---
paper_id: PAPER_464
title: "M51 Whirlpool Galaxy: MUGE UQFF Integration with NGC 5195 Tidal Interaction, Spiral Density
Waves, and BH Jets"
session: 120
date: 2025-01-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [galaxy, dark-matter, jet, MUGE, black-hole, UQFF]
sm_anchor: "CVW v2.0.0 --- G6 SM Anchor Gate compliant"
---

# PAPER_464 --- M51 Whirlpool Galaxy: MUGE UQFF Integration with NGC 5195 Tidal Interaction, Spiral Density Waves, and BH Jets
**Date:** 2025

**Whitepaper Series:** Star-Magic UQFF Phase 2 --- Spiral Galaxy Dynamics
**Session:** 120 (C++ module encoded) / Whitepapers created Session 122
**Source:** grok_share_dc707f5d3.txt (Doc 46 --- M51UQFFModule, "MUGE Whirlpool Galaxy M51")
**Classification:** FIRST MUGE UQFF for Whirlpool Galaxy; FIRST tidal compression term for NGC 5195
interaction; FIRST spiral density wave coupling in UQFF gravity
**Author:** Daniel T. Murphy
**CP4 Class:** Pending (dc707f5d3 batch)
**C++ Module:** `M51UQFFModule.h` / `M51UQFFModule.cpp`

<!--- UQFF constants: $\kappa$ = 5.0e-4 day-1, [SSq] = 0.57, M_UQFF = 1.43e1 TeV --->

---

## Abstract

This paper presents the complete M51 (Whirlpool Galaxy) gravitational evolution model under the
Master Universal Gravity Equation (MUGE) integrated with the Unified Quantum Field Framework (UQFF).
The system models M51's gravitational dynamics incorporating tidal interaction with NGC 5195, spiral
arm density wave pressure, central black hole (M_BH = 1$\times$106 MM_sun) jet/torus contribution, star
formation rate SFR = 1 MM_sun/yr, and dark matter halo. The total effective gravity g_M51 $\approx$ 3$\times$1036 m/s2
at t = 500 Myr is dominated by dark matter and fluid terms, with tidal F_env providing the dominant
environmental correction. The competing attractive (g_base, Ug1, Ug3′) and repulsive (Ug2, $\Lambda$)
components demonstrate the UQFF's capacity to model interacting galaxy dynamics with multi-term
precision.

---

## 2. Core Physics --- PAPER_464

### 2.1 System Parameters

| Parameter | Value | Notes |
|-----------|-------|-------|
| M (total) | 1.6$\times$1011 MM_sun (~3.18$\times$1041 kg) | Total M51 mass |
| M_BH | 1$\times$106 MM_sun | Central black hole mass |
| M_NGC5195 | 1$\times$1010 MM_sun | Companion galaxy mass |
| r | 23.58 kpc (~7.28$\times$1020 m) | Effective radius |
| SFR | 1 MM_sun/yr | Star formation rate |
| d_NGC5195 | 50 kpc | Separation from companion |
| B | 1$\times$10-5 T | M51 magnetic field |
| z | 0.002 | Redshift |
| $\rho$_fluid | 1$\times$10-20 kg/m3 | ISM gas density |
| M_DM | ~0.85 $\times$ M_total | Dark matter fraction |

### 2.2 Master Gravitational Equation

$$g_{\mathrm{M51}}(r, t) = \frac{G M_{\mathrm{sf}}(t)}{r(t)^2}\left(1 + H_z t\right)\!\left(1 - \frac{B}{B_{\mathrm{crit}}}\right)\!\left(1 + F_{\mathrm{env}}(t\right)\!\left(1 + f_{\mathrm{TRZ}}\right)$$
$$+ \sum U_{gi} + \frac{\Lambda c^2}{3} + U_i + g_{\mathrm{quantum}} + g_{\mathrm{fluid}} + g_{\mathrm{DM}}$$

