# PAPER_026c: Sterile Neutrino Mass Derivation in the UQFF Framework (short variant)

> **RE-ID NOTE (B52 ruling, Daniel, 2026-09-03):** this 90-line late-era variant
> previously claimed the PAPER_026 identifier. The Session-0 sequential file
> (PAPER_026_Sterile_Neutrino_Mass_Generation_UQFF.md, 821 lines, 7.1 keV RGE
> spectrum) is canonical PAPER_026. This file is re-identified as PAPER_026c.
> Known issue (disclosed, content unchanged): the m_s = rho_SCm*S_26^(3)*Phi_res/c^2
> chain as written yields 5.4e8 eV, not 5.4 keV (rho_SCm is J/m^3, not J - a 1e5
> unit-dimension issue). Preserved as history per Rule 9.

**Star Magic UQFF Framework**

**Author:** Daniel Murphy
**Date:** 2026

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## Abstract

We derive sterile neutrino mass bounds within the Unified Quantum Field Framework (UQFF) using the 26-dimensional vacuum density series (VDS), the $F_{U,Bi,i}$ buoyancy integral, and negative-time modulation $\cos(\pi t_n)$. The SCm vacuum density $\rho_{\text{vac,SCm}} = 7.09 \times 10^{-37}\ \text{J/m}^3$ provides the effective mass-generation potential for sterile neutrino mixing, yielding mass estimates in the keV range consistent with dark matter candidates. Oscillation probabilities are computed via the SCm phonon resonance at 1.25 THz.

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## 1. SCm Vacuum as Mass-Generation Potential

The sterile neutrino mass $m_s$ arises from the SCm vacuum energy density interacting with the buoyancy force:

$$m_s = \frac{\rho_{\text{vac,SCm}} \cdot S_{26}^{(3)} \cdot \Phi_{\text{res}}}{c^2}$$

where $\rho_{\text{vac,SCm}} = 7.09 \times 10^{-37}\ \text{J/m}^3$, $S_{26}^{(3)} = 1.4531 \times 10^{26}$, and $\Phi_{\text{res}} = 0.84$.

Numerically:

$$m_s c^2 = 7.09 \times 10^{-37} \times 1.4531 \times 10^{26} \times 0.84 \approx 8.66 \times 10^{-11}\ \text{J} \approx 5.4\ \text{keV}$$

This falls within the Tremaine-Gunn lower bound ($m_s > 0.5\ \text{keV}$) and the X-ray line observation range ($\sim 3.5\ \text{keV}$ from XMM-Newton, arXiv:1402.4119).

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## 2. Buoyancy-Suppressed Mixing Angle

The active-sterile mixing angle $\theta$ is modulated by $F_{U,Bi,i}$ buoyancy:

$$\sin^2(2\theta)_{\text{eff}} = \sin^2(2\theta_0) \cdot \left|\cos(\pi t_n)\right| \cdot (1 + \beta_i \cdot F_{TRZ})$$

where $\beta_i = 0.6$, $F_{TRZ} = 0.1$, and $t_n < 0$ (negative-time gate). This suppresses the mixing angle in active-sector interactions while enhancing it during the negative-time resonance window, providing a natural mechanism for the MSW (Mikheyev-Smirnov-Wolfenstein) suppression pattern in stellar environments.

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## 3. Oscillation Probability via SCm Phonon

The sterile neutrino oscillation probability between mass eigenstates $\nu_1$ and $\nu_s$ is:

$$P(\nu_1 \to \nu_s) = \sin^2(2\theta) \cdot \sin^2\!\left(\frac{\Delta m^2 L}{4E}\right) \cdot S_{26}^{(3)} \cdot \Phi_{\text{res}} \cdot \left|\cos(\pi t_n)\right|$$

The additional $S_{26}^{(3)} \cdot \Phi_{\text{res}}$ factor represents the 26D vacuum resonance amplification of the oscillation probability in the SCm framework. At 1.25 THz phonon resonance this coupling is maximized.

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## 4. Dark Matter Production Rate

The sterile neutrino dark matter production rate via the Dodelson-Widrow mechanism, enhanced by SCm resonance:

$$\Gamma_{\text{DW-SCm}} = \frac{G_F^2 T^5}{\pi} \cdot \sin^2(2\theta)_{\text{eff}} \cdot \frac{S_{26}^{(3)} \cdot \Phi_{\text{res}}}{\rho_{\text{vac,UA}}/\rho_{\text{vac,SCm}}}$$

where $\rho_{\text{vac,UA}} = 7.09 \times 10^{-36}\ \text{J/m}^3$ provides the UA vacuum density ratio, $G_F = 1.166 \times 10^{-5}\ \text{GeV}^{-2}$ is the Fermi constant, and $T$ is the plasma temperature.

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## 5. VDS Convergence and Mass Bound

The 26-dimensional vacuum density series:

$$\text{VDS}([SSq]) = \sum_{n=1}^{\infty} \frac{[SSq]^n}{n^{26}} = \text{Li}_{26}(0.57)$$

converges absolutely since $|[SSq]| = 0.57 < 1$. The Ramanujan acceleration operator $S_{26}^{(3)}$ amplifies the raw $\text{Li}_{26}(0.57)$ value to the physical scale $S_{26}^{(3)} = 1.4531 \times 10^{26}$.

The sterile neutrino mass window consistent with UQFF is:

$$1\ \text{keV} \lesssim m_s \lesssim 100\ \text{keV}$$

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## 6. Observational Consistency

- **3.5 keV X-ray line** (XMM-Newton, Chandra): consistent with $m_s \approx 7\ \text{keV}$, arXiv:1402.4119
- **Tremaine-Gunn bound**: $m_s > 0.5\ \text{keV}$ satisfied
- **Lyman-$\alpha$ forest constraints**: $m_s > 2\ \text{keV}$ (arXiv:0703516) consistent with SCm prediction of $5.4\ \text{keV}$
- **UQFF calibrated constants**: $\kappa = 5.0 \times 10^{-4}\ \text{day}^{-1}$, $[SSq] = 0.57$, $\beta_i = 0.6$

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## References

1. Dodelson, S. & Widrow, L.M. (1994). Sterile neutrinos as dark matter. *Phys. Rev. Lett.* **72**, 17.
2. Boyarsky, A. et al. (2014). Unidentified line in X-ray spectra. arXiv:1402.4119.
3. SCm vacuum manifold: `scm_vacuum_manifold.py`
4. VDS derivation: `vds_dvp_bsh_symbolic_proofs.py`
