---
paper_id: PAPER_276
title: "UQFF"
session: 76
date: 2026-03-01
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [galaxy, AGN, Hubble, dark-matter, BEC, UQFF]
sm_anchor: "CVW v2.0.0 — G6 SM Anchor Gate compliant"
---

# PAPER_276: UQFF
**Date:** March 2026
    $n = [int]# UQFF PAPER_276
## Andromeda Friedmann-UQFF Gravity Coupling: H(z)t Expansion Term and H_UQFF Near-Unity Resonance

**Author:** Daniel T. Murphy  
**Session:** 76 (March 2026)  
**Module:** ANDROMEDA_UQFF_MODULE.cpp (M31 Master, UQFF 2.0)  
**WOLFRAM_TERM:** `AndromedaUQFF:g_exp=G*M/r^2*H(z)*t; H(z)=H0*Sqrt[Om_m*(1+z)^3+Om_L]/Mpc2m;
H_UQFF=H(z)*t_H~0.987 [PAPER_276]`

---

## Abstract

This paper introduces a new UQFF gravity term for the Andromeda Galaxy (M31): the **Friedmann-UQFF
Expansion Coupling**, defined as g_expansion = ($\mu$_s$\nabla$(M_s/r))  H(z)  t. For Andromeda's blueshift redshift z
= -0.001, the Friedmann H(z) equals 69.969 km/s/Mpc = 2.269$\times$10?8 s-1. Evaluated at the Hubble
timescale t = t_H = 4.352$\times$10-7 s, the dimensionless Friedmann-UQFF Resonance Coefficient H_UQFF =
H(z)$\cdot$t_H  **0.987**  a near-unity value representing gravitational doubling over cosmological time.
This paper also introduces the M_visible/M_DM_mass cascade (explicit DM split tracking) and a minor
ISM dust drag acceleration term. Together these constitute three additive UQFF 2.0 enhancements
derived from the Andromeda module upgrade.

---

## 1. Background

The UQFF framework computes total gravitational acceleration g_total(r,t) as a sum of DPM-seeded,
Triadic (26-layer), cosmological, quantum, electromagnetic, fluid, resonant, and dark matter terms.
PAPER_273275 established the Andromeda-specific physics: blueshift approach amplifier ?, HI 21-cm
resonance, and DM 80/20 NFW partition.

The original Andromeda implementation (pre-Session 76) lacked a time-dependent expansion coupling
that accounts for the Friedmann-Lemaitre cosmological evolution of Hubble parameter H(z). This is
significant because over a Hubble timescale, H(z)t  1  implying that expansion-mediated
gravitational coupling is of the same order as the Step 10 Newton observational projection term.

---

## 2. Friedmann H(z) Expansion Coupling

### 2.1 Friedmann Equation

The Hubble parameter as a function of redshift in the standard ?CDM cosmology:

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

For Andromeda parameters: H0 = 70.0 km/s/Mpc, O_m = 0.3, O_? = 0.7, z = -0.001.

$$H(z=-0.001) = 70.0 \times \sqrt{0.3 \times (0.999)^3 + 0.7}$$

$$= 70.0 \times \sqrt{0.3 \times 0.997003 + 0.7} = 70.0 \times \sqrt{0.9991}$$

$$= 70.0 \times 0.99955 = 69.969 \text{ km/s/Mpc}$$

In SI units: H_SI = 69.969 $\times$ 10 / 3.086$\times$10 = **2.269$\times$10?8 s-1**

### 2.2 UQFF Expansion Term

The new UQFF gravity coupling term is:

$$g_{\text{expansion}}(r, t) = \frac{G \cdot M}{r^2} \cdot H(z) \cdot t$$

This represents the gravitational effect of cosmological expansion flow coupling into UQFF buoyancy
at timescale t.

**Numerical values for Andromeda:**
- g_base = $\mu$_s$\nabla$(M_s/r) = 6.674$\times$10?  1.989$\times$104 / (1.04$\times$10) = **1.227$\times$10? m/s**
- At t = 1 Gyr (3.156$\times$10-6 s): g_expansion = 1.227$\times$10?  2.269$\times$10?8 $\times$ 3.156$\times$10-6 = **8.79$\times$10? m/s**
- At t = t_Hubble (4.352$\times$10-7 s): g_expansion = 1.227$\times$10? $\approx$ 0.987 = **1.211$\times$10? m/s**

---

## 3. H_UQFF Near-Unity Resonance Coefficient

### 3.1 Definition

$$H_{\text{UQFF}} = H(z) \times t_H$$

where t_H = 4.352$\times$10-7 s (Hubble time  13.8 Gyr).

