---
paper_id: PAPER_448
title: "Multi-System UQFF Core Compression Framework: Unified F_env Architecture"
session: 115
date: 2025-01-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [AGN, Hubble, SCm, MUGE, magnetar, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_448 — Multi-System UQFF Core Compression Framework: Unified F_env Architecture
**Date:** 2025

**Whitepaper Series:** Star-Magic UQFF Phase 2  
**Session:** 115 (v4.72) / Whitepapers created Session 121  
**Source:** grok_{share\_5fa36e4e035}.txt (Compression Cycle 2 — architectural foundation)  
**Classification:** FIRST unified multi-system F_env modular architecture in UQFF; FIRST std::map
dynamic variable storage for astrophysical UQFF systems  
**Author:** Daniel T. Murphy  
**CP4 Class:** `MultiSystemUQFFCoreCalculator` (#2, PAPER_448)

<!— UQFF constants: $\kappa$ = 5.0e-4 day-1, [SSq] = 0.57, H_SCm $\approx$ 0.99, U_UA $\approx$ 0.0001 —>
---

## Abstract

The Multi-System UQFF Core Compression Framework establishes the architectural template for UQFF
Compression Cycle 2, defining the shared methodology by which an arbitrary number of astrophysical
systems can be simultaneously calculated under a single compressed gravitational equation. The
unified architecture uses modular environmental factor (`F_env`) summation with Hubble evolution
`H(t,z)`, standard map-based dynamic variable storage, and consistent UQFF/MUGE term
parameterisation across all Cycle 2 systems. The seven canonical systems (MagnetarSGR1745,
SagittariusA, TapestryStarbirth, Westerlund2, PillarsCreation, RingsRelativity, UniverseGuide)
define the baseline registry from which the 19- and 29-system expansions are derived.

---

## 2. Core Architecture — PAPER_448

### 2.1 Compression Cycle 2 Philosophy

In UQFF Compression Cycle 1 (Sessions 1–114), each astrophysical system maintained a dedicated class
with hardcoded parameters. Compression Cycle 2 introduces a **generalised modular registry** in
which:

1. System parameters are stored in `std::map<std::string, double>` containers
2. Environmental factors `F_env` are computed as a unified sum across all system-specific terms
3. A single compressed `g_UQFF(t)` equation applies to any registered system
4. Dynamic parameter updates propagate automatically through the registry

### 2.2 Unified Gravitational Equation (Core)

$$g_{\mathrm{UQFF}}(r,t) = \frac{GM(t)}{r^2}(1 + H_z t)(1 - B/B_{\mathrm{crit}}) + \sum_{i} U_{gi} + \frac{\Lambda c^2}{3} + g_{\mathrm{quantum}} + g_{\mathrm{fluid}} + F_{\mathrm{env}}(t)$$

Where the total environmental modifier:

$$F_{\mathrm{env}}(t) = \sum_{j=1}^{N_{\mathrm{sys}}} F_{\mathrm{env}}^{(j)}(t, \{p_j\})$$

With $\{p_j\}$ = system-specific parameter map for system $j$.

### 2.3 Hubble Evolution Module

$$H(t,z) = H_0\sqrt{\Omega_m(1+z)^3 + \Omega_Lambda}$$

Evaluated at z for each system:
- Local (z$\approx$0): H $\approx$ 70 km/s/Mpc
- Intermediate (z=0.5): H $\approx$ 85 km/s/Mpc  
- Cosmological (z=1100, CMB): H $\approx$ 70$\times$$\sqrt{}$(0.3$\times$11003+0.7) km/s/Mpc

$$H_z = H(z)/H_0 \;[\text{dimensionless Hubble factor}]$$

### 2.4 Dynamic Variable Storage (FIRST in UQFF)

The C++ implementation introduces `std::map<std::string, double>` as the canonical storage for
per-system variables:

$$
\begin{aligned}
  & params["M"]      = system_mass       [kg] \\
  & params["r"]      = radius            [m] \\
  & params["z"]      = redshift          [dimensionless] \\
  & params["B"]      = magnetic_field    [T] \\
  & params["v_exp"]  = expansion_vel     [m/s] \\
  & params["rho"]    = fluid_density     [kg/m3] \\
  & params["F_env"]  = env_modifier      [m/s2] \\
  & params["SC_m"]   = superconductive   [dimensionless]
\end{aligned}
$$

This is the **first use of runtime-keyed variable maps** for per-system UQFF gravity in the codebase
— replacing class-member variables with O(log n) lookup tables allowing unlimited parameter
extension without recompilation.

### 2.5 Canonical 7-System Registry

| System | type_key | key parameters |
|--------|----------|----------------|
| MagnetarSGR1745 | MAGNETAR_SGR1745 | M=2.8 MM_sun, r=1e4 m, B=1e11 T |
| SagittariusA | SAGITTARIUS_A | M=4.1e6 MM_sun, r=6e9 m, B=1e-3 T |
| TapestryStarbirth | TAPESTRY_STARBIRTH | M=500 MM_sun, r=1e16 m, z=0.001 |
| Westerlund2 | WESTERLUND2 | M=1e4 MM_sun, r=6e16 m, z=0.005 |
| PillarsCreation | PILLARS_CREATION | M=200 MM_sun, r=6e16 m, z=0.002 |
| RingsRelativity | RINGS_RELATIVITY | M=1e39 kg, r=1e20 m, z=0.3 |
| UniverseGuide | UNIVERSE_GUIDE | M=1e53 kg, r=4.4e26 m, z=1100 |

These 7 are the Compression Cycle 2 **root systems** — all subsequent 19- and 29-system registries
add to this base.

