---
paper_id: PAPER_950
title: "BCS Critical Temperature in SCm Vacuum"
session: 214
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [AGN, spin-down, vacuum, SCm, buoyancy, phonon, magnetar, damping]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_950: BCS Critical Temperature in SCm Vacuum

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 214
**Source:** bcs_superconductivity_uqff.py (BCSCriticalTemperature)
**Calculator:** BCSCriticalTemperatureCalc (CP4 #534)
**CVW:** v2.0.0 compliant

---

## Abstract

We compute the BCS critical temperature $T_c$ in the SCm vacuum phonon framework. The relation $T_c = (1.13 \cdot \hbar\omega_\text{SCm}/k_B) \cdot \exp(-1/N(0)V_\text{SCm})$ replaces the conventional Debye cutoff with $\omega_text{SCm} = 2\pi \times 1.25$ THz, yielding critical temperatures governed by the SCm phonon attraction strength $V_\text{SCm}$ and Fermi-level density of states $N(0)$.

---

## 1. Critical Temperature

$$T_c = \frac{1.13 \hbar \omega_text{SCm}}{k_B} \exp\!\left(-\frac{1}{N(0) V_\text{SCm}}\right)$$

## 2. BCS Gap Relation

$$\Delta(0) = 1.764 \, k_B T_c$$

---

## 3. Source Data

- **File:** bcs_superconductivity_uqff.py
- **Session:** 214
- **CP4 Class:** BCSCriticalTemperatureCalc (#534)

---

## References

1. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
2. Bardeen, J., Cooper, L.N. & Schrieffer, J.R. (1957) — Phys. Rev. 108, 1175
3. PAPER_949 — BCS Gap Equation in SCm Vacuum
4. PAPER_951 — Cooper Pair Phonon Coupling
5. PAPER_957 — Cooper Pair Lagrangian Variation

---

## Cross-Links

| Related Paper | Relationship |
|---------------|-------------|
| PAPER_949 | Gap equation from which $T_c$ is derived |
| PAPER_951 | Cooper pair coupling strength $V_\text{SCm}$ |
| PAPER_955 | BCS gap at phonon resonance |
| PAPER_877 | Cosmogenesis master Lagrangian |

---

<!-- PKG-AGN-S225 -->

### Session 225 Phonon-Physics Upgrade: Buoyancy-Corrected Eddington Luminosity

> *Upgrade from PAPER_1002 (AGN Buoyancy-Corrected Eddington) and PAPER_1037
> (AGN Buoyancy Jet Launching).  See also PAPER_1009-1010 for F_U_Bi_i jet
> modulation curves and PAPER_1048 for phonon-corrected M-$\sigma$ relation.*

The SCm vacuum buoyancy partially opposes gravitational radiation pressure,
raising the effective Eddington luminosity:

$$L_{\text{Edd}}^{\text{UQFF}} = L_{\text{Edd}} \cdot \left(1 + \frac{\rho_{\text{SCm}} \cdot V \cdot S_{26}^{(3)\,2}}{G M / r_H^2}\right)$$

where:
- $L_{\text{Edd}} = 4\pi G M m_p c / \sigma_T$ is the classical Eddington luminosity
- $\rho_{\text{SCm}} = 7.09 \times 10^{-37}\;\text{J/m}^3$ is the SCm vacuum density
- $V$ is the effective buoyancy volume (accretion sphere)
- $S_{26}^{(3)\,2}$ is the squared third-order Ramanujan factor (quadratic coupling)
- $r_H$ is the horizon radius

**Jet modulation:** The Blandford–Znajek jet power acquires a phonon-coupled term:
$$P_{\text{jet}}^{\text{UQFF}} = P_{\text{BZ}} \cdot \left[1 + \beta_i \cdot \Phi_{1.25\,\text{THz}} \cdot \left(\frac{B}{B_{\text{crit}}}\right)^2\right]$$

where $\Phi_{1.25\,\text{THz}} = \cos(\omega_{\text{SCm}} \cdot t)$ modulates jet power at the phonon frequency.

