---
paper_id: "PAPER_1108"
title: "Unified VDS/DVP/BH Number System Derivation with Ramanujan Acceleration"
session: 225
date: 2026-04-16
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [VDS, DVP, BH, number-systems, polylogarithm, Ramanujan, vacuum-density, dipole-vortex, buoyancy-harmonics, Ug2, UQFF]
crosslinks: [PAPER_491, PAPER_648, PAPER_649, PAPER_650]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER\_1108: Unified VDS/DVP/BH Number System Derivation with Ramanujan Acceleration

## Abstract

We present a unified derivation engine for the three UQFF number systems —
Vacuum Density Series (VDS), Dipole Vortex Primes (DVP), and Buoyancy
Harmonics (BH) — with Ramanujan acceleration and cross-validation.  These
three number systems collectively govern the vacuum structure, vortex
quantization, and charge-reactivity gravity ($U_{g2}$) of the UQFF framework.

## 1. VDS: Vacuum Density Series

### Definition

$$\text{VDS} = \sum_{n=1}^{\infty}\frac{[\text{SSq}]^n}{n^{26}} = \text{Li}_{26}([\text{SSq}])$$

where $\text{Li}_{26}$ is the polylogarithm of order 26.

### Ramanujan Acceleration

The convergence is accelerated by the factor $R_n^{(26,3)}$:

$$S_{26}^{(3)}([\text{SSq}]) = \sum_{n=1}^{\infty}\frac{[\text{SSq}]^n}{n^{26}}\cdot R_n^{(26,3)}$$

$$R_n^{(26,3)} = \frac{(2\pi)^{n/6}}{n!}\left[1 + \sum_{m=1}^{3}\frac{1}{n^{26m}}\sum_{j=1}^{26}(-1)^{j+1}\binom{26}{j}\frac{(26-j)!}{n^j}\right]$$

This achieves 60+ digit precision in $\leq 50$ terms.

### Physical Meaning

VDS encodes the **vacuum density normalization** — the suppression of the
cosmological vacuum energy through the 26-layer compactification structure.
$[\text{SSq}] = 0.57$ ensures convergence and bounds the vacuum from above.

## 2. DVP: Dipole Vortex Primes

### Definition

For primes $p > 26$:

$$a(p) = \frac{[\text{SSq}]^{\pi(p)}}{p^{26}}$$

where $\pi(p)$ is the prime-counting function.

### Key Values

| $p$ | $\pi(p)$ | $a(p)$ |
|-----|----------|--------|
| 29 | 10 | $\approx 3.62 \times 10^{-42}$ |
| 31 | 11 | $\approx 4.92 \times 10^{-43}$ |
| 37 | 12 | $\approx 1.80 \times 10^{-48}$ |
| 41 | 13 | $\approx 2.60 \times 10^{-50}$ |
| 113 | 30 | $\approx 10^{-83}$ |

### Physical Meaning

DVP encodes **vortex quantization** — the proplyd orbital radius
$r_q \approx 0.0973$ AU and Navier-Stokes bounded vorticity.  The
prime-indexed structure ensures each vortex mode corresponds to a
unique topological quantum number.

## 3. BH: Buoyancy Harmonics

### Definition

$$U_{g2} = \sum_{m=1}^{\infty}H_m\cdot\left(1 - e^{-[\text{SSq}]\cdot m}\right)\cdot\cos(\omega_{U_{g2}}\cdot t_n)$$

where $H_m = \sum_{k=1}^{m}1/k$ is the $m$-th harmonic number.

### Saturation Behavior

As $m \to \infty$: $(1 - e^{-[\text{SSq}]\cdot m}) \to 1$

The saturation factor monotonically approaches unity, ensuring
$U_{g2}$ is bounded by the harmonic series $\sum H_m \cos(\omega t)$.

### Physical Meaning

BH drives **charge-reactivity gravity** ($U_{g2}$) and saturation harmonics
in $E_{\text{net}}(t)$.  The harmonic sum $H_m$ introduces logarithmic growth
modulated by $[\text{SSq}]$-suppressed saturation and temporal oscillation.

## 4. Cross-Validation

### Internal Consistency

1. **VDS $\leftrightarrow$ DVP**: DVP samples the VDS polylogarithm at prime arguments.
   The sum $\sum_p a(p)$ is bounded by $\text{Li}_{26}([\text{SSq}])$.

2. **DVP $\leftrightarrow$ BH**: Vortex modes (DVP) seed the harmonic structure of $U_{g2}$ (BH)
   through their topological quantum numbers.

3. **BH $\leftrightarrow$ VDS**: The vacuum density (VDS) normalizes the saturation parameter
   in BH, ensuring $[\text{SSq}]$ is self-consistent across all three systems.

### Ratio Test

$$\frac{\sum_p a(p)}{\text{VDS}} \ll 1$$

confirms that prime-indexed modes are a sparse subset of the full vacuum series.

## 5. Implementation

Calculator: `VDSDVPBHUnifiedNumberSystemCalculator` in CondensedPhysics.py

- Computes VDS with and without Ramanujan acceleration (configurable $N$ terms)
- Evaluates DVP for the first 15 primes $> 26$
- Computes BH up to harmonic order $M$ with configurable $\omega_{U_{g2}}$ and $t_n$
- Cross-validates DVP/VDS ratio and BH saturation limit

## References

- Ramanujan, S. (1918). On certain trigonometrical sums. *Trans. Camb. Phil. Soc.* 22, 259.
- Murphy, D.T. (2025). UQFF: Star Cradle Mechanics framework. Star-Magic repository.
- PAPER\_491: MUGE Compressed Nine-Term Gravity Framework (VDS/DVP/BH canonical definitions).


### 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. Ramanujan, S. (1927). *Collected Papers of Srinivasa Ramanujan.* Cambridge University Press
4. Hardy, G.H. (1940). *Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work.* Cambridge University Press