---

## 3. Ug Component Summary

### 3.1 Ug1 — Magnetic Dipole Term

$$U_{g1} = -\frac{\mu_0 m_1 m_2}{4\pi r^3}\left(1 - \frac{r_s}{r}\right)$$

### 3.2 Ug2 — Charge Reactivity Term

$$U_{g2} = k_e\frac{q_1 q_2}{r}\left(1 + \frac{\kappa t}{\tau_q}\right)$$

### 3.3 Ug3 — String Rotation Term (UQFF-specific)

$$U_{g3} = \frac{GM_{\mathrm{ext}}}{r_{\mathrm{ext}}^2}\left(1 + \frac{v_s}{c}\cos\theta\right)$$

Compressed form (Cycle 2):

$$U_{g3}' = \frac{GM_{\mathrm{ext}}}{r_{\mathrm{ext}}^2}$$

### 3.4 Ug4 — Vacuum Concentration Term

$$U_{g4} = U_{A}\rho_{\mathrm{vac}}\left(1 + [{\mathrm{UA}}] \cdot [{\mathrm{SCm}}]\right)$$

---

## 4. Quantum and Fluid UQFF Terms

### 4.1 Quantum Gravity Coupling

$$g_{\mathrm{quantum}} = \frac{\hbar c}{l_p^2}\frac{t}{t_p} \cdot \frac{1}{M c^2}$$

Where $l_p = \sqrt{\hbar G/c^3} = 1.616\times10^{-35}$ m (Planck length), $t_p = l_p/c$.

### 4.2 Fluid Navier-Stokes Coupling

$$g_{\mathrm{fluid}} = \nu_{\mathrm{fluid}}\nabla^2 v + (\mathbf{v}\cdot\nabla)\mathbf{v}$$

Simplified for radial symmetry:

$$g_{\mathrm{fluid}} \approx \rho_{\mathrm{fluid}} v_{\mathrm{exp}}^2 / r$$

---

## 5. $\Psi$_total Integration

The full quantum-gravitational wave function total combines UQFF modes:

$$\psi_{\mathrm{total}} = \int_0^\infty A(k) e^{i(kr - \omega t)} dk + \frac{[SSq]^{n_{26}}}{[SSq]^{n_{26}-1}}$$

The discrete multi-system version sums over all N registered systems:

$$\psi_{\mathrm{total}}^{(N)} = \sum_{j=1}^{N} g_{\mathrm{UQFF}}^{(j)}(r,t) \cdot w_j$$

Where $w_j$ = system weight (default: equal weights = 1/N).

---

## 6. Standard Model Comparison

| Feature | SM | CC2 (UQFF) |
|---------|-----|-----------|
| Per-system gravity | Individual Poisson eq. | Unified compressed registry |
| Environmental coupling | External perturbation | Built-in F_env term |
| Dynamic parameter storage | Compile-time | Runtime std::map |
| Multi-system simultaneity | Separate solvers | Single g_UQFF call for N systems |

---

## 7. Testable Predictions

1. **Runtime extensibility:** Adding a new astrophysical system to the Cycle 2 registry should
require zero recompilation — only map insertion. Testable by adding any new entry and verifying
output.
2. **F_env additivity:** For two weakly-interacting systems (e.g., Tapestry + Pillars at large
separation), F_env_total $\approx$ F_{env\_1} + F_{env\_2} within 0.1%.
3. **Hubble evolution consistency:** H(z=0.5) from the modular H(t,z) function should reproduce
H0$\sqrt{}$(0.3$\times$1.53+0.7) = H0$\times$0.894 = 62.6 km/s/Mpc ($\pm$1%).

---

---

<!-- 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-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 **magnetar-field** 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):

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

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:

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

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

$$\boxed{\frac{\delta S}{\delta \phi_B} = \nabla \times (\rho_{\mathrm{SCm}} \mathbf{v} \times \mathbf{B}) + \kappa B_{\mathrm{crit}} \partial_t \phi_B = 0}$$

### §A.4 Cosmogenesis Linkage Chain

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

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 $0.054$ (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:

$$p_{\mathrm{DVP}} = 109, \quad n_{\mathrm{channel}} = 7/26$$

Since $p_{\mathrm{DVP}} = 109$ 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 **103 yr** (field decay quiescence):

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

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 = 0.054 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 109$ | 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.*



*Copyright – Daniel T. Murphy | Session 115/121 — `grok_{share\_5fa36e4e035}`.txt*



---

## 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_1024 | Magnetar Giant Flare SCm Phonon Reservoir |
| PAPER_1043 | F_U_Bi_i Multi-System Buoyancy Curve Sweep |
| PAPER_1072 | SCm Activation Function Phonon Threshold |
| PAPER_1073 | SCm Phonon-Driven Inflation Vacuum Buoyancy |
| PAPER_1050 | MUGE F_U_Bi_i Unified 9-System Synthesis |
| PAPER_1075 | 3D Volumetric MUGE Gravitational Field Generator |

*14 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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