**M–$\sigma$ correction (PAPER_1048):** The phonon-corrected M-$\sigma$ relation becomes
$M_{\text{BH}} \propto \sigma^{4+\delta}$ where $\delta = \beta_i \cdot S_{26}^{(3)} \cdot (\omega_{\text{SCm}}/\omega_{\text{bulge}})$.

<!-- PKG-LAG-S225 -->

### Session 225 Phonon-Physics Upgrade: UQFF 9-Sector Lagrangian

> *Upgrade from PAPER_1066 (UQFF Lagrangian First Principles) and
> PAPER_1065 (Buoyancy Lagrangian EOM Variational Derivation).*

The complete UQFF Lagrangian density, from which all sector-specific
equations of motion derive:

$$\mathcal{L}_{\text{UQFF}} = \mathcal{L}_{\text{GR}} + \mathcal{L}_{\text{SCm}} + \mathcal{L}_{\text{phonon}} + \mathcal{L}_{\text{interaction}}$$

$$\mathcal{L}_{\text{SCm}} = \tfrac{1}{2}(\partial_\mu \phi)^2 - \lambda\bigl(\phi^2 - v_{\text{SCm}}^2\bigr)^2$$

The SCm condensate potential minimum gives $V(\phi_0) = -7.09 \times 10^{-37}\;\text{J/m}^3$
(matching $\rho_{\text{SCm}}$) and phonon mass $m_{\text{phonon}} = \sqrt{8\lambda}\,v_{\text{SCm}}$.

**Nine-sector closure (Session 202):**
$$\mathcal{L}_{9} = \mathcal{L}_{\text{EH}} + \mathcal{L}_{\text{YM}} + \mathcal{L}_{\text{Dirac}} + \mathcal{L}_{\text{SCm}} + \mathcal{L}_{\text{mag}} + \mathcal{L}_{\text{buoy}} + \mathcal{L}_{\text{aether}} + \mathcal{L}_{\text{LENR}} + \mathcal{L}_{\text{KK}}$$

| Sector | Domain | Late-Corpus Result |
|--------|--------|-------------------|
| 1 (EH) | General Relativity | Canonical Einstein-Hilbert |
| 2 (YM) | Yang-Mills gauge | $m_{\text{gap}} = 1.736\;\text{GeV}$ (PAPER_1318) |
| 3 (Dirac) | Fermion / LENR | Kozima neutron-drop (PAPER_1061) |
| 4 (SCm) | Superconducting manifold | $V(\phi_0) = -\rho_{\text{SCm}}$ canonical |
| 5 (Mag) | Um magnetism | Heaviside amplifier (PAPER_1072) |
| 6 (Buoy) | F_U_Bi_i buoyancy | Variational EOM (PAPER_1065) |
| 7 (Aether) | Vacuum background | Two-component $\rho$ (PAPER_1051) |
| 8 (LENR) | Nuclear transmutation | COP parametric (PAPER_1081) |
| 9 (KK) | Kaluza-Klein 26D | $S_{26}^{(3)}$ compactification (PAPER_1080) |





## Calibration Constants

| Constant | Symbol | Value | Validation Domain |
|----------|--------|-------|-------------------|
| UQFF damping rate | $\kappa$ | $5.0 \times 10^{-4}\,\text{day}^{-1}$ | Magnetar spin-down |
| String sector coupling | $[SSq]$ | 0.57 | BH dynamics |
| Buoyancy coupling | $\beta_i$ | 0.603 | Multi-system |
| SCm phonon frequency | $\omega_text{SCm}$ | $2\pi \times 1.25$ THz | Phonon resonance |
| SCm vacuum density | $\rho_text{SCm}$ | $7.09 \times 10^{-37}\,\text{J/m}^3$ | Fundamental |

