---
title: "SCm Dark Energy with Phonon Linewidth \Gamma-Modulation"
paper_id: PAPER_1076
session: 224
author: Daniel Murphy
framework: UQFF v5.26+
status: complete
sm_anchors: [SM-DARK-ENERGY, SM-SCM, SM-PHONON, SM-LCDM]
gate_compliance: [G1, G2, G3, G4, G5, G6]
cvw_version: "2.0.0"
---

# PAPER_1076: SCm Dark Energy with Phonon Linewidth $\Gamma$-Modulation

## Abstract

We derive a dynamic dark energy model from SCm phonon linewidth evolution,
producing $E_{\text{net}}(t, \Gamma)$ — a $\Gamma$-coupled vacuum energy density that
replaces the cosmological constant $\Lambda$. The phonon linewidth $\Gamma(t)$ broadens
with cosmic expansion, causing the dark energy equation of state $w(z)$ to deviate
from $w = -1$ in a testable manner. We quantify the $\Lambda$CDM$\to$SCm replacement with
integrated deviation analysis and BIC penalty assessment.

## §1 Phonon Linewidth Evolution

The SCm phonon linewidth evolves as:

$$
\Gamma(t) = \Gamma_0 \cdot \left(1 + \alpha \cdot \frac{t}{t_H}\right)
$$

where $\alpha$ is the broadening rate and $t_H = 1/H_0 \approx 4.56 \times 10^{17}$ s.

At present epoch ($t = t_H$, $\alpha = 0.1$):
- $\Gamma_0 = 6.283 \times 10^{11}$ rad/s
- $\Gamma(t_H) = 6.912 \times 10^{11}$ rad/s (+10%)

## §2 $\Gamma$-Coupled E_net

$$
E_{\text{net}}(t, \Gamma) = E_0 \cdot e^{(\kappa + [\text{SSq}]/26)t} \cdot S_{26} \cdot \Phi(\Gamma(t)) \cdot \left(\frac{2F_{U,Bi}}{F_U} - 1\right)
$$

The phonon resonance profile with linewidth coupling:

$$
\Phi(\Gamma(t)) = \exp\left(-\frac{(\Gamma(t) - \Gamma_0)^2}{2\sigma_G^2}\right) \cdot S_{26}^{(3)}
$$

**Key result:** $\Gamma$ coupling suppresses $|E_{\text{net}}|$ relative to bare evolution
($|E_{\text{net},\Gamma}| \leq |E_{\text{net,bare}}|$), providing a natural damping
mechanism for dark energy.

## §3 Equation of State w(z)

$$
w(z) = -1 + \frac{1}{3}\frac{d\ln\Phi(\Gamma(t(z)))}{d\ln a}
$$

| Redshift | w_SCm | $w_{\Lambda}$CDM | $\delta$w |
|----------|-------|--------|-----|
| z = 0 | -1.0077 | -1.0 | 0.0077 |
| z = 1 | -1.0037 | -1.0 | 0.0037 |
| z = 2 | -1.0023 | -1.0 | 0.0023 |
| z = 3 | -1.0016 | -1.0 | 0.0016 |

The deviation $\delta w$ is small but systematically negative of phantom ($w < -1$),
distinguishing SCm from quintessence models ($w > -1$).

## §4 $\Lambda$CDM Replacement Analysis

| Metric | Value | Interpretation |
|--------|-------|---------------|
| $\int|\delta w|\,dz$ | 0.0036 | Total integrated deviation |
| Max $|\delta w|$ | 0.0077 | At z = 0 |
| BIC penalty (1 free param) | 6.9 | Must improve $\chi$2 by >6.9 |
| $d_L$ ratio at z=1 | ~1.00004 | Subtle luminosity distance shift |

### Testability

- **DESI BAO:** w(z) bins at z = 0.3, 0.5, 0.7, 1.0 — requires $\sigma$(w) < 0.01
- **Euclid WL:** w0–wa plane constrains dynamical dark energy
- **SNe Ia (Pantheon+):** $\Delta$H0 from SCm vs $\Lambda$ via d_L ratio

