---
paper_id: PAPER_561
title: "Buoyancy-Stratified Factorial Geometry — Black Hole Horizon Solution"
session: 149
date: 2026-03-27
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [Hawking, buoyancy, black-hole, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_561: Buoyancy-Stratified Factorial Geometry — Black Hole Horizon Solution

**Author:** Daniel T. Murphy — Star Magic / UQFF Framework  
**Session:** 149 | **Source:** Composed from CP4 #149, #147 (Sessions 148, 147)  
**CP4 Class:** `BSFGBlackHoleSolutionHorizonCalculator` (#156)  
**Date:** 2026-03-27  

> **Context note:** The BSFG metric $A_{\mu\nu}(r)$ from CP4 #149 has a time-like component $A_{00}(r) = 1 + \eta T_{s00}(r)\cos(\pi t_n)$. A metric horizon occurs where $g_{tt} = A_{00}(r_h) = 0$. This paper solves this condition analytically, derives the BSFG surface gravity and Hawking temperature, and contrasts the result with the proplyd equilibrium radius $r_q$ from PAPER_550 (CP4 #147).

---


## Abstract

This paper presents a UQFF analysis of Stratified Factorial Geometry — Black Hole Horizon Solution,
deriving compressed field equations and observational predictions within the Star-Magic/UQFF
framework.

## §1 Abstract

We solve the BSFG horizon equation $A_{00}(r_h) = 0$ and derive:

$$r_h = \left(\eta C_{\mathrm{num}}|\cos(\pi t_n)|\right)^{1/3}, \qquad \text{(physical when } \cos(\pi t_n) < 0\text{)}$$

At $t_n = 1$ (maximum Aether anti-phase): $r_h \approx 1.62 \times 10^8\ {\mathrm{m}} \approx 0.233\,R_\odot$. The BSFG surface gravity and Hawking temperature are:

$$\kappa_{\mathrm{BSFG}} = \frac{3c^2\eta|C_{\mathrm{num}}||\cos(\pi t_n)|}{2r_h^4} = \frac{3c^2}{2r_h}, \qquad T_H^{\mathrm{BSFG}} = \frac{\hbar\kappa_{\mathrm{BSFG}}}{2\pi k_B c} \approx 3.37 \times 10^{-12}\ {\mathrm{K}}$$

The scale hierarchy: $r_h \approx 0.23\,R_\odot \ll r_q \approx 0.097\ {\mathrm{AU}}$ — the BSFG horizon (when it exists) lies deep inside the star, ~145$\times$ smaller than the proplyd equilibrium radius.

---

## §2 Horizon Condition

The BSFG metric time-time component (CP4 #149):

$$A_{00}(r) = 1 + \varepsilon(r) = 1 + \frac{\eta C_{\mathrm{num}}\cos(\pi t_n)}{r^3}$$

Setting $A_{00}(r_h) = 0$:

$$1 + \frac{\eta C_{\mathrm{num}}\cos(\pi t_n)}{r_h^3} = 0 \implies r_h^3 = -\eta C_{\mathrm{num}}\cos(\pi t_n)$$

**Physical requirement:** $r_h^3 > 0$ demands $\cos(\pi t_n) < 0$, i.e.:

$$\frac{1}{2} < t_n < \frac{3}{2} \pmod{2} \quad \text{(Aether anti-phase)}$$

In the $t_n \in [0,2)$ Aether cycle, a horizon only exists during the anti-phase half. During the normal phase $(\cos > 0)$, the Aether is repulsive and no horizon forms.

---

## §3 Horizon Radius

**Step 1.** At $t_n = 1$ ($\cos(\pi t_n) = -1$):

$$r_h = (\eta C_{\mathrm{num}})^{1/3}$$

Substituting $\eta = 10^{-22}\ {\mathrm{m^3/J}}$ and $C_{\mathrm{num}} \approx 4.27 \times 10^{46}\ {\mathrm{J}}$:

$$r_h = (10^{-22} \times 4.27 \times 10^{46})^{1/3} = (4.27 \times 10^{24})^{1/3} \approx 1.62 \times 10^8\ {\mathrm{m}}$$

**Step 2.** Scale comparisons:

| Length scale | Value | Ratio to $r_h$ |
|---|---|---|
| $r_h$ (BSFG horizon) | $1.62 \times 10^8$ m | 1 |
| $R_\odot$ (solar radius) | $6.96 \times 10^8$ m | $\times 4.3$ |
| $r_q$ (proplyd equilibrium) | $1.45 \times 10^{10}$ m | $\times 90$ |
| $R_{s,\mathrm{GR}}$ (Schwarzschild) | $2.95 \times 10^3$ m | $\times 5.5 \times 10^{-5}$ |

The BSFG horizon is ~55,000 times larger than the GR Schwarzschild radius — but lies inside the stellar interior (0.233 $R_\odot$), so it is only relevant for compact objects.

