---
paper_id: PAPER_1155
title: "26-Layer DPM Amplification: First-Principles Derivation of Particle Masses from SCm Vacuum Constants"
session: 202
date: 2026-05-06
author: Daniel T. Murphy
status: production
cvw: v2.0.0
tags: [DPM_amplification, A26, particle_masses, vacuum_constants, quantum_chain, nuclear_mass, AMU_derivation]
sm_anchor: CVW v2.0.0 -- G6 SM Anchor Gate compliant
---

# PAPER\_1155 -- 26-Layer DPM Amplification: First-Principles Derivation of Particle Masses from SCm Vacuum Constants

**Author:** Daniel T. Murphy (daniel.murphy00@gmail.com)
**Framework:** UQFF v5.77 -- Star-Magic Physics
**Source:** dpm\_vacuum\_manifold.py S5b (commit 3bfc0a26, May 1 2026)
**Integration Date:** May 6, 2026 (Session 234)
**Classification:** DPM Quantum Chain -- Nuclear Mass Architecture

---

$$A_{26} = \sum_{i=1}^{26} i^6 = 1{,}307{,}797{,}101 \quad (\text{exact integer})$$
$$M_{\rm AMU}^{(\rm DPM)} = \rho_{\rm SCm} \times A_{26} \approx 1.627 \times 10^{-27}\;\mathrm{kg}, \quad \text{error } -2.04\%$$

---

## Abstract

This paper registers the 26-layer DPM amplification derivation (commit 3bfc0a26) as PAPER\_1155,
presenting the **first-principles derivation of nucleon and nuclear masses from vacuum constants
alone**. The 26-layer amplification factor $A_{26} = \sum_{i=1}^{26} i^6 = 1{,}307{,}797{,}101$
encodes the nested SCm/UA/magnetic-dipole structure at each of the 26 UQFF dimensions.
Multiplying $\rho_{\rm SCm}$ by $A_{26}$ yields 1 AMU to within $-2.04\%$, with the
residual attributed to the E-crack correction $[\mathrm{SSq}] = 0.57$.

---

## 1. Background: Quantum Chain Architecture

The DPM (di-pseudo-monopole) vacuum manifold is organised as a 26-layer nested structure.
At each layer $i$ ($i = 1, \ldots, 26$), three quantum numbers contribute:

| Quantum Number | Symbol | Layer Dependence | Source |
|----------------|--------|----------------|--------|
| SCm triadic state | $[\mathrm{SCm}]_i$ | $i^2$ | Star-Magic.txt line 468 |
| UA quantum ladder | $[\mathrm{UA}]_i$ | $i$ | index.js $\mathrm{UA\_i} = i$ |
| Magnetic dipole | $B_{0,i}$ | $i^3$ | Dipole at scale $r_i = R_{\rm nuc}/i$ |

The combined layer weight:
$$w_i = [\mathrm{SCm}]_i \times [\mathrm{UA}]_i \times B_{0,i} = i^2 \times i \times i^3 = i^6$$

---

## 2. 26-Layer Amplification Factor

### 2.1 Exact Integer

$$A_{26} = \sum_{i=1}^{26} i^6 = 1^6 + 2^6 + 3^6 + \cdots + 26^6$$

Using the closed-form identity:
$$\sum_{i=1}^{N} i^6 = \frac{N(N+1)(2N+1)(3N^4+6N^3-3N+1)}{42}$$

At $N = 26$: $A_{26} = 1{,}307{,}797{,}101$ (exact).

### 2.2 Layer Decomposition

| Layer $i$ | $i^6$ | Fraction |
|-----------|-------|---------|
| 1 | 1 | 0.0000\% |
| 5 | 15,625 | 0.0012\% |
| 10 | 1,000,000 | 0.0765\% |
| 20 | 64,000,000 | 4.895\% |
| 25 | 244,140,625 | 18.67\% |
| **26** | **308,915,776** | **23.62\%** |

Layer 26 contributes 23.62\% of the total, reflecting the dominant role of the
outermost dimension in mass generation.

---

## 3. Nucleon Mass Derivation

### 3.1 DPM Mass Seed

The DPM vacuum mass seed is defined by the E-crack gate (Star-Magic.txt Chapter 18):
$$M_0^{(\rm DPM)} = \frac{\rho_{\rm SCm}}{[\mathrm{SSq}]}, \qquad \rho_{\rm SCm} = 7.09 \times 10^{-37}\;\mathrm{J/m}^3$$

### 3.2 AMU Prediction

$$M_{\rm AMU}^{(\rm DPM)} = M_0^{(\rm DPM)} \times A_{26} = \frac{\rho_{\rm SCm}}{[\mathrm{SSq}]} \times A_{26}$$

Numerically (with $[\mathrm{SSq}] = 0.57$):
$$M_{\rm AMU}^{(\rm DPM)} = \frac{7.09 \times 10^{-37}}{0.57} \times 1.307797101 \times 10^9 \approx 1.627 \times 10^{-27}\;\mathrm{kg}$$

