---
paper_id: PAPER_444
title: "Hubble Ultra Deep Field \"Galaxies Galore\": Per-System MUGE at Cosmic Scale z=3.5"
session: 119
date: 2025-01-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [AGN, Hubble, MUGE, BEC, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_444 — Hubble Ultra Deep Field "Galaxies Galore": Per-System MUGE at Cosmic Scale z=3.5
**Author:** Daniel T. Murphy
**Date:** 2025

**Source:** grok_{share\_68eb34022}.txt — Document 17: "Master Universal Gravity
Equation_{HUDF\_Galaxies\_Galore\_03May2025}.docx" (lines 5154–5538)
**Session:** 119
**CP4 Class:** `HUDFGalaxiesGaloreMUGE_CosmicScale_HighRedshift_Calculator` (#99)

---


## Abstract

This paper presents a UQFF analysis of Hubble Ultra Deep Field "Galaxies Galore": Per-System MUGE at
Cosmic Scale z=3.5, deriving compressed field equations and observational predictions within the
Star-Magic/UQFF framework.

## 1. Overview

PAPER_444 delivers the **complete per-system MUGE** for the Hubble Ultra Deep Field (HUDF) — representing the cosmic field around epoch $z \approx 3.5$, total mass $M_0 = 10^{12} \, M_\odot$, FoV extent $r = 1.3 \times 10^{11}$ ly $= 1.230 \times 10^{27}$ m (co-moving scale of the UDF at co-moving depth to $z \sim 7$). Magnetic field $B = 10^{-10}$ T (CMB-era primordial seed field — the weakest in the per-system series).

**Novel claim (Q1):** First UQFF MUGE representing a **full cosmic survey scale** — the HUDF is not a single object but a cosmological field spanning $\sim 10$ billion years of lookback time. PAPER_444 introduces the **cosmic interaction boost** $I(t) = 0.05 \, e^{-t/\tau_text{inter}}$ (weaker than Antennae's 10% because interactions in the UDF are stochastically averaged over $\sim 3000$ galaxies), applied as $(1+I(t))$ to $T_1$ and $T_2$. The $z_\text{avg} = 3.5$ Einstein expansion sets $H(z) \approx 1.201 \times 10^{-17}$ s-1 — the largest $H(z)$ in the per-system series — and $B = 10^{-10}$ T is the **smallest magnetic field** among all 17 documents.

---

## 2. System Parameters

| Parameter | Symbol | Value |
|-----------|--------|-------|
| Co-moving field mass | $M_0$ | $10^{12} \, M_\odot = 1.989 \times 10^{42}$ kg |
| Co-moving FoV scale | $r$ | $1.3 \times 10^{11}$ ly $= 1.230 \times 10^{27}$ m |
| Mean redshift | $z_\text{avg}$ | 3.5 |
| $H(z)$ | | $H_0 \sqrt{\Omega_m(1+z)^3 + \Omega_Lambda}$ |
| $H(z=3.5)$ | | $\approx 370.6$ km/s/Mpc $= 1.201 \times 10^{-17}$ s-1 |
| Primordial B field | $B$ | $10^{-10}$ T |
| SFR factor | $\text{SFR}_f$ | 1.0 (normalized) |
| SF timescale | $\tau_text{SF}$ | 1 Gyr $= 3.156 \times 10^{16}$ s |
| Interaction factor | $I_0$ | 0.05 |
| Interaction timescale | $\tau_text{inter}$ | 1 Gyr $= 3.156 \times 10^{16}$ s |
| Wind density | $\rho_w$ | $10^{-22}$ kg/m3 (early universe, lower density) |
| Wind velocity | $v_w$ | $10^6$ m/s |
| Fluid density | $\rho_f$ | $10^{-22}$ kg/m3 |

---

## 3. Cosmological H(z) Computation

$$H(z=3.5) = H_0\sqrt{\Omega_m(1+z_\text{avg})^3+\Omega_Lambda}$$
$$= 70 \, \text{km/s/Mpc}\sqrt{0.3\times(4.5)^3+0.7}$$
$$= 70\sqrt{0.3\times91.125+0.7} = 70\sqrt{27.3375+0.7} = 70\sqrt{28.0375}$$
$$= 70 \times 5.295 = 370.6 \, \text{km/s/Mpc}$$
$$\boxed{H(z=3.5) = \frac{370.6\times10^3}{3.086\times10^{22}} \approx 1.201\times10^{-17} \, \text{s}^{-1}}$$

This is $\approx 5.5\times$ larger than the local $H_0 = 2.184 \times 10^{-18}$ s-1 — the Hubble drag term $H(z)t$ is commensurately larger at high-$z$.

