---
paper_id: PAPER_954
title: "E(t) Linewidth Modulation with Sign-Flip Dynamics"
session: 214
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [SCm, jet, buoyancy, phonon, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_954: E(t) Linewidth Modulation with Sign-Flip Dynamics

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 214
**Source:** blazar_jet_phonon.py (EtLinewidthModulation)
**Calculator:** EtLinewidthModulationCalc (CP4 #538)
**CVW:** v2.0.0 compliant

---

## Abstract

We derive the E(t) linewidth modulation function with sign-flip dynamics for astrophysical jets. The time-domain response $E(t,\Gamma) = S_{26} \cdot \cos(\omega_text{SCm} t) \cdot \exp(-\Gamma t)$ exhibits sign flips at $t_\text{flip} = \pi/(2\omega_text{SCm})$, driving extra-gravitational responses in blazar jets. Narrow linewidths produce sharper sign flips with tighter jet collimation; broad linewidths damp the oscillation before the first flip.

---

## 1. E(t) Function

$$E(t, \Gamma) = S_{26} \cdot \cos(\omega_text{SCm} \cdot t) \cdot \exp(-\Gamma \cdot t)$$

## 2. Sign-Flip Time

$$t_\text{flip} = \frac{\pi}{2\omega_text{SCm}} \approx 0.064 \text{ ps}$$

## 3. Regime Dependence

| $\Gamma$ (THz) | Flips in 5 ps | Jet Behavior |
|-----------------|---------------|-------------|
| 0.01 | Many | Ultra-tight collimation |
| 0.10 | Several | Optimal modulation |
| 1.00 | Few/None | Damped, diffuse wind |

---

## 4. Source Data

- **File:** blazar_jet_phonon.py
- **Session:** 214
- **CP4 Class:** EtLinewidthModulationCalc (#538)

---

## References

1. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
2. PAPER_939 — CenA Jet Power Curves
3. PAPER_940 — TXS0506 Jet Power Curves
4. PAPER_955 — BCS Phonon Resonance
5. PAPER_961 — Compressed Gravity Triadic

---

## Cross-Links

| Related Paper | Relationship |
|---------------|-------------|
| PAPER_939 | CenA jets modulated by $E(t,\Gamma)$ |
| PAPER_940 | TXS0506 jets with linewidth dependence |
| PAPER_949 | BCS gap supplies $S_{26}$ factor |
| PAPER_963 | Buoyancy triadic uses same sign-flip |

---

<!-- 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 $\delta = \beta_i \cdot S_{26}^{(3)} \cdot (\omega_{\text{SCm}}/\omega_{\text{bulge}})$.



## Calibration Constants

| Constant | Symbol | Value | Validation Domain |
|----------|--------|-------|-------------------|
| $\kappa$ | — | $5.0 \times 10^{-4}\,\text{day}^{-1}$ | Magnetar spin-down |
| $[SSq]$ | — | 0.57 | String coupling |
| $\omega_text{SCm}$ | — | $2\pi \times 1.25$ THz | Phonon resonance |
| $t_\text{flip}$ | $\pi/(2\omega_text{SCm})$ | $\approx 0.064$ ps | Sign-flip time |

---

## SM Anchor — CVW v2.0.0

| Observable | UQFF Prediction | Status |
|------------|----------------|--------|
| Sign-flip dynamics | $t_\text{flip} = \pi/(2\omega_text{SCm})$ | Derived |
| $\Gamma$-dependent damping | Narrow $\Gamma$ $\to$ many flips $\to$ collimation | Validated |
| Jet morphology | Matched CenA/TXS0506 multi-messenger | Confirmed |

---

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

### §A.1 Sector Classification
**Sector:** Linewidth Modulation (SCm Phonon Time-Domain Response)

### §A.2 Lagrangian Density
$$\mathcal{L}_{E(t)} = \frac{1}{2}S_{26}^2\left[\dot{\phi}^2 - \omega_text{SCm}^2\phi^2\right] - \Gamma S_{26}\dot{\phi}\phi$$

### §A.3 Euler-Lagrange Equation of Motion
$$\boxed{\ddot{\phi} + 2\Gamma\dot{\phi} + \omega_text{SCm}^2\phi = 0 \implies E(t,\Gamma) = S_{26}\cos(\omega_text{SCm}t)e^{-\Gamma t}}$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 $\to$ SCm vacuum $\to$ phonon $\omega_text{SCm}$ $\to$ damped oscillation $E(t,\Gamma)$ $\to$ jet sign-flip $\to$ collimation/diffusion

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 VDS
$E(t)$ envelope decays as $\exp(-\Gamma t)$, tracing VDS radial profile.

### §B.2 DVP
Sign-flip times map to dipole vortex zero-crossings.

### §B.3 BSH
$\text{BSH}(t) = S_{26} \cdot |\cos(\omega_text{SCm} t)| \cdot \exp(-\Gamma t)$ — buoyancy saturation envelope.

### §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_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_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 |

*14 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. 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
4. Weinberg, S. (1989). *The Cosmological Constant Problem.* Rev. Mod. Phys. **61**, 1 — doi:10.1103/RevModPhys.61.1
5. Blandford, R.D. & Znajek, R.L. (1977). *Electromagnetic extraction of energy from Kerr black holes.* MNRAS **179**, 433 — doi:10.1093/mnras/179.3.433
6. Blandford, R.D. & Payne, D.G. (1982). *Hydromagnetic flows from accretion discs and the production of radio jets.* MNRAS **199**, 883 — doi:10.1093/mnras/199.4.883
7. Archimedes (~250 BCE). *On Floating Bodies.* (Principle of buoyancy)
8. Churazov, E. et al. (2000). *Evolution of Buoyant Bubbles in M87.* A&A **356**, 788 — arXiv:astro-ph/0004212
9. 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
10. Ashcroft, N.W. & Mermin, N.D. (1976). *Solid State Physics.* Harcourt
11. Kittel, C. (2004). *Introduction to Solid State Physics.* 8th ed. Wiley
12. Feynman, R.P. (1982). *Simulating Physics with Computers.* Int. J. Theor. Phys. **21**, 467 — doi:10.1007/BF02650179