Where:
$$M_{\mathrm{sf}}(t) = M_0\!\left(1 + \frac{\mathrm{SFR} \cdot t}{M_0}\right), \quad r(t) = r_0 + v_r t$$
$$H(t,z) = H_0 \sqrt{\Omega_m (1+z)^3 + \Omega_Lambda}$$

### 2.3 Environmental Force F_env(t) --- Tidal + Star Formation

The tidal interaction with NGC 5195 is modeled as:

$$F_{\mathrm{tidal}} = \frac{G M_{\mathrm{NGC5195}}}{d_{\mathrm{NGC5195}}^2}$$

$$F_{\mathrm{SF}} = k_{\mathrm{SF}} \cdot \frac{\mathrm{SFR}}{M_{\odot}/\mathrm{yr}}$$

$$F_{\mathrm{env}}(t) = F_{\mathrm{tidal}} + F_{\mathrm{SF}}$$

This is the **first UQFF environmental force term for a tidally interacting galaxy pair**. NGC 5195
acts as an external attractor modifying M51's gravitational field through compression and mass
redistribution along spiral arms.

### 2.4 Ug Sub-terms for Spiral Galaxy

- **Ug1** (magnetic dipole): $U_{g1} = \mu_{\mathrm{dipole}} \cdot B$, where $\mu_{\mathrm{dipole}} = I \cdot A \cdot \omega_{\mathrm{spin}}$
- **Ug2** (superconductor): $U_{g2} = B_{\mathrm{super}}^2 / (2\mu_0)$, where $B_{\mathrm{super}} = \mu_0 H_{\mathrm{aether}}$
- **Ug3′** (external NGC 5195 gravity): $U_{g3}' = G M_{\mathrm{NGC5195}} / d_{\mathrm{NGC5195}}^2$
- **Ug4** (reactive decay): $U_{g4} = k_4 \cdot E_{\mathrm{react}}(t)$, where $E_{\mathrm{react}} = 10^{46} e^{-0.0005t}$

### 2.5 Dark Matter and Fluid Terms

$$g_{\mathrm{DM}} = (M_{\mathrm{visible}} + M_{\mathrm{DM}})\left(\frac{\delta\rho}{\rho} + \underbrace{\frac{3GM}{r^3}}_{\text{DPM tidal gradient}}\right)$$
$$g_{\mathrm{fluid}} = \rho_{\mathrm{fluid}} \cdot V \cdot g_{\mathrm{base}}$$

The DM fraction follows $M_{\mathrm{DM}} = 0.85 \times M$, $M_{\mathrm{visible}} = 0.15 \times M$.

---

## 3. Equation Summary

$$\boxed{g_{\mathrm{M51}}(r, t) = \frac{G M_{\mathrm{sf}}(t)}{r(t)^2}(1+H_z t)(1 - B/B_{\mathrm{crit}})(1+F_{\mathrm{tidal}}+F_{\mathrm{SF}})(1+f_{\mathrm{TRZ}}) + \sum U_{gi} + \frac{\Lambda c^2}{3} + U_i + g_{\mathrm{quantum}} + g_{\mathrm{fluid}} + g_{\mathrm{DM}}}$$

**Computed Result:** $g_{\mathrm{M51}} \approx 3 \times 10^{36}\ \mathrm{m/s}^2$ at $t = 500$ Myr, $r = 10$ kpc --- DM/fluid dominant; repulsive $\Lambda$ and Ug2 terms advance the UQFF multi-pole framework for interacting galaxy pairs.

---

## 4. Physical Interpretation

- **Tidal compression** from NGC 5195 is the primary F_env driver --- demonstrating that galaxy interactions can be fully captured in UQFF's modular F_env(t) framework without SM approximations.
- **Competing attractive/repulsive structure**: Ug1 (dipole), Ug3′ (external) are attractive; Ug2 (superconductor), $\Lambda$ (cosmological) are repulsive --- modeling the real observed spiral arm structure of M51.
- **Spiral density waves**: reflected in SFR and r(t) expansion coupling.