---

## 4. Time-Dependent Functions

**Mass with cosmic SF:**
$$M(t) = M_0\left(1 + \text{SFR}_f \cdot e^{-t/\tau_text{SF}}\right) = M_0\left(1 + e^{-t/\tau_text{SF}}\right)$$

At $t=0$: $M(0) = 2M_0 = 3.978 \times 10^{42}$ kg (field actively forming stars)  
At $t = 1$ Gyr: $M = M_0(1 + 1/e) \approx 1.368 M_0$  
At $t = 3$ Gyr: $M \approx 1.05 M_0$ (star formation essentially complete)

**Cosmic interaction boost:**
$$\boxed{I(t) = 0.05 \, e^{-t/\tau_text{inter}}}$$

At $t=0$: $I = 0.05$ — 5% boost from early-universe elevated merger/interaction rate  
At $t = 1$ Gyr: $I = 0.0184$ — interaction rate declining as cosmic merger rate falls  
At $t = 3$ Gyr: $I \approx 0.0025$ — essentially gone by cosmic noon ($z \sim 2$)

---

## 5. Complete 10-Term MUGE

$$\boxed{g_\text{HUDF}(r,t) = T_1(1+I) + T_2(1+I) + T_3 + T_4 + T_5 + T_6 + T_7 + T_8 + T_9 + T_{10}}$$

**T1 — DPM-seeded + H(z)t + B + interaction:**
$$T_1 = \frac{GM(t)}{r^2}(1+H(z)t)(1-B/B_\text{crit})(1+I(t))$$
$$\frac{GM_0}{r^2} = \frac{6.674\times10^{-11}\times1.989\times10^{42}}{(1.230\times10^{27})^2} = \frac{1.327\times10^{32}}{1.513\times10^{54}} \approx 8.77\times10^{-23} \, \text{m/s}^2$$
$$T_1(t=0) = \frac{GM(0)}{r^2}(1+I_0) = \frac{6.674\times10^{-11}\times3.978\times10^{42}}{1.513\times10^{54}}\times1.05 \approx 1.84\times10^{-22} \times 1.05 \approx 1.93\times10^{-22} \, \text{m/s}^2$$

**T3 — $\Lambda$ dark energy (large at co-moving scales):**
$$T_3 = \frac{\Lambda c^2}{3}r = \frac{1.11\times10^{-52}\times 9times10^{16}}{3}\times1.230\times10^{27} \approx 4.11\times10^{-9} \, \text{m/s}^2$$

Note: At cosmic co-moving scale $r = 1.23 \times 10^{27}$ m, the cosmological constant term becomes the **third largest contribution**.

**T2 — UQFF Ug with interaction:**
$$T_2 = 2\times\frac{GM(0)}{r^2}\times f_\text{TRZ}\times(1+I_0) \approx 2\times1.84\times10^{-22}\times1.1\times1.05 \approx 4.24\times10^{-22} \, \text{m/s}^2$$

**T9 — Early-universe galactic wind:**
$$T_9 = \frac{\rho_w v_w^2}{\rho_f \cdot r} = \frac{10^{-22}\times10^{12}}{10^{-22}\times1.230\times10^{27}} = \frac{10^{12}}{1.230\times10^{27}} \approx 8.13\times10^{-16} \, \text{m/s}^2 \quad [\text{negligible}]$$

---

## 6. Canonical Numerical Result

At $t = 0$ (redshift $z = 3.5$, early universe):

| Term | Value (m/s2) | Fraction |
|------|-------------|---------|
| $T_3$ $\Lambda$ (cosmological) | $4.11 \times 10^{-9}$ | **$\gg$ all other terms** |
| $T_1$ DPM-seeded$\times$(1+I) | $1.93 \times 10^{-22}$ | negligible |
| $T_2$ UQFF$\times$(1+I) | $4.24 \times 10^{-22}$ | negligible |
| $T_9$ Wind | $8.13 \times 10^{-16}$ | trace |

$$\boxed{g_\text{HUDF}(t=0) \approx 4.11\times10^{-9} \, \text{m/s}^2} \quad [\Lambda \text{ cosmological term dominant at co-moving scale}]$$

**Remark — $\Lambda$ dominance at cosmic scale:** This result is the **first occurrence in the 17-paper series** where the dark energy ($T_3$) term dominates over all other gravitational terms. At co-moving $r = 10^{27}$ m, $T_3$ exceeds $T_1$ by 13 orders of magnitude. This reflects the HUDF FoV being a cosmological scale — the MUGE correctly predicts that dark energy governs on scales larger than the matter power spectrum coherence length.

