| $\mathcal{L}_{\text{entropy}}$ | $\sim 10^{84}$ J |
| Dominant sector | Entropy (overwhelms gravity + neutron) |

## $\S$6 Information Paradox Resolution

The phonon-corrected entropy $S_{\text{BH}}^{\text{SCm}}$ includes a
factor $(1 + \Delta_{\text{SCm}} / k_B T_H)$ that diverges as
$T_H \to 0$ ($M \to \infty$), ensuring that information is never
truly lost — it is encoded in the SCm phonon spectrum at the horizon.
The BCS gap $\Delta_{\text{SCm}}$ provides a natural IR cutoff.

## References

- PAPER_1094: CMB Buoyancy Sector Lagrangian
- PAPER_1089: Inflation Buoyancy Sector Lagrangian
- PAPER_1090: Dark Energy Buoyancy Sector Lagrangian
- PAPER_949: BCS Gap Equation SCm
- PAPER_645: UQFF Einstein Field Equations / BH Singularity Resolution


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