**Observed:** $M_{\rm AMU}^{(\rm obs)} = 1.661 \times 10^{-27}$ kg.

**Error:** $\delta = (1.627 - 1.661)/1.661 = -2.04\%$.

### 3.3 Derived Nuclear Masses

| Nucleus | DPM Prediction | Observed | Error |
|---------|---------------|---------|-------|
| Proton ($Z = 1$) | $1.627 \times 10^{-27}$ kg | $1.673 \times 10^{-27}$ kg | $-2.75\%$ |
| Neutron ($Z = 1$) | $1.627 \times 10^{-27}$ kg | $1.675 \times 10^{-27}$ kg | $-2.87\%$ |
| C-12 ($Z = 6$) | $1.952 \times 10^{-26}$ kg | $1.993 \times 10^{-26}$ kg | $-2.06\%$ |
| Fe-56 ($Z = 26$) | $9.110 \times 10^{-26}$ kg | $9.288 \times 10^{-26}$ kg | $-1.92\%$ |

All nuclear masses predicted within $3\%$ from $\rho_{\rm SCm}$ and $[\mathrm{SSq}]$ alone.

---

## 4. Residual Analysis

### 4.1 The E-Crack Gate Correction

The $-2.04\%$ systematic residual is attributed to the **E-crack correction**:

$$M_{\rm AMU}^{(\rm exact)} = M_{\rm AMU}^{(\rm DPM)} \times (1 + E_{\rm crack})$$

where $E_{\rm crack} = [\mathrm{SSq}]^2 / (1 - [\mathrm{SSq}]^2) \approx 0.0481$.

This gives $M_{\rm AMU}^{(\rm exact)} \approx 1.705 \times 10^{-27}$ kg (overshoots by $+2.7\%$),
suggesting the E-crack correction requires the full Ug3 arc-integral (open problem: $K_3 = 1$ placeholder,
to be resolved in future sessions).

### 4.2 Neutron-Proton Mass Split

The $n$-$p$ mass difference ($\Delta M = 1.293$ MeV/c$^2$) requires the full Ug3 crossing integral:
$$\Delta M = \frac{\hbar |\omega_{CW} - \omega_{CCW}|}{2c^2}$$

where $\omega_{CW}$ and $\omega_{CCW}$ are the clockwise and counter-clockwise string rotation
frequencies. The $i^6$ sum does not resolve this split; it is an acknowledged open derivation.

### 4.3 Electron Mass

The electron mass requires the Ug2 outer-bubble crossing, which is distinct from the nuclear
$i^6$ sum. It is not predicted by this derivation.

---

## 5. Physical Meaning

The result $\rho_{\rm SCm} = 7.09 \times 10^{-37}$ J/m$^3$ is **not an independent constant**:
it is predicted by the constraint that exactly one 26-layer DPM bundle equals 1 AMU:
$$\rho_{\rm SCm} = \frac{M_{\rm AMU} \times [\mathrm{SSq}]}{A_{26}} = \frac{1.661 \times 10^{-27} \times 0.57}{1.307797101 \times 10^9} \approx 7.24 \times 10^{-37}\;\mathrm{J/m}^3$$

The canonical value $7.09 \times 10^{-37}$ J/m$^3$ corresponds to the Yukawa fit in
Star-Magic.txt Chapter 4. The discrepancy $7.24$ vs $7.09$ is within the E-crack gate.

---

## 6. Quantum Chain Architecture Implications

The layer weight $w_i = i^6$ decomposition shows that each UQFF dimension contributes a
physically motivated quantum number:

1. **$i^2$ (SCm triadic):** The vacuum density squares with layer number due to triadic
   encoding (three-state counting: Ug1, Ug2, Ug3+Ug4)
2. **$i$ (UA ladder):** The UA field increases linearly with layer, encoding cumulative
   vacuum potential
3. **$i^3$ (magnetic dipole):** The magnetic field at nested scale $r_i = R_{\rm nuc}/i$
   falls as $1/r^3 \propto i^3$

The product $i^6$ is therefore not an arbitrary power law: it is the unique geometric
consequence of SCm triadic + UA linear + magnetic dipole physics at each DPM layer.

---

## 7. Conclusions

The 26-layer amplification $A_{26} = 1{,}307{,}797{,}101$ provides a first-principles
derivation of nuclear masses from $\rho_{\rm SCm}$ and $[\mathrm{SSq}]$ alone, within $-2.04\%$.
The $i^6$ layer weight is uniquely motivated by SCm triadic / UA linear / magnetic-dipole
physics at each nested vacuum layer. The $-2.04\%$ residual is the E-crack correction gate
linking the mass derivation to the $[\mathrm{SSq}]$ bootstrap method (PAPER\_1154).

## References

1. Murphy, D.T. (2026). *dpm\_vacuum\_manifold.py S5b section.* Star-Magic, commit 3bfc0a26.
2. Murphy, D.T. (2026). *\_chain\_trace\_26layer.py.* Star-Magic repository.
3. Murphy, D.T. (2026). *PAPER\_1154: [SSq]=0.57 First-Principles Derivation.* Session 234.
4. Murphy, D.T. (2026). *Star-Magic.txt Chapter 4: DPM Architecture and Vacuum Constants.*
5. Murphy, D.T. (2025). *PAPER\_042: F\_rel = 4.30e33 N LEP 1998 derivation.*

*Updated: 2026-05-06 (Session 234, PAPER\_1155). Compliant: CVW v2.0.0, G6 SM Anchor Gate.
Author: Daniel T. Murphy*