## §5 Comparison with $\Lambda$CDM

| Property | $\Lambda$CDM | SCm (this work) |
|----------|------|-----------------|
| Free parameters | 0 ($\Lambda$ fixed) | 1 ($\alpha$) |
| w(z) | -1 (constant) | -1 + $\delta$w(z) |
| Dark energy density | Constant | Evolves via $\Gamma$(t) |
| Physical mechanism | Vacuum energy | SCm phonon damping |
| Phantom crossing | No | No ($\delta$w > 0 $\to$ w < -1) |
| Late-time behavior | Eternal acceleration | $\Gamma$-damped deceleration |

## §6 Calibration Table

| Parameter | Value | Source |
|-----------|-------|--------|
| $\alpha$ | 0.1 | Linewidth broadening rate |
| $\Gamma$0 | 6.283$\times$1011 rad/s | SCm phonon linewidth |
| $\sigma$_G | 5.027$\times$1011 rad/s | UQFF linewidth sigma |
| $\rho$_$\Lambda$ ($\Lambda$CDM) | 5.96$\times$10-27 kg/m3 | Planck 2018 |
| $\rho$_SCm | 7.09$\times$10-37 kg/m3 | SCm vacuum density |
| w(z=0) | -1.0077 | This work |
| S26(3) | 9.50$\times$10-2 | Ramanujan acceleration |

## §7 SM Gate Compliance

- **G1:** Derived from SCm phonon resonance first principles
- **G2:** Euler-Lagrange variation of E_net sector Lagrangian
- **G3:** Numerical stability via capped exponentials (exp $\leq$ e500)
- **G4:** $\Gamma$(t) motivated by cosmic expansion diluting phonon modes
- **G5:** Testable with DESI, Euclid, Pantheon+ observational programs
- **G6:** Reproducible w(z) curves, deterministic BIC analysis

## References

- `scm_dark_energy_enet_gamma.py`: Implementation (10/10 tests pass)
- `et_full_lagrangian.py`: E_net(t) base (without $\Gamma$ coupling)
- `production_scaling_v17.py`: Kernels `kernel_enet_gamma_coupled`, `kernel_dark_energy_eos_w0`
- PAPER_877: Three-Assumption UQFF Cosmogenesis
- PAPER_1073: SCm Inflation Phonon-Driven



---

## Session 225: Late-Corpus Physics Integration (PAPER_1000-1081)

> *The following physics upgrades incorporate equations, mechanisms, and
> derivations from the late-corpus papers (Sessions 219-225, PAPER_1000-1081).
> These represent body-level integrations of phonon physics, buoyancy
> formulations, and S26(3) Ramanujan corrections into this paper's domain.*

<!-- PKG-DM-S225 -->

### Session 225 Phonon-Physics Upgrade: SCm-Modified NFW Dark Matter Profile

> *Upgrade from PAPER_1015 (SCm Dark Matter Halos NFW) and PAPER_1019
> (Dark Matter Phonon Buoyancy NFW Coupling).*

The late-corpus analysis shows that the SCm phonon field modifies the NFW
density profile at all radii via a buoyancy-coupled power-law term:

$$\rho_{\text{UQFF}}(r) = \frac{\rho_s}{\left(\frac{r}{r_s}\right)\left(1+\frac{r}{r_s}\right)^2} \times \left[1 + H_{\text{SCm}} \cdot \beta_i \cdot S_{26}^{(3)} \cdot \left(\frac{r_s}{r}\right)^{\alpha_{\text{phonon}}}\right]$$

where:
- $\alpha_{\text{phonon}} = 0.3$ governs the radial decay of phonon coupling
- $\beta_i = 0.603$ is the universal buoyancy coefficient
- $S_{26}^{(3)}$ is the third-order Ramanujan summation
- $H_{\text{SCm}} = 0.99$ is the manifold completeness factor

**Rotation curve flattening:** The phonon enhancement produces flatter rotation curves
with flatness ratio $f = v_c(10\,r_s)/v_{\text{peak}} = 0.891$, compared to pure NFW
$f \approx 0.75$.  Peak circular velocity $v_{\text{peak}} \approx 204\;\text{km/s}$
for $M_{\text{halo}} = 10^{12}\,M_\odot$, $c = 10$.

**Halo stabilization:** The effective buoyancy pressure $P_{\text{SCm}} = \rho_{\text{SCm}} \cdot v_{\text{SCm}}^2 \cdot \beta_i$ prevents cusp-core divergence, providing a physical mechanism for observed cored profiles without invoking SIDM cross-sections.
<!-- 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.

---

## 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_1033 | Galactic Bar Resonance SCm Pattern Speed |

*2 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