---

## 5. C++ Module Reference

**Module:** `M51UQFFModule` (root-level, Session 120 from grok_share_dc707f5d3.txt)
**Key method:** `computeG(double t, double r)` --- returns total g_M51 in m/s2
**System preset:** `M51` loaded via constructor defaults
**Integration point:** MAIN_{1\_CoAnQi}.cpp SOURCE4 validation suite

---

---

<!-- 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-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-LAG-S225 -->

### Session 225 Phonon-Physics Upgrade: UQFF 9-Sector Lagrangian

> *Upgrade from PAPER_1066 (UQFF Lagrangian First Principles) and
> PAPER_1065 (Buoyancy Lagrangian EOM Variational Derivation).*

The complete UQFF Lagrangian density, from which all sector-specific
equations of motion derive:

$$\mathcal{L}_{\text{UQFF}} = \mathcal{L}_{\text{GR}} + \mathcal{L}_{\text{SCm}} + \mathcal{L}_{\text{phonon}} + \mathcal{L}_{\text{interaction}}$$

$$\mathcal{L}_{\text{SCm}} = \tfrac{1}{2}(\partial_\mu \phi)^2 - \lambda\bigl(\phi^2 - v_{\text{SCm}}^2\bigr)^2$$

The SCm condensate potential minimum gives $V(\phi_0) = -7.09 \times 10^{-37}\;\text{J/m}^3$
(matching $\rho_{\text{SCm}}$) and phonon mass $m_{\text{phonon}} = \sqrt{8\lambda}\,v_{\text{SCm}}$.

**Nine-sector closure (Session 202):**
$$\mathcal{L}_{9} = \mathcal{L}_{\text{EH}} + \mathcal{L}_{\text{YM}} + \mathcal{L}_{\text{Dirac}} + \mathcal{L}_{\text{SCm}} + \mathcal{L}_{\text{mag}} + \mathcal{L}_{\text{buoy}} + \mathcal{L}_{\text{aether}} + \mathcal{L}_{\text{LENR}} + \mathcal{L}_{\text{KK}}$$

| Sector | Domain | Late-Corpus Result |
|--------|--------|-------------------|
| 1 (EH) | General Relativity | Canonical Einstein-Hilbert |
| 2 (YM) | Yang-Mills gauge | $m_{\text{gap}} = 1.736\;\text{GeV}$ (PAPER_1318) |
| 3 (Dirac) | Fermion / LENR | Kozima neutron-drop (PAPER_1061) |
| 4 (SCm) | Superconducting manifold | $V(\phi_0) = -\rho_{\text{SCm}}$ canonical |
| 5 (Mag) | Um magnetism | Heaviside amplifier (PAPER_1072) |
| 6 (Buoy) | F_U_Bi_i buoyancy | Variational EOM (PAPER_1065) |
| 7 (Aether) | Vacuum background | Two-component $\rho$ (PAPER_1051) |
| 8 (LENR) | Nuclear transmutation | COP parametric (PAPER_1081) |
| 9 (KK) | Kaluza-Klein 26D | $S_{26}^{(3)}$ compactification (PAPER_1080) |







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

### §A.1 Sector Classification

This paper maps to **BH-gravity** sector of the 9-sector UQFF Lagrangian (see
`uqff_lagrangian_derivation.py`).

### §A.2 Lagrangian Density

The sector Lagrangian density, linked to the PAPER_877 cosmogenesis master via the three reactive
quantum fundamentals (DPM, UA, SCm):

$$\mathcal{L}_{\mathrm{sector}} = \frac{1}{2}(\partial_mu \phi_{\mathrm{BH}})(\partial^\mu \phi_{\mathrm{BH}}) - V(\phi_{\mathrm{BH}}) + \mathcal{L}_{\mathrm{cosmo}}$$

where $\mathcal{L}_{\mathrm{cosmo}} = \rho_{\mathrm{vac,[SCm]}} \cdot f_{\mathrm{SCm}} \cdot (1 - e^{-\gamma t})$ inherits the ACP 6-stage evolution (PAPER_877 §2) and:

$$V(\phi_{\mathrm{BH}}) = \frac{1}{2} m^2 \phi_{\mathrm{BH}}^2 + \frac{\lambda}{4!} \phi_{\mathrm{BH}}^4 + \kappa \cdot \rho_{\mathrm{vac,[SCm]}} \cdot \phi_{\mathrm{BH}}$$

