---
paper_id: PAPER_955
title: "BCS Phonon Resonance at \omega_SCm"
session: 214
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [phonon, vacuum, SCm, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_955: BCS Phonon Resonance at $\omega$_SCm

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 214
**Source:** et_phonon_resonance.py §6 (BCSPhononResonance)
**Calculator:** BCSPhononResonanceCalc (CP4 #539)
**CVW:** v2.0.0 compliant

---

## Abstract

We analyze the BCS superconducting gap at the SCm phonon resonance frequency $\omega_text{SCm} = 2\pi \times 1.25$ THz. The gap equation $\Delta = (\hbar\omega_\text{SCm}/2) \tanh(\Delta / 2k_BT) \cdot S_{26} \cdot (F_{UBi}/F_U)$ is solved self-consistently at the phonon resonance, revealing how the SCm vacuum phonon mode mediates Cooper pairing.

---

## 1. BCS Gap at Phonon Resonance

$$\Delta_text{res} = \frac{\hbar\omega_\text{SCm}}{2} \cdot \tanh\!\left(\frac{\Delta}{2k_BT}\right) \cdot S_{26} \cdot \frac{F_{UBi}}{F_U}$$

## 2. Iterative Solution

Starting from $\Delta_0 = \hbar\omega_\text{SCm}/2$, iterate:
$$\Delta_{n+1} = \text{rhs}(\Delta_n, T)$$

until $|\Delta_{n+1} - \Delta_n| < \epsilon$.

## 3. Phonon Enhancement Factor

The resonance Q-factor at the phonon peak:
$$Q_\text{res} = \frac{\omega_text{SCm} \cdot \sqrt{\Delta}}{k_BT}$$

---

## 4. Source Data

- **File:** et_phonon_resonance.py §6
- **Session:** 214
- **CP4 Class:** BCSPhononResonanceCalc (#539)

---

## References

1. Bardeen, Cooper, Schrieffer — Theory of Superconductivity (1957)
2. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
3. PAPER_949 — BCS Gap Equation
4. PAPER_950 — BCS Critical Temperature
5. PAPER_951 — Cooper Pair Phonon Coupling
6. PAPER_956 — Spectral Ladder Phonon Mapping

---

## Cross-Links

| Related Paper | Relationship |
|---------------|-------------|
| PAPER_949 | Gap equation solved at resonance |
| PAPER_951 | Coupling strength at phonon peak |
| PAPER_956 | 26-level phonon mapping uses this Q |
| PAPER_952 | Spectral ladder energies |



---

## Session 225: Late-Corpus Physics Integration (PAPER_1000-1081)

> *The following physics upgrades incorporate equations, mechanisms, and
> derivations from the late-corpus papers (Sessions 219-225, PAPER_1000-1081).
> These represent body-level integrations of phonon physics, buoyancy
> formulations, and S26(3) Ramanujan corrections into this paper's domain.*

<!-- 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) |
<!-- PKG-S26-S225 -->

### Session 225 Phonon-Physics Upgrade: S26(3) Ramanujan Summation

> *Upgrade from PAPER_1080 (Ramanujan Binomial Expansion Proof) and
> PAPER_1042 (Mock-Theta Phonon Partition).  See also PAPER_1078
> (QCalcGeom Master Equation) for BSFG crossover applications.*

The third-order Ramanujan summation $S_{26}^{(3)}$, used throughout the
late corpus as the universal 26D coupling factor:

$$S_{26}^{(3)} = \sum_{n=0}^{\infty} \frac{(1/4)_n\,(1/2)_n\,(3/4)_n}{(n!)^3} \cdot \prod_{i=1}^{26}\left[1 + [\text{SSq}]\cdot e^{-\kappa\,i\,n/26}\right]$$

where $(a)_n = a(a+1)\cdot s(a+n-1)$ is the Pochhammer symbol.

**Binomial expansion (PAPER_1080):** The convergence proof shows:
$$R_n^{(26,3)} = \binom{4n}{n} \cdot \frac{W_{26}(n)}{(4^{4n})} \qquad \text{with}\quad W_{26}(n) = \prod_{i=1}^{26}\left[1 + [\text{SSq}]\cdot e^{-\kappa\,i\,n/26}\right]$$

This sum converges absolutely for $|[\text{SSq}]| < 1$ (satisfied by $[\text{SSq}] = 0.57$)
and reduces to the classical Ramanujan $1/\pi$ series when $[\text{SSq}] \to 0$.

