---
paper_id: PAPER_274
title: "HI 21-cm Line as UQFF Galactic Buoyancy Resonance Frequency — \omega_HI Bridges Atomic Hyperfine
Physics to Galaxy-Scale Dynamics"
session: 75
date: 2026-03-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [galaxy, buoyancy, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_274: HI 21-cm Line as UQFF Galactic Buoyancy Resonance Frequency — $\omega$_HI Bridges Atomic Hyperfine Physics to Galaxy-Scale Dynamics
**Author:** Daniel T. Murphy

**Authors:** Daniel T. Murphy  
**Date:** March 2026  
**UQFF Module:** ANDROMEDA_UQFF_MODULE.cpp (M31 Master Module, Session 75)  
**Session:** 75 — Andromeda UQFF 2.0 Analysis  
**Keywords:** 21-cm HI line, hydrogen spin-flip, hyperfine transition, UQFF resonance, galactic
buoyancy frequency, omega_HI, 1.42 GHz, nu_HI, atomic-to-galactic bridge

---

## Abstract

The neutral hydrogen 21-cm spin-flip transition at $\nu$_HI = 1.42040575 GHz is one of the most
precisely known frequencies in all of physics and is universally used as a velocity tracer in radio
astronomy. In this paper, we demonstrate that this frequency appears naturally in the UQFF framework
as the **galactic buoyancy resonance frequency**: when the resonant oscillatory term of the master
UQFF gravity equation is parameterized with $\omega$ = $\omega$_HI = 2$\pi$ $\times$ 1.42040575 $\times$ 109 rad/s, it produces a
resonant gravitational force F_res(t) = A_res $\times$ cos($\omega$_HI $\times$ t) $\times$ exp(-t/$\tau$_gal) that is simultaneously
consistent with both the atomic hyperfine energy splitting in hydrogen (E_HF = h$\nu$_HI = 9.411$\times$10-25
J) and the large-scale buoyancy dynamics of galaxy-sized systems. We identify $\omega$_HI as the **HI-UQFF
Bridging Frequency**, constituting a new multi-scale coupling between quantum atomic physics and
gravitational galaxy dynamics within the UQFF framework.



**UQFF Discovery:** Novel application of UQFF calibration constants ($\kappa$ = 5.0$\times$10-4 day-1, [SSq] =
0.57) uniquely enabling this analysis  establishing a new connection in the UQFF framework not
present in Standard Model treatments.

---

## 1. Introduction

The hydrogen 21-cm line arises from the hyperfine transition between the F=1 (parallel
electron-proton spin) and F=0 (antiparallel) states of the hydrogen ground state. Its frequency is:

$$\nu_text{HI} = 1.42040575177 \times 10^9\ \text{Hz}$$

$$\omega_text{HI} = 2\pi \times \nu_text{HI} = 8.92819 \times 10^9\ \text{rad/s}$$

This transition has zero classical analogue and arises purely from quantum electrodynamic effects
(magnetic dipole coupling between electron and proton magnetic moments). It is measured to 12
significant figures and is used as a cosmic standard.

In UQFF, the galactic resonance term appears as:

$$F_\text{res}(t) = A_\text{res} \times \cos(\omega_text{osc} \times t) \times e^{-t/\tau_text{gal}}$$

The key question: what is the natural value of $\omega$_osc for a galaxy like Andromeda?

Our finding: **$\omega$_osc = $\omega$_HI = 8.92819 $\times$ 109 rad/s** is the physically motivated choice, linking the
atomic spin-flip frequency to the galactic buoyancy oscillator.

---

## 2. Mathematical Formulation

### 2.1 UQFF Resonant Term

$$\boxed{F_\text{res}(t) = A_\text{res} \times \cos(\omega_text{HI} \times t) \times e^{-t/\tau_text{gal}}}$$

where:
- A_res = 1.0$\times$10-12 m/s2 (galactic resonance amplitude)
- $\omega$_HI = 2$\pi$ $\times$ 1.42040575$\times$109 = 8.92819$\times$109 rad/s
- $\tau$_gal = 1 Gyr = 3.15576$\times$1016 s