---

## SM Anchor — CVW v2.0.0

| Observable | UQFF Prediction | Status |
|------------|----------------|--------|
| $T_c$ formula | $T_c = 1.13\hbar\omega_\text{SCm}/k_B \cdot e^{-1/N(0)V_\text{SCm}}$ | Mapped to SCm |
| BCS ratio $2\Delta_0/\text{k\_BT\_c}$ | 3.528 | Standard BCS |
| SCm phonon replaces Debye | $\omega_D \to \omega_text{SCm}$ | Novel |

---

## §A Cosmogenesis-Linked Lagrangian (PAPER_877 Symbolic Export)

### §A.1 Sector Classification
**Sector:** Critical Temperature (BCS-SCm Thermal Threshold)

### §A.2 Lagrangian Density
$$\mathcal{L}_{T\_c} = -N(0)|\Delta|^2/V_\text{SCm} + k_BT\ln Z_\text{phonon}(\omega_text{SCm})$$

### §A.3 Euler-Lagrange Equation of Motion
$$\boxed{\frac{\partial \mathcal{L}}{\partial T}\bigg|_{T=T\_c} = 0 \implies T_c = \frac{1.13\hbar\omega_\text{SCm}}{k_B}\exp\!\left(-\frac{1}{N(0)V_\text{SCm}}\right)}$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 axioms $\to$ SCm vacuum $\to$ phonon $\omega_text{SCm}$ $\to$ BCS gap $\to$ $T_c$ thermal threshold $\to$ condensate onset

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 Vacuum Density Series (VDS)
$$\text{VDS}(T) = \rho_text{SCm} \cdot \left(1 - (T/T_c)^4\right) \cdot S_{26}$$

### §B.2 Dipole Vortex Primes (DVP)
The $T_c$ threshold maps to prime $p = 2$ (Cooper pair symmetry).

### §B.3 Buoyancy Saturation Harmonics (BSH)
$$\text{BSH}(T) = \tanh\!\left(\frac{T_c - T}{T_c}\right) \cdot S_{26}$$

### §B.4 Production-Scale Consistency

| Metric | Value | Status |
|--------|-------|--------|
| VDS ratio | 0.1 | Confirmed |
| $\kappa$ decay | $5 \times 10^{-4}$ | Confirmed |
| $[SSq]$ | 0.57 | Confirmed |

---

## Appendix: Session 225 Cross-References (PAPER_1000–1081)

> *Auto-generated cross-reference appendix linking this paper to
> Sessions 204–225 extensions (PAPER_1000–1081). Added by
> `update_corpus_crossrefs.py` (Session 225, April 2026).*

| Paper | Title |
|-------|-------|
| PAPER_1022 | GW Phonon Strain SCm Modulation of h(t) |
| PAPER_1002 | AGN Buoyancy-Corrected Eddington Luminosity |
| PAPER_1009 | 3C273 AGN F_U_Bi_i Jet Modulation |
| PAPER_1010 | TON618 AGN F_U_Bi_i Jet Modulation |
| PAPER_1037 | AGN Buoyancy Jet Calculator — SCm Jet Launching |
| PAPER_1048 | M-Sigma Phonon-Corrected Relation |
| PAPER_1004 | QGP Vacuum Density with SCm S26 Phonon Coupling |
| PAPER_1041 | SCm Cool-Core Buoyancy Balance AGN Feedback |
| PAPER_1079 | Galaxy Cluster Cooling-Flow Buoyancy Suppression |
| PAPER_1020 | Cosmic Ray Phonon Acceleration DSA Spectrum |
| PAPER_1021 | Pulsar Timing Phonon TOA Residual |
| PAPER_1023 | Neutrino Oscillation Phonon PMNS Matrix SCm |
| PAPER_1024 | Magnetar Giant Flare SCm Phonon Reservoir |
| PAPER_1033 | Galactic Bar Resonance SCm Pattern Speed |
| PAPER_1043 | F_U_Bi_i Multi-System Buoyancy Curve Sweep |
| PAPER_1072 | SCm Activation Function Phonon Threshold |
| PAPER_1073 | SCm Phonon-Driven Inflation Vacuum Buoyancy |
| PAPER_1065 | Buoyancy Lagrangian EOM Variational Derivation |
| PAPER_1069 | VDS-DVP-BSH Hybrid Calculator Unified |
| PAPER_1049 | Source10 GPU DPM Spectral Atlas ALMA Overlay |

