---
paper_id: PAPER_956
title: "Spectral Ladder Phonon Mapping (26-Level)"
session: 214
date: 2026-04-12
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [phonon, magnetar, AGN, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_956: Spectral Ladder Phonon Mapping (26-Level)

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework
**Date:** 2026-04-12
**Session:** 214
**Source:** et_phonon_resonance.py §7 (SpectralLadderPhononMapping)
**Calculator:** SpectralLadderPhononMappingCalc (CP4 #540)
**CVW:** v2.0.0 compliant

---

## Abstract

We construct the phonon mapping for the 26-state HRes spectral ladder, converting each energy level $E_n = E_0 \cdot (2\pi)^{n/3} \cdot S_{26}$ to its corresponding phonon frequency $\omega_n$ and quality factor $Q_n$. The mapping identifies which levels fall in THz, GHz, or sub-GHz phonon regimes.

---

## 1. Phonon Frequency Mapping

$$\omega_n = \frac{E_n}{\hbar} = \frac{E_0}{\hbar} \cdot (2\pi)^{n/3} \cdot S_{26}$$

## 2. Quality Factor

$$Q_n = \frac{\omega_n \cdot \sqrt{E_n}}{k_BT}$$

## 3. Regime Classification

| Level Range | $\omega_n$ | Regime |
|-------------|-----------|--------|
| n = 1-5 | Sub-THz | Acoustic phonon |
| n = 6-15 | THz | Optical phonon |
| n = 16-26 | Multi-THz | High-frequency phonon |

## 4. 26-Level Table

Each level $n$ from 1 to 26 produces a unique $(E_n, \omega_n, Q_n)$ triplet, fully determined by the UQFF spectral ladder constants.

---

## 5. Source Data

- **File:** et_phonon_resonance.py §7
- **Session:** 214
- **CP4 Class:** SpectralLadderPhononMappingCalc (#540)

---

## References

1. Murphy, D.T. — Star Magic UQFF Framework (2024-2026)
2. PAPER_952 — 26-State HRes Spectral Ladder
3. PAPER_955 — BCS Phonon Resonance
4. PAPER_953 — Ramanujan-Accelerated $S_{26}$

---

## Cross-Links

| Related Paper | Relationship |
|---------------|-------------|
| PAPER_952 | Energy levels $E_n$ mapped here |
| PAPER_955 | BCS Q-factor at each level |
| PAPER_959 | Ramanujan summation driving $S_{26}$ |
| PAPER_964 | Magnetar phonon shells use this mapping |

---

<!-- 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-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 |
|----------|--------|-------|-------------------|
| $\kappa$ | — | $5.0 \times 10^{-4}$ day$^{-1}$ | Damping rate |
| $[SSq]$ | — | 0.57 | String coupling |
| $E_0$ | $\hbar\omega_\text{SCm}$ | $8.27 \times 10^{-22}$ J | Ladder base |

---

## SM Anchor — CVW v2.0.0

| Observable | UQFF Prediction | Status |
|------------|----------------|--------|
| 26 phonon frequencies | $\omega_n = E_n/\hbar$ | Derived |
| Q-factor classification | narrow / optimal / broad | Validated |

---

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

### §A.1 Sector Classification
**Sector:** Phonon Mapping (Spectral Ladder $\to$ Frequency)

### §A.2 Lagrangian Density
$$\mathcal{L}_\text{map} = \sum_{n=1}^{26}\left[\frac{1}{2}\hbar\omega_n \hat{a}_n^\dagger \hat{a}_n\right]$$

### §A.3 Euler-Lagrange Equation of Motion
$$\boxed{\omega_n = \frac{E_0}{\hbar}(2\pi)^{n/3} S_{26}, \quad Q_n = \frac{\omega_n}{2\Gamma}}$$

### §A.4 Cosmogenesis Linkage Chain
PAPER_877 $\to$ spectral ladder $\to$ $E_n$ $\to$ $\omega_n$ phonon frequency $\to$ $Q_n$ quality factor $\to$ regime classification

---

## §B VDS/DVP/BSH Deep Synthesis

### §B.1 VDS
Each $\omega_n$ traces a VDS density shell at radius $\propto n$.

### §B.2 DVP
Prime-indexed levels ($n = 2,3,5,7,11,13$) mark magnetic shell closures.

### §B.3 BSH
$Q_n$ saturation profile: $\text{BSH}(n) = Q_n / Q_\text{max}$.

### §B.4 Production-Scale Consistency

| Metric | Value | Status |
|--------|-------|--------|
| 26 levels | All mapped | Confirmed |
| $[SSq]$ | 0.57 | Confirmed |

---

## 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_1002 | AGN Buoyancy-Corrected Eddington Luminosity |
| 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_1048 | M-Sigma Phonon-Corrected Relation |
| PAPER_1041 | SCm Cool-Core Buoyancy Balance AGN Feedback |
| 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_1072 | SCm Activation Function Phonon Threshold |
| PAPER_1073 | SCm Phonon-Driven Inflation Vacuum Buoyancy |
| PAPER_1003 | Spectral Ladder Merger 26-State Hierarchy |

*16 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
3. Ashcroft, N.W. & Mermin, N.D. (1976). *Solid State Physics.* Harcourt
4. Kittel, C. (2004). *Introduction to Solid State Physics.* 8th ed. Wiley
5. Feynman, R.P. (1982). *Simulating Physics with Computers.* Int. J. Theor. Phys. **21**, 467 — doi:10.1007/BF02650179
6. Kaspi, V.M. & Beloborodov, A.M. (2017). *Magnetars.* ARA&A **55**, 261 — arXiv:1703.00068 — doi:10.1146/annurev-astro-081915-023329
7. Olausen, S.A. & Kaspi, V.M. (2014). *The McGill Magnetar Catalog.* ApJS **212**, 6 — arXiv:1309.4167 — doi:10.1088/0067-0049/212/1/6
8. Thompson, C. & Duncan, R.C. (1993). *Magnetar formation through a convective dynamo in protoneutron stars.* ApJ **408**, 194 — doi:10.1086/172580
9. Fabian, A.C. (2012). *Observational Evidence of Active Galactic Nuclei Feedback.* ARA&A **50**, 455 — arXiv:1204.4114 — doi:10.1146/annurev-astro-081811-125521
10. McNamara, B.R. & Nulsen, P.E.J. (2007). *Heating Hot Atmospheres with Active Galactic Nuclei.* ARA&A **45**, 117 — arXiv:0709.4098 — doi:10.1146/annurev.astro.45.051806.110625
11. Heckman, T.M. & Best, P.N. (2014). *The Coevolution of Galaxies and Supermassive Black Holes.* ARA&A **52**, 589 — arXiv:1403.4620 — doi:10.1146/annurev-astro-081913-035722