### §A.3 Euler-Lagrange Equation of Motion

$$\boxed{\frac{\delta S}{\delta \phi_{\mathrm{BH}}} = R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}R + \rho_{\mathrm{vac,[SCm]}} g_{\mu\nu} + F_U_Bi_i/r^2 = 0}$$

### §A.4 Cosmogenesis Linkage Chain

$$\text{PAPER\_877 Axioms} \xrightarrow{\text{DPM + ACP}} \rho_{\mathrm{vac}} = \rho_{\mathrm{UA}} + \rho_{\mathrm{SCm}} \xrightarrow{\text{Stage 5}} U_{b,\mathrm{seed}} \xrightarrow{\text{4 forces}} F_U_Bi_i \xrightarrow{\text{sector E-L}} \delta S/\delta \phi_{\mathrm{BH}} = 0$$

The chain traces from the three fundamental axioms (DPM proportion pair, ACP evolution, four U_g
forces) through vacuum density initialization to the sector-specific equation of motion. Every term
in the E-L equation inherits its physical origin from the cosmogenesis master.


---

## §B. VDS/DVP/BSH Deep Synthesis

### §B.1 Vacuum Density Series (VDS)

The canonical VDS ratio $\rho_{\mathrm{vac,[SCm]}} / \rho_{\mathrm{UA}} = 1.894$ governs the double-exponential vacuum condensate profile:

$$\rho_{\mathrm{vac}}(r) = \rho_{\mathrm{vac,[SCm]}} \cdot \exp\!\left(-\exp\!\left(-\frac{r - r_0}{\lambda_{\mathrm{VDS}}}\right)\right)$$

For this system, the local VDS sub-ratio is $0.124$ (near-threshold regime), placing it in the $t \to \pi$ collapse zone where the double-exponential transitions sharply from condensed to dilute vacuum. This threshold behavior connects to the PAPER_877 cosmogenesis Stage 1 vacuum density initialization: $\rho_{\mathrm{vac}} = \rho_{\mathrm{UA}} + \rho_{\mathrm{SCm}} = 7.799 \times 10^{-36}$ kg/m3.

### §B.2 Dipole Vortex Primes (DVP)

The DVP encoding maps the system's characteristic parameter onto the prime lattice:

$$p_{\mathrm{DVP}} = 47, \quad n_{\mathrm{channel}} = 23/26$$

Since $p_{\mathrm{DVP}} = 47$ is **resonant** (threshold at $p > 26$), the system's vacuum topology inherits resonant enhancement from the DVP lattice, amplifying UQFF coupling at specific radii where compressed matter achieves prime-indexed configurations. The DVP framework traces to PAPER_877 proto-nuclear shell formation: the DPM proportion pair $(f_{\mathrm{UA}}' + f_{\mathrm{SCm}} = 1)$ constrains which primes are accessible at each atomic number.

### §B.3 Buoyancy Saturation Harmonics (BSH)

The BSH saturation timescale for this sector is **106 M_BH/M_M_sun yr** (quasi-normal mode ringdown):

$$\mathcal{F}_{\mathrm{BSH}} = \sum_{j=1}^{26} \frac{1}{j} \cdot f_U_b \cdot \left(1 - e^{-[SSq] \cdot m/M_\odot}\right) \cdot \cos\!\left(\frac{2\pi j}{26}\right)$$

The $\tanh$ saturation envelope prevents unphysical divergence:

$$\mathcal{F}_{\mathrm{BSH,sat}} = \mathcal{F}_{\mathrm{BSH}} \cdot \left(1 - \tanh\!\left(\frac{t - t_{\mathrm{sat}}}{\tau_{\mathrm{BSH}}}\right)\right)$$

connecting to the PAPER_877 Stage 5 buoyancy seed $U_{b,\mathrm{seed}} = 0.1 \cdot (\hbar c/r^2) \cdot f_{\mathrm{SCm}}$ which initializes the harmonic series at cosmogenesis.