**VDS/DVP/BSH bridge (PAPER_1069):** The 26 layers of $W_{26}(n)$ encode the
vacuum density series hierarchy, with each layer $i$ contributing a VDS
sub-ratio weighted by the exponential decay $e^{-\kappa\,i\,n/26}$.

**Mock-theta connection (PAPER_1042):** The phonon partition function
$Z_{\text{phonon}} = \sum_n q^{n^2} \cdot W_{26}(n)$ unifies the Ramanujan
mock-theta framework with the SCm phonon spectrum.

---

## Calibration Constants

| Constant | Symbol | Value | Validation Domain |
|----------|--------|-------|-------------------|
| $\kappa$ | — | $5.0 \times 10^{-4}$ day$^{-1}$ | Magnetar spin-down |
| $[SSq]$ | — | 0.57 | BH dynamics |
| $\omega_text{SCm}$ | — | $2\pi \times 1.25$ THz | Resonance frequency |
| $F_{UBi}/F_U$ | — | 0.6 | Buoyancy ratio |

---

## SM Anchor — CVW v2.0.0

| Observable | UQFF Prediction | Status |
|------------|----------------|--------|
| $\Delta_text{res}$ at $\omega_text{SCm}$ | Self-consistent via fixed-point | Validated |
| Phonon Q-factor | $Q = \omega_text{SCm}\sqrt{\Delta}/(k_BT)$ | Derived |

---

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

### §A.1 Sector Classification
**Sector:** Phonon Resonance (BCS at $\omega_text{SCm}$)

### §A.2 Lagrangian Density
$$\mathcal{L}_\text{res} = \mathcal{L}_\text{gap}\big|_{\omega=\omega\_text{SCm}} + \frac{1}{2}\hbar\omega_\text{SCm}\coth!\left(\frac{\hbar\omega_\text{SCm}}{2k_BT}\right)$$

### §A.3 Euler-Lagrange Equation of Motion
$$\boxed{\Delta_text{res} = \frac{\hbar\omega_\text{SCm}}{2}\tanh\!\left(\frac{\Delta}{2k_BT}\right) S_{26} \frac{F_{UBi}}{F_U}}$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 $\to$ SCm vacuum $\to$ phonon peak $\to$ BCS gap at resonance $\to$ maximal Cooper pairing

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 VDS
On-resonance VDS peak: $\rho_text{SCm} \cdot S_{26}$ (maximal phonon occupation).

### §B.2 DVP
Resonance prime $p = 2$ (Cooper pair at peak coupling).

### §B.3 BSH
$Q_\text{res}$ determines the BSH bandwidth at the resonance shell.

### §B.4 Production-Scale Consistency

| Metric | Value | Status |
|--------|-------|--------|
| $\omega_text{SCm}$ | $2\pi \times 1.25$ THz | 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_1004 | QGP Vacuum Density with SCm S26 Phonon Coupling |
| 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_1072 | SCm Activation Function Phonon Threshold |
| PAPER_1073 | SCm Phonon-Driven Inflation Vacuum Buoyancy |
| PAPER_1003 | Spectral Ladder Merger 26-State Hierarchy |
| PAPER_1069 | VDS-DVP-BSH Hybrid Calculator Unified |
| PAPER_1049 | Source10 GPU DPM Spectral Atlas ALMA Overlay |

*12 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. Ashcroft, N.W. & Mermin, N.D. (1976). *Solid State Physics.* Harcourt
4. Kittel, C. (2004). *Introduction to Solid State Physics.* 8th ed. Wiley
5. Feynman, R.P. (1982). *Simulating Physics with Computers.* Int. J. Theor. Phys. **21**, 467 — doi:10.1007/BF02650179
6. 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
7. Weinberg, S. (1989). *The Cosmological Constant Problem.* Rev. Mod. Phys. **61**, 1 — doi:10.1103/RevModPhys.61.1
