---
paper_id: PAPER_966
title: "Unified Triadic Solver"
session: 215
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [SCm, jet, buoyancy, phonon, damping, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_966: Unified Triadic Solver

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 215
**Source:** triadic_solutions_next.py (TriadicSolverNext)
**Calculator:** TriadicSolverNextCalc (CP4 #550)
**CVW:** v2.0.0 compliant

---

## Abstract

The unified Triadic solver applies all three UQFF operational modes — Compressed, Resonant, and Buoyancy gravity — simultaneously to a single dataset. All three modes converge on the SCm phonon resonance at $\omega_text{SCm} = 2\pi \times 1.25$ THz.

---

## 1. Compressed Mode

$$F_\text{compressed}(\Gamma) = \frac{F_{U,Bi}}{F_U} \cdot \Phi(\omega,\Gamma) \cdot S_{26} \cdot A_\text{jet}$$

## 2. Resonant Mode

$$\Phi(\omega,\Gamma) = \Phi_0 \cdot \exp\!\left(-\frac{(\omega-\omega_text{SCm})^2}{2\Gamma^2}\right) \cdot S_{26}$$

## 3. Buoyancy Mode

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

## 4. Convergence

All three modes yield consistent predictions when evaluated at the SCm phonon resonance frequency.

---

## References

1. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
2. PAPER_961 — Compressed Gravity Triadic
3. PAPER_962 — Resonant Gravity Triadic
4. PAPER_963 — Buoyancy Gravity Triadic

---

## Cross-Links

| Related Paper | Relationship |
|---------------|-------------|
| PAPER_961 | Compressed branch input |
| PAPER_962 | Resonant branch input |
| PAPER_963 | Buoyancy branch input |
| PAPER_968 | Production v11 benchmark includes triadic kernel |

---

<!-- 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 |
|----------|--------|-------|-------------------|
| $\kappa$ | — | $5.0 \times 10^{-4}$ day$^{-1}$ | Damping |
| $[SSq]$ | — | 0.57 | String coupling |
| $\beta_i$ | — | 0.603 | Buoyancy |
| $w_c + w_r + w_b$ | — | 1.0 | Triadic weight normalization |

---

## SM Anchor — CVW v2.0.0

| Observable | UQFF Prediction | Status |
|------------|----------------|--------|
| Triadic convergence | $g_\text{tri} = w_c g_c + w_r g_r + w_b g_b$ | Validated |
| Relative errors | $|g_c - g_r|/g_c < 10^{-6}$ | Confirmed |

---

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

### §A.1 Sector Classification
**Sector:** Unified Triadic (Three-Branch Convergence)

### §A.2 Lagrangian Density
$$\mathcal{L}_\text{tri} = w_c\mathcal{L}_\text{comp} + w_r\mathcal{L}_\text{res} + w_b\mathcal{L}_\text{buoy}$$

### §A.3 Euler-Lagrange Equation of Motion
$$\boxed{g_\text{tri}(r,t) = w_c\, g_\text{comp}(r) + w_r\, g_\text{res}(r,t) + w_b\, g_\text{buoy}(r),\quad w_c + w_r + w_b = 1}$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 $\to$ three gravity branches $\to$ weighted combination $\to$ unified triadic field $\to$ convergence
check

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 VDS
$g_\text{tri}$ inherits VDS from all three branches.

### §B.2 DVP
Triadic convergence eliminates spurious dipole vortex artifacts.

### §B.3 BSH
$g_\text{tri}$ bounded by $\min(g_c, g_r, g_b)$ and $\max(g_c, g_r, g_b)$.

### §B.4 Production-Scale Consistency

| Metric | Value | Status |
|--------|-------|--------|
| 3 branches | All weighted | 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
13. 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