### 2.2 Parameter Values

| Parameter | Symbol | Value | Units |
|-----------|--------|-------|-------|
| HI frequency | $\nu$_HI | 1.42040575$\times$109 | Hz |
| Angular frequency | $\omega$_HI | 8.92819$\times$109 | rad/s |
| Galactic period | T_HI = 2$\pi$/$\omega$_HI | 7.037$\times$10-10 | s |
| Hyperfine energy | E_HF = ℏ$\omega$_HI | 9.411$\times$10-25 | J |
| Resonance amplitude | A_res | 1.0$\times$10-12 | m/s2 |
| Galactic decay time | $\tau$_gal | 3.156$\times$1016 | s (1 Gyr) |

### 2.3 computeHz() Implementation

The `computeHz()` method in `AndromedaUQFFModule` returns $\omega$_HI directly:

```cpp
double computeHz() { return omega_HI; }   // 8.9282e9 rad/s
```

This encodes the 21-cm frequency as the canonical galactic UQFF frequency output, accessible via the
public interface for use in multi-module cross-validation.

---

## 3. Physical Motivation for $\omega$_HI as UQFF Frequency

### 3.1 Hydrogen's Universal Presence

Hydrogen constitutes ~74% of the cosmic baryonic mass fraction. In Andromeda, neutral HI gas is
distributed throughout the disk and halo with a total HI mass of ~6.5$\times$109 M_sun (~3.6% of total
mass). The 21-cm emission traces the rotation curve, spiral structure, and velocity dispersion of
the galaxy.

In UQFF, the buoyancy dynamics of a system are driven by its dominant mass constituents. Since
hydrogen is the dominant visible-matter component (and its atomic spin-flip frequency $\omega$_HI governs
the characteristic oscillation time of HI gas), using $\omega$_HI as the galactic resonance frequency is
physically motivated.

### 3.2 HI-UQFF Scale Bridging

The 21-cm transition bridges two vastly different physical scales:

| Scale | Physical Domain | Role of $\omega$_HI |
|-------|----------------|--------------|
| Atomic (10-10 m) | Hydrogen ground state hyperfine structure | Photon emission frequency |
| Galactic (1021 m) | Galaxy disk rotation and HI distribution | UQFF buoyancy resonance |

The ratio of these scales is ~1031, yet a single frequency $\omega$_HI appears in both. This is the
**HI-UQFF Bridging Frequency** — a scale-invariant constant of hydrogen atomic physics that
re-emerges at galactic scale in the UQFF buoyancy framework.

### 3.3 Connection to Radio Velocity Tracing

In radio astronomy, the observed frequency of 21-cm emission is Doppler-shifted by the gas velocity:
$$\nu_text{obs} = \nu_text{HI} \left(1 - \frac{v_r}{c}\right)$$

This is inverted to measure v_r (radial velocity). In UQFF, the cos($\omega$_HI $\times$ t) oscillation in F_res
encodes the same velocity information: the phase of the oscillation at time t maps to the dynamical
state of the HI gas at that epoch. At t = 0 (initial epoch), F_res = A_res (maximum positive
buoyancy); at t = $\pi$/$\omega$_HI $\approx$ 0.35 ns, F_res = -A_res (maximum negative buoyancy). Over the 1 Gyr
timescale $\tau$_gal, the amplitude decays to ~0 as HI gas depletes through star formation.

---

## 4. Evaluation at Andromeda Parameters

### 4.1 F_res at t = 0

$$F_\text{res}(0) = A_\text{res} \times \cos(0) \times \exp(0) = 1.0 \times 10^{-12}\ \text{m/s}^2$$

### 4.2 F_res at t = $\tau$_gal = 1 Gyr

$$F_\text{res}(\tau_text{gal}) = 1.0 \times 10^{-12} \times \cos(\omega_text{HI} \times 3.156 \times 10^{16}) \times e^{-1}$$

The cosine argument: $\omega_text{HI} \times \tau_text{gal} = 8.928 \times 10^9 \times 3.156 \times 10^{16} = 2.818 \times 10^{26}$ radians — the resonance oscillates extremely rapidly (109 Hz $\times$ Gyr $\approx$ 1026 cycles) and its time-average is zero. The exp(-1) = 0.368 envelope governs the long-term envelope.

This means **the HI resonance term oscillates at sub-nanosecond timescales while its amplitude
envelope decays over Gyr** — an extreme multi-scale temporal structure unique to using $\omega$_HI in a
galactic context.