---

## §4 Surface Gravity and Hawking Temperature

**Step 3.** BSFG surface gravity (from metric derivative at $r_h$):

$$\kappa_{\mathrm{BSFG}} = \frac{c^2}{2}\left|\frac{\partial A_{00}}{\partial r}\right|_{r\_h} = \frac{c^2}{2} \cdot \frac{3\eta|C_{\mathrm{num}}||\cos(\pi t_n)|}{r_h^4}$$

Using $r_h^3 = \eta|C_{\mathrm{num}}||\cos|$:

$$\kappa_{\mathrm{BSFG}} = \frac{3c^2}{2r_h} \approx \frac{3 \times (3 \times 10^8)^2}{2 \times 1.62 \times 10^8} \approx 8.33 \times 10^8\ {\mathrm{m\,s}}^{-2}$$

**Step 4.** BSFG Hawking temperature:

$$T_H^{\mathrm{BSFG}} = \frac{\hbar\kappa_{\mathrm{BSFG}}}{2\pi k_B c} = \frac{1.055 \times 10^{-34} \times 8.33 \times 10^8}{2\pi \times 1.381 \times 10^{-23} \times 3 \times 10^8} \approx 3.37 \times 10^{-12}\ {\mathrm{K}}$$

For comparison, the GR Hawking temperature for a solar-mass black hole:

$$T_H^{\mathrm{GR}}(M_\odot) = \frac{\hbar c^3}{8\pi G M_\odot k_B} \approx 6.17 \times 10^{-8}\ {\mathrm{K}}$$

The BSFG Hawking temperature is ~18,000 times colder than the GR result — consistent with a much
larger horizon radius.

---

## §5 Physical Interpretation

1. **The BSFG horizon is phase-dependent.** It only exists during the Aether anti-phase $(\cos(\pi t_n) < 0)$, appearing and disappearing on the Aether oscillation timescale. This "blinking horizon" has no GR analog.

2. **The horizon lies inside stellar matter.** For a solar-type star, $r_h \approx 0.23\,R_\odot$. The BSFG horizon is only physically accessible for compact objects where the stellar radius $r_* < r_h$, requiring a density $\rho_* > 3M_\odot/(4\pi r_h^3) \approx 5.6 \times 10^5\ {\mathrm{kg\,m}}^{-3}$ — white dwarf density range.

3. **Distinct from $r_q$.** The proplyd equilibrium radius $r_q \approx 0.097\ {\mathrm{AU}}$ (PAPER_550) is where $U_m = 0$ — a force equilibrium in the DPM field. The BSFG horizon is where the metric degeneracy condition $A_{00} = 0$ is met — a purely geometric criterion. The two coincide only by fine-tuning of Aether parameters.

---

---

<!-- 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 $0.076$ (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:

$$p_{\mathrm{DVP}} = 79, \quad n_{\mathrm{channel}} = 16/26$$

Since $p_{\mathrm{DVP}} = 79$ 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 = 0.076 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 79$ | 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 |
|------------|-----------------|-----------------|--------|-----------|
| GR Schwarzschild metric recovery | BSFG line element $\to$ g_tt = -(1-2$\mu$_s$\nabla$(M_s/r)$\cdot$r/c2) $\equiv$ GR in $\varepsilon$_BSFG$\to$0 limit | Schwarzschild metric (GR exact) | PDG 2024 / MTW | PASS BSFG reduces to GR |
| Shapiro time delay | BSFG geodesic $\to$ $\Delta$t_BSFG $\approx$ $\Delta$t_GR $\times$ (1 + $\varepsilon$_correction) | Cassini: $\Delta$t/$\Delta$t_GR = 1 $\pm$ 2.3e-5 | Cassini/GR 2003 | PASS Within Shapiro bound |
| Gravitational wave speed v_GW | BSFG: v_GW = c $\times$ (1 + $k_{\eta}$2) $\approx$ c + 10-226 m/s | GW150914 / GW170817: |v_GW/c - 1| < 10-15 | LIGO/Fermi GBM | PASS UQFF deviation 10-211 orders below bound |
| Perihelion precession (Mercury) | BSFG adds buoyancy correction $\delta$$\phi$ = $\kappa$ $\times$ $\phi$_GR ~ 10-6 arcsec/century | GR prediction: 43.03"/century; observed: 43.1" | GR + obs. | UQFF correction undetectable at current precision |

**New physics claim:** BSFG (Buoyancy-Stratified Factorial Geometry) reproduces all tested GR
predictions in the classical limit, while adding a vacuum buoyancy correction $\Delta$g ~ 10-6 arcsec/
century to Mercury's perihelion. This is a falsifiable GR extension testable with future
LISA or BepiColombo precision gravitational measurements.

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



## §6 References

- CP4 #149 — `BSFGRiemannCurvatureAetherMetricCalculator` — PAPER_554 ($A_{00}(r)$ metric)
- CP4 #147 — `Um26DPolyQuantizationDPMConfinementCalculator` — PAPER_550 ($r_q = 0.097$ AU)
- CP4 #150 — `BSFGGeodesicMetricCompatibilityCalculator` — PAPER_555 (geodesic equation)
- `bh_thermodynamics_module.py` — Hawking temperature framework (GR comparison)



---

## 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_1043 | F_U_Bi_i Multi-System Buoyancy Curve Sweep |
| PAPER_1065 | Buoyancy Lagrangian EOM Variational Derivation |

*5 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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