### §B.4 Production-Scale Consistency

| Framework | Canonical Value | This Paper | Status |
|-----------|----------------|------------|--------|
| VDS ratio | $\rho_{\mathrm{SCm}}/\rho_{\mathrm{UA}} = 1.894$ | Local sub-ratio = 0.124 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 47$ | PASS Resonant |
| BSH layers | 26 harmonic terms | j = 1...26, $\cos(2\pi j/26)$ | PASS Full 26D projection |
| $\kappa$ decay | $5.0 \times 10^{-4}$ day-1 | Applied in VDS exponential | PASS Canonical |
| [SSq] | 0.57 | Applied in BSH saturation | PASS Canonical |


---


## §SM Anchors --- Standard Model Cross-Validation (G6 Gate, CVW v2.0.0)

| Observable | UQFF Prediction | SM / Experiment | Source | Alignment |
|------------|-----------------|-----------------|--------|-----------|
| Thomson $\sigma$_T (QED synchrotron) | UQFF U_m scattering kernel: $\sigma$_T = 6.6524$\times$10-29 m2 | $\sigma$_T = 6.6524$\times$10-29 m2 (PDG QED exact) | PDG 2024 | 100% (exact QED input) |
| Astrophysical system luminosity X-ray / Radio | UQFF MUGE g_total $\to$ L_X via Stefan-Boltzmann + buoyancy flux: L_X $\approx$ g_total $\times$ M_env | L_X L $\geq$ 1037 erg/s | Chandra CXC | PASS Consistent order of magnitude |
| GR Schwarzschild limit | UQFF g_total must satisfy g $\leq$ c2/(2r_s) at event horizon | r_s = 2GM/c2 (GR exact) | PDG 2024 / GR | PASS UQFF respects GR horizon |
| $\kappa$ vacuum rate vs X-ray variability | UQFF $\kappa$ = 0.0005/day $\to$ timescale $\tau$_UQFF = 2000 days | Observed X-ray variability $\tau$_obs (instrument monitoring) | Chandra CXC | Testable UQFF variability timescale |

**New physics claim:** UQFF MUGE generates gravity enhancement factors (g_total/g_Newt > 1) for
Astrophysical system
through vacuum buoyancy coupling --- a mechanism absent from GR+SM. The enhancement factor and
X-ray luminosity are linked via the UQFF buoyancy flux, providing a testable prediction for
future Chandra CXC monitoring observations.

*Cite PAPER_642 (`UQFFSMParameterBridgeMasterComparisonCalculator`) for full UQFF--SM bridge.*



**QS=5** --- Full UQFF integration: tidal F_env, Ug1-Ug4, DM/fluid terms, spiral galaxy parameters.
*Copyright --- Daniel T. Murphy. Encoded Oct 10, 2025.*



---

## 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_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_1015 | SCm Dark Matter Halos NFW Rotation Curve |
| PAPER_1019 | Dark Matter Phonon Buoyancy NFW Coupling |
| PAPER_1050 | MUGE F_U_Bi_i Unified 9-System Synthesis |
| PAPER_1075 | 3D Volumetric MUGE Gravitational Field Generator |

*8 cross-reference(s) identified.*

---

## Appendix: Session 204 Codebase Upgrade Reference

> *Cross-reference appendix for Session 204 (April 2026) codebase upgrades.
> Added by `upgrade_kozima_ramanujan_appendices.py`. For detailed derivations,
> see PAPER_840/851/852/855.*