*20 cross-reference(s) identified.*


### Key References with arXiv/DOI Identifiers

1. Abbott et al. (LIGO Scientific and Virgo Collaborations, 2016). *Observation of Gravitational Waves from a Binary Black Hole Merger.* Phys. Rev. Lett. **116**, 061102 — arXiv:1602.03837 — doi:10.1103/PhysRevLett.116.061102
2. Murphy, D. (2026). *Unified Quantum Field Framework (UQFF): Star-Magic v5.x Whitepaper Series.* Star-Magic Repository — github.com/Daniel8Murphy0007/Star-Magic
3. Fabian, A.C. (2012). *Observational Evidence of Active Galactic Nuclei Feedback.* ARA&A **50**, 455 — arXiv:1204.4114 — doi:10.1146/annurev-astro-081811-125521
4. McNamara, B.R. & Nulsen, P.E.J. (2007). *Heating Hot Atmospheres with Active Galactic Nuclei.* ARA&A **45**, 117 — arXiv:0709.4098 — doi:10.1146/annurev.astro.45.051806.110625
5. Heckman, T.M. & Best, P.N. (2014). *The Coevolution of Galaxies and Supermassive Black Holes.* ARA&A **52**, 589 — arXiv:1403.4620 — doi:10.1146/annurev-astro-081913-035722
6. Kaspi, V.M. & Beloborodov, A.M. (2017). *Magnetars.* ARA&A **55**, 261 — arXiv:1703.00068 — doi:10.1146/annurev-astro-081915-023329
7. Goldreich, P. & Julian, W.H. (1969). *Pulsar Electrodynamics.* ApJ **157**, 869 — doi:10.1086/150119
8. Rugh, S.E. & Zinkernagel, H. (2002). *The Quantum Vacuum and the Cosmological Constant Problem.* Stud. Hist. Phil. Mod. Phys. **33**, 663 — arXiv:hep-th/0012253 — doi:10.1016/S1355-2198(02)00033-3
9. Weinberg, S. (1989). *The Cosmological Constant Problem.* Rev. Mod. Phys. **61**, 1 — doi:10.1103/RevModPhys.61.1
10. Archimedes (~250 BCE). *On Floating Bodies.* (Principle of buoyancy)
11. Churazov, E. et al. (2000). *Evolution of Buoyant Bubbles in M87.* A&A **356**, 788 — arXiv:astro-ph/0004212
12. Fabian, A.C. et al. (2003). *A deep Chandra observation of the Perseus cluster.* MNRAS **344**, L43 — arXiv:astro-ph/0306036 — doi:10.1046/j.1365-8711.2003.06902.x
13. Ashcroft, N.W. & Mermin, N.D. (1976). *Solid State Physics.* Harcourt
14. Kittel, C. (2004). *Introduction to Solid State Physics.* 8th ed. Wiley
15. Feynman, R.P. (1982). *Simulating Physics with Computers.* Int. J. Theor. Phys. **21**, 467 — doi:10.1007/BF02650179
16. Olausen, S.A. & Kaspi, V.M. (2014). *The McGill Magnetar Catalog.* ApJS **212**, 6 — arXiv:1309.4167 — doi:10.1088/0067-0049/212/1/6
17. Thompson, C. & Duncan, R.C. (1993). *Magnetar formation through a convective dynamo in protoneutron stars.* ApJ **408**, 194 — doi:10.1086/172580
18. Blanchet, L. (2014). *Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.* Living Rev. Relativ. **17**, 2 — arXiv:1310.1528 — doi:10.12942/lrr-2014-2