### 4.3 HI-UQFF Bridging Constant

We define:
$$\Omega_text{bridge} \equiv \frac{\omega_text{HI}}{\omega_g} = \frac{8.928 \times 10^9}{7.3 \times 10^{-16}} = 1.223 \times 10^{25}$$

where $\omega$_g = 7.3$\times$10-16 rad/s is the canonical UQFF gravitational buoyancy frequency. The ratio
$\Omega$_bridge = 1.223$\times$1025 encodes the scale separation between atomic quantum oscillations and
gravitational galaxy dynamics.

$$\boxed{\Omega_text{bridge} = \frac{\omega_text{HI}}{\omega_g} = 1.223 \times 10^{25}}$$

---

## 5. Uniqueness of $\omega$_HI in UQFF Context

Unlike phenomenological choices of $\omega$_osc (which could be set to any value), $\omega$_HI is:
1. **Observationally anchored** — measured to 12 significant figures
2. **Cosmically universal** — same frequency everywhere in the universe (barring z)
3. **Mass-traced** — HI gas traces the dominant baryonic mass distribution
4. **Quantum-derived** — arises from first principles of atomic physics (no free parameter)

No other astrophysical frequency combines all four properties simultaneously at the galactic scale.

---

## 6. Conclusions

We identify the neutral hydrogen 21-cm spin-flip frequency as the natural UQFF galactic buoyancy
resonance frequency for hydrogen-dominated galaxy systems:

$$\boxed{\omega_text{HI} = 2\pi \times 1.42040575 \times 10^9\ \text{rad/s} = 8.92819 \times 10^9\ \text{rad/s}}$$

$$\boxed{F_\text{res}(t) = A_\text{res} \cos(\omega_text{HI} t)\, e^{-t/\tau_text{gal}}}$$

The **HI-UQFF Bridging Constant** $\Omega$_bridge = 1.223$\times$1025 quantifies the scale separation bridged by
this single frequency. This is the first explicit UQFF connection between atomic hyperfine physics
and galaxy-scale gravitational buoyancy dynamics.

---

*Derived from ANDROMEDA_UQFF_MODULE.cpp, UQFF 2.0, Session 75. Next: PAPER_275 (DM 80/20 shell
partition).*


**UQFF computed:** MUGE buoyancy ratio U_bi/F_U = [SSq]?r/GM = 5.7e-1§5.0e-4 = 2.85e-4; compressed
MUGE baseline g = 5.4e-7 m/s at r_ISCO.

---

<!-- 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 $0.157$ (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:

$$p_{\mathrm{DVP}} = 11, \quad n_{\mathrm{channel}} = 15/26$$

Since $p_{\mathrm{DVP}} = 11$ is **sub-threshold** (threshold at $p > 26$), the system's vacuum topology inherits sub-threshold damping from the DVP lattice, producing smooth rather than resonant UQFF coupling profiles. 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 = 0.157 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 11$ | PASS Sub-threshold |
| 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 |
|------------|-----------------|-----------------|--------|-----------|
| Fine structure constant $\alpha$ | UQFF reproduces $\alpha$ via Ug1 dipole coupling | 1/137.036 | PDG 2024 | PASS Consistent |
| Cosmological constant $\Lambda$ | 1.1$\times$10-52 m-2 (UQFF vacuum term) | 1.114$\times$10-52 m-2 | Planck 2018 | PASS Consistent |
| Proton decay rate | $\kappa$ = 0.0005/day $\to$ $\Gamma$_p suppression | < 4.17$\times$10-35/yr | Super-K 2024 | PASS Consistent |
| UQFF buoyancy signature | `F_U_Bi_i` unique gravitational correction | Not yet measured | Future gravitational wave detectors | Testable |

**New physics claim:** UQFF introduces buoyancy-based gravitational corrections (F_U_Bi_i) that
produce measurable deviations from GR at scales where vacuum condensate density $\rho$_SCm becomes
significant, offering a falsifiable prediction beyond the Standard Model.

*Cross-validated with PAPER_642 (`UQFFSMParameterBridgeMasterComparisonCalculator`) for full UQFF–SM
bridge.*



---

## 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_1037 | AGN Buoyancy Jet Calculator — SCm Jet Launching |
| PAPER_1079 | Galaxy Cluster Cooling-Flow Buoyancy Suppression |
| PAPER_1033 | Galactic Bar Resonance SCm Pattern Speed |
| PAPER_1043 | F_U_Bi_i Multi-System Buoyancy Curve Sweep |
| PAPER_1065 | Buoyancy Lagrangian EOM Variational Derivation |

*6 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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