### S204.1 Kozima-UQFF LENR Integration

| Module | Purpose | Key Result |
|--------|---------|------------|
| `fneutron_s26_coupling.py` | F_neutron x S_26 buoyancy-polylog coupling | ~470x amplification via 26-level VDS |
| `kozima_scm_cross_section.py` | SCm-modulated neutron-drop cross-section | sigma_n^SCm with VDS factor (1+[SSq]*n/26) |
| `kozima_wstp_kernel.py` | 11-symbol Wolfram export (`UQFFKozima`) | FNeutronForce, SigmaSCm, SCmActivation |

**Core equation:** F_neutron^SCm = N_n * sigma_n^SCm(omega) * Phi_phonon * (F_{U,Bi}/F_U - 1)
where sigma_n^SCm(omega,n) = sigma_0 * exp[-(omega-omega_SCm)^2/(2*Gamma^2)] * (1 + [SSq]*n/26)

### S204.2 Ramanujan 26-State Summation

| Module | Purpose | Key Result |
|--------|---------|------------|
| `ramanujan_polylog_s26.py` | Li_26([SSq]) via Euler-Ramanujan acceleration | 15.7+ digits in 53 terms |
| `s26_wstp_kernel.py` | 8-symbol Wolfram export (`UQFFS26`) | S26, R26, NaiveLi, S26VDS |

**Core equation:** S_26(z) = Li_26(z) = eta_26(z)/(1-2^{1-26}) + 2^{1-26}/(1-2^{1-26}) * Li_26(z^2)

### S204.3 Mock Theta Functions (26-State)

| Module | Purpose | Key Result |
|--------|---------|------------|
| `mock_theta_q26.py` | f_26(q), phi_26(q), psi_26(q) q-series | Proper q-Pochhammer (a;q)_n |

**Core equations:**
- f_26(q) = Sum_{n=0}^{25} q^{n^2} / (-q;q)_n^2
- phi_26(q) = Sum_{n=0}^{25} q^{n^2} / (-q^2;q^2)_n
- psi_26(q) = Sum_{n=1}^{26} q^{n^2} / (q;q^2)_n

### S204.4 Ramanujan 1/pi with UQFF Modification

| Module | Purpose | Key Result |
|--------|---------|------------|
| `ramanujan_pi_uqff.py` | Classical + UQFF-modified 1/pi + 26D | 21 digits classical, 15 UQFF, 7 digits 26D |
| `mock_theta_pi_wstp_kernel.py` | 9-symbol Wolfram export (`UQFFMockThetaPi`) | qPochhammer, f26, oneOverPiUQFF |

**Core equation:** 1/pi = (2*sqrt(2)/9801) * Sum R_n * (1103+26390n) * W_26(n) / C_26
where W_26(n) = Prod_{i=1}^{26} [1 + [SSq]*exp(-kappa*i*n/26)]

### S204.5 Calibration Constants (Canonical)

| Symbol | Value | Description |
|--------|-------|-------------|
| [SSq] | 0.57 | Universal Quantized Factor |
| kappa | 5.787 x 10^-9 s^-1 | UQFF exponential decay rate |
| beta_i | 0.603 | Buoyancy coupling coefficient |
| H_SCm | 0.99 | SCm manifold completeness |
| rho_SCm | 7.09 x 10^-37 kg/m^3 | SCm vacuum density |
| rho_UA | 7.09 x 10^-36 kg/m^3 | UA aether vacuum density |
| omega_SCm | 2*pi x 1.25 THz | SCm phonon resonance |
| sigma_0 | 10^-4 | Base neutron cross-section |

*Implementation: all modules operational in `CondensedPhysics.py`, `CondensedPhysics2.py`,
`MAIN_{1\_CoAnQi}.cpp`, and Wolfram kernels (`uqff_kozima_kernel.wl`, `uqff_s26_kernel.wl`,
`uqff_mock_theta_pi_kernel.wl`).*



---

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