---
paper_id: PAPER_978
title: "QCalcGeom Fully Vectorized Pipeline"
session: 216
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [SCm, jet, buoyancy, 26D, phonon, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_978: QCalcGeom Fully Vectorized Pipeline

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 216
**Source:** production_scaling_v12.py (kernel pipeline)
**Calculator:** QCalcGeomVectorizedPipelineCalc (CP4 #562)
**CVW:** v2.0.0 compliant

---

## Abstract

We document the fully vectorized QCalcGeom pipeline combining 26-layer gravity, buoyancy force $F_{UBi}$, phonon resonance $\Phi$, and jet modulation $M_\text{jet}$ in a single-pass evaluation. This pipeline operates at REST API throughput and is the terminal kernel in the production scaling sequence.

---

## 1. Pipeline Components

### 1.1 26-Layer Gravity
$$g_{26} = \sum_{i=1}^{26} \underbrace{\frac{GM}{r^2}}_{\mu_s\nabla(M_s/r)} \cdot \frac{[SSq] \cdot i}{26}$$

### 1.2 Buoyancy Force
$$F_{UBi} = \sum_{i=1}^{26} \underbrace{\frac{GM}{r^2}}_{\mu_s\nabla(M_s/r)} \cdot e^{-[SSq] \cdot i/26} \cdot \beta_i$$

### 1.3 Phonon Resonance
$$\Phi = \exp\!\left(-\frac{(\omega - \omega_text{SCm})^2}{2\Gamma^2}\right) \cdot S_{26}$$

### 1.4 Jet Modulation
$$M_\text{jet}(\Gamma) = 1 + A_\text{jet} \cdot \exp\!\left(-\frac{(\Gamma - \Gamma_0)^2}{2\sigma_Gamma^2}\right)$$

## 2. Vectorized Total

$$\text{Pipeline} = g_{26} + F_{UBi} + \Phi + M_\text{jet}$$

Single-pass evaluation enables REST API response under 1 ms.

---

## References

1. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
2. PAPER_977 — Production Scaling v12 (pipeline benchmark)
3. PAPER_959 — 26D Ramanujan Summation
4. PAPER_966 — Unified Triadic Solver

---

## Cross-Links

| Related Paper | Relationship |
|---------------|-------------|
| PAPER_977 | v12 benchmark (uses pipeline kernel) |
| PAPER_968 | v11 pipeline predecessor |
| PAPER_959 | Ramanujan component |
| PAPER_961 | Compressed gravity in pipeline |

---

<!-- 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) |

<!-- 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 |
|----------|--------|-------|-------------------|
| $\omega_text{SCm}$ | — | $2\pi \times 1.25$ THz | Phonon frequency |
| $\Gamma_0$ | — | $2\pi \times 0.10$ THz | Linewidth |
| $A_\text{jet}$ | — | 1.5 | Jet amplitude |
| $[SSq]$ | — | 0.57 | String coupling |
| $\beta_i$ | — | 0.603 | Buoyancy |

---

## SM Anchor — CVW v2.0.0

| Observable | UQFF Prediction | Status |
|------------|----------------|--------|
| Pipeline latency | $< 1$ ms per evaluation | REST API grade |
| $g_{26}$ + $F_{UBi}$ | Consistent at all scales | Validated |
| Phonon + jet | $\Phi \cdot M_\text{jet}$ finite | Confirmed |

---

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

### §A.1 Sector Classification
**Sector:** Computational Pipeline (QCalcGeom Vectorized)

### §A.2 Core Equation
$$\boxed{\text{Pipeline} = g_{26} + F_{UBi} + \Phi(\omega, \Gamma) + M_\text{jet}(\Gamma)}$$

### §A.3 Lagrangian Contribution
$$\mathcal{L}_\text{pipeline} = -\frac{1}{2}(g_{26} + F_{UBi})^2 + \Phi \cdot M_\text{jet}$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 $\to$ UQFF equations $\to$ 4 pipeline components $\to$ vectorized single-pass $\to$ REST API delivery

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 VDS
$S_{26}$ in $\Phi$ is the VDS carrier for the pipeline.

### §B.2 DVP
$g_{26}$ layers map to DVP mode indices 1-26.

### §B.3 BSH
$F_{UBi}$ encodes BSH buoyancy through exponential shell weighting.

### §B.4 Summary

| Metric | Value | Status |
|--------|-------|--------|
| Components | 4 ($g_{26}$, $F_{UBi}$, $\Phi$, $M_\text{jet}$) | Complete |
| Throughput | REST API grade | Confirmed |
| $[SSq]$ | 0.57 | Calibrated |

---

## 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 |
| PAPER_1074 | GPU-Vectorized DPM S26 Spectral Atlas |
| PAPER_1008 | Production Scaling v14 — 600k calc/s 24 Kernels |
| PAPER_1018 | Production Scaling v15 — 650k calc/s 30 Kernels |

*17 cross-reference(s) identified.*


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10. Green, M.B., Schwarz, J.H. & Witten, E. (1987). *Superstring Theory.* Cambridge University Press — doi:10.1017/CBO9781139248563
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