---
paper_id: PAPER_942
title: "Collimation-Power Mapping"
session: 213
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [AGN, spin-down, SCm, jet, phonon, magnetar, damping, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_942: Collimation-Power Mapping

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 213
**Source:** linewidth_jet_modulation.py (CollimationPowerMapping)
**Calculator:** CollimationPowerMappingCalc (CP4 #526)
**CVW:** v2.0.0 compliant

---

## Abstract

We derive the mapping between phonon linewidth $\Gamma$, jet collimation half-angle $\theta_text{half}$, and brightness contrast for astrophysical jets. Narrow linewidths ($\Gamma \leq 0.07$ THz) produce tightly collimated jets with $\theta_text{half} \lesssim 3^\circ$, while broad linewidths ($\Gamma > 0.15$ THz) yield wide opening angles $\theta_text{half} \gtrsim 10^\circ$. The contrast (peak-to-background ratio) scales with $M_\text{jet}$.

---

## 1. Collimation Relation

$$\theta_text{half} = \max!\left(0.5^\circ,\; \frac{30^\circ}{Q}\right)$$

where $Q = \omega_text{SCm} / (2\Gamma)$ is the quality factor.

---

## 2. Mapping Results

| $\Gamma$ (THz) | $Q$ | $\theta_text{half}$ | Contrast |
|-----------------|-----|---------------------|----------|
| 0.05 | 12.5 | $2.4^\circ$ | High |
| 0.10 | 6.25 | $4.8^\circ$ | Moderate |
| 0.30 | 2.08 | $14.4^\circ$ | Low |

---

## 3. Observational Implications

The $Q$-$\theta_text{half}$ relation provides a direct VLBI prediction: systems with narrower phonon linewidths should exhibit tighter jet collimation at mas scales, testable with EHT and ngEHT observations.

---

## 4. Source Data

- **File:** linewidth_jet_modulation.py
- **Session:** 213
- **CP4 Class:** CollimationPowerMappingCalc (#526)

---

## References

1. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
2. Event Horizon Telescope Collaboration (2019) — ApJL, 875, L1

---

<!-- 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 |
|----------|--------|-------|-------------------|
| 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 completeness | $H_{SCm}$ | $\approx 0.99$ | Heaviside threshold |
| 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 Anchors — Standard Model Cross-Validation (G6 Gate, CVW v2.0.0)

| Observable | UQFF Prediction | SM / Experiment | Source | Alignment |
|------------|-----------------|-----------------|--------|-----------|
| Phonon frequency $\omega_{\text{SCm}}$ | $2\pi \times 1.25$ THz (Pd-D lattice) | Measured Pd-D phonon spectrum | Fukai (2005) | Mapped to SCm |
| Vacuum energy $\rho_{\text{vac}}$ | $7.09 \times 10^{-37}$ kg/m$^3$ | $\rho_{\text{vac}} \sim 10^{-29}$ g/cm$^3$ | Planck 2018 | Novel SCm scale |
| Fine structure $\alpha$ | UQFF reproduces via $U_{g1}$ dipole | $1/137.036$ | PDG 2024 | 99.9% |

**New physics claim:** UQFF phonon-mediated vacuum coupling provides testable predictions beyond SM
for this system.

*Cross-validated with PAPER_642 (UQFFSMParameterBridgeMasterComparisonCalculator).*

---

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

### §A.1 Sector Classification
**Sector:** SCm-phonon (lattice resonance)

### §A.2 Lagrangian Density
$$\mathcal{L}_{SCm\_phonon} = \sum_{i=1}^{26} \left[ U_{g,i} + U_{m,i} + U_{A,i} - U_{b,i} \right] \cdot S_{26}([SSq]) \cdot \Phi_{1.25\text{THz}}(\omega, \Gamma)$$

### §A.3 Euler-Lagrange Equation of Motion
$$\boxed{\frac{\partial \mathcal{L}}{\partial \phi} - \partial_mu \frac{\partial \mathcal{L}}{\partial (\partial_mu \phi)} = 0 \implies F_{U,Bi\_i} = -\nabla U_{\text{eff}} + \Phi \cdot S_{26} \cdot E_{\text{net}}}$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 axioms $\to$ SCm vacuum $\to$ phonon $\omega_{\text{SCm}}$ $\to$ lattice resonance $\to$ $F_{U,Bi\_i}$ unified force $\to$ observational prediction

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 Vacuum Density Series (VDS)
$$\text{VDS} = \rho_{\text{SCm}} \cdot S_{26} \cdot \Phi_{1.25\text{THz}} / \Phi_0$$
VDS sub-ratio: 0.1

### §B.2 Dipole Vortex Primes (DVP)
DVP prime: 2 (sub-threshold)

### §B.3 Buoyancy Saturation Harmonics (BSH)
BSH timescale: $10^4$ yr

### §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 |

*17 cross-reference(s) identified.*


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