**Interaction boost contribution:** $I_0 = 0.05 \Rightarrow 5\%$ of $T_1$:
$$\Delta g_I = 0.05 \times 1.84\times10^{-22} \approx 9.2\times10^{-24} \, \text{m/s}^2$$

---

## 7. Uniqueness vs Prior Papers

| Prior Paper | Overlap | New in PAPER_444 |
|-------------|---------|-----------------|
| PAPER_441 (Antennae) | $I(t)$ boost | $I_0=0.05$ vs 0.1, $\tau=1$ Gyr vs 400 Myr, cosmic avg |
| All others | Local/cluster | Only paper with $z > 1$ |
| None | $T_3$ dominance | **First paper where $\Lambda$ term is THE dominant term** |
| None | $H(z)=1.2\times10^{-17}$ s-1 | **Highest H(z) in per-system series** |
| None | $B=10^{-10}$ T | **Weakest magnetic field — primordial seed** |

---

## 8. Comparison to Standard Model

Cosmological N-body simulations (IllustrisTNG, EAGLE) model the UDF statistically, not as individual per-system physics. The SM treats dark energy as a negative-pressure background fluid ($w = -1$) uniformly accelerating expansion. UQFF re-parameterizes $\Lambda$ as the $T_3$ term in the gravitational field equation, making its dominance at cosmic scale explicit and calculable per-system. The SF mass evolution $M(t) = M_0(1+e^{-t/\tau_text{SF}})$ matches the $z \sim 3.5$ "cosmic noon" star formation peak seen in Madau-Dickinson curve — but expressed as a direct mass multiplier on the UQFF gravitational terms rather than a statistical ensemble rate.

---

---

<!-- 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 $\delta = \beta_i \cdot S_{26}^{(3)} \cdot (\omega_{\text{SCm}}/\omega_{\text{bulge}})$.



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

### §A.1 Sector Classification

This paper maps to **NS-compact** 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):

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

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:

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

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

$$\boxed{\frac{\delta S}{\delta \phi_{\mathrm{NS}}} = \nabla^2 \phi_{\mathrm{NS}} - (4\pi G \rho_{\mathrm{NS}}/c^2)\phi_{\mathrm{NS}} + \Omega_{\mathrm{spin}} \partial_t \phi_{\mathrm{NS}} = 0}$$

### §A.4 Cosmogenesis Linkage Chain

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

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 $0.094$ (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:

$$p_{\mathrm{DVP}} = 97, \quad n_{\mathrm{channel}} = 3/26$$

Since $p_{\mathrm{DVP}} = 97$ 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 **104 yr** (spin-down equilibrium):

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

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 = 0.094 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 97$ | 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 = 6.6524e-29 m2 | $\sigma$_T = 6.6524e-29 m2 (PDG QED exact) | PDG 2024 | 100% (exact QED input) |
| HUDF Deep Field luminosity UV/optical/IR z>1 | UQFF MUGE g_total $\to$ L_X via Stefan-Boltzmann + buoyancy flux: L_X $\approx$ g_total $\times$ M_env | L_X n_gal ~ 104 per arcmin2 | HST/JWST | 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) | HST/JWST | Testable UQFF variability timescale |

**New physics claim:** UQFF MUGE generates gravity enhancement factors (g_total/g_Newt > 1) for HUDF
Deep Field
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 HST/JWST monitoring observations.

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



## 9. Testable Predictions

**Q5 Prediction 1:** $1 + H(z=3.5) \cdot t$ for $t = 10^{16}$ s (0.32 Gyr): $H(z)t = 1.201\times10^{-17}\times10^{16} = 0.1201$, predicting 12% Hubble-drag enhancement of $T_1$ within the first 300 Myr after $z=3.5$. UQFF predicts this as a 12% excess in the power spectrum amplitude at $k \sim 0.1$ Mpc-1 accessible in JWST/NIRCam deep-field power spectrum analyses of JADES-Deep-GOODS-S.

**Q5 Prediction 2:** $I_0 = 0.05$ with $\tau_text{inter} = 1$ Gyr predicts that the galaxy pair fraction at $z=3.5$ provides a 5% gravitational enhancement averaged over the UDF field — declining to $\sim 0.5\%$ by $z \sim 1$ ($t \sim 3\tau$). Observable as a $\Delta n(z)$ excess in close galaxy pair counts ($r_\text{proj} < 30$ kpc) in HUDF WFC3/ACS mosaics: specifically, 5% more merging pairs at $z = 3.5$ than single-disk galaxies relative to $z = 1$ — testable via morphological classification with JWST/CEERS or COSMOS-Web.

**Q5 Prediction 3:** $B = 10^{-10}$ T (primordial seed field at $z=3.5$) predicts that the Faraday rotation measure (RM) averaged over UDF lines of sight is $(1/B_\text{crit}) \times B \approx 2.27\times10^{-24}$ — the gravitational suppression factor is completely negligible, meaning the UQFF Ug at $z=3.5$ operates at full strength ($f_B \approx 1.0000$). This predicts no magnetic-field suppression of early structure formation — testable via SKA RM synthesis maps of UDF field radio sources at $z > 3$.



---

## 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_1036 | Primordial Nucleosynthesis BBN Phonon |
| PAPER_1050 | MUGE F_U_Bi_i Unified 9-System Synthesis |
| PAPER_1075 | 3D Volumetric MUGE Gravitational Field Generator |

*11 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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