**For Andromeda (z = -0.001):**

$$H_{\text{UQFF}} = 2.269 \times 10^{-18} \times 4.352 \times 10^{17} = \mathbf{0.987}$$

### 3.2 Near-Unity Discovery

H_UQFF $\approx$ 0.987  **1** is a remarkable result. It means:

> Over a Hubble timescale, the Friedmann expansion coupling adds an amount of gravitational acceleration equal to **98.7% of the Step 10 Newton observational projection term**  nearly doubling g_base.

This near-unity value is not coincidental. In a flat ?CDM universe (O_m + O_? = 1):

$$H_{\text{UQFF}} = H_0 \times t_H \approx 1.0$$

because H0  t_H is the dimensionless Hubble number $\approx$ 0.96§1.0 for observationally consistent H0.

**New UQFF Constant:** H_UQFF = H(z)$\cdot$t_H  **Friedmann-UQFF Near-Unity Resonance Coefficient**

| Parameter | Value |
|---|---|
| H0 | 70.0 km/s/Mpc |
| O_m | 0.3 |
| O_? | 0.7 |
| z (Andromeda) | -0.001 |
| H(z) | 2.269$\times$10?8 s-1 |
| t_H | 4.352$\times$10-7 s |
| **H_UQFF** | **0.987** (~1) |
| g_expansion(t_H) | 1.211$\times$10? m/s |
| g_base | 1.227$\times$10? m/s |
| Ratio g_exp/g_base | 98.7% |

### 3.3 Blueshift Sensitivity

For blueshift z < 0: H(z) is slightly **lower** than H0 (matter term O_m(1+z) is suppressed below
O_m for z < 0), meaning:

$$H(z < 0) < H_0 \Rightarrow H_{\text{UQFF}}(z < 0) < H_{\text{UQFF}}(z=0)$$

For Andromeda (z = -0.001): H_UQFF = 0.987 vs flat-universe H_UQFF0 = 0.9985. The blueshift
suppresses H_UQFF by 0.15%.

This is the inverse of the redshift behaviour (z > 0 ? H(z) > H0 for moderate z), making Andromeda's
approaching trajectory a unique laboratory for probing blueshift Friedmann-UQFF coupling.

---

## 4. ISM Dust Drag Acceleration

A second minor additive term captures ISM dust ram-pressure drag:

$$a_{\text{dust}} = \frac{\rho_{\text{dust}} \cdot v_{\text{orbit}}^2}{c^2 \cdot \rho_{\text{mean}}} \cdot g_{\text{base}}$$

where ?_mean = M/V_fluid = 1.989$\times$104 / 106 = 1.989$\times$10?8 kg/m.

**Numerical value for Andromeda:**
$$a_{\text{dust}} = \frac{10^{-20} \times (2.5 \times 10^5)^2}{(2.998 \times 10^8)^2 \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{8.988 \times 10^{16} \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{0.1788} \times 1.227 \times 10^{-10} = 4.29 \times 10^{-19} \text{ m/s}^2$$

At ~9 orders below g_base, the dust drag is physically negligible but dimensionally consistent and
additive within the UQFF sum.

---

## 5. M_visible / M_DM_mass Cascade

When M is updated via `updateVariable("M", value)`, the derived quantities are now automatically
recomputed:

$$M_{\text{visible}} = (1 - f_{\text{DM}}) \times M$$
$$M_{\text{DM,mass}} = f_{\text{DM}} \times M$$

For Andromeda defaults: M = 1.989$\times$104 kg, f_DM = 0.80:
- M_visible = 0.20 $\times$ 1.989$\times$104 = **3.978$\times$104 kg** (20% visible baryons)
- M_DM_mass = 0.80 $\times$ 1.989$\times$104 = **1.591$\times$104 kg** (80% dark matter)

These are tracked in `updateCache()` and exported via `exportState()` (params 2632), providing
explicit bookkeeping of the M31 baryonic/DM split consistent with PAPER_275.

---

## 6. Updated ANDROMEDA_UQFF_MODULE.cpp g_total

The full UQFF 2.0 equation for Andromeda with all PAPER_273276 terms:

$$g_{\text{total}}(r, t) = \left[ g_{\text{grav}} + U_{g,\text{sum}}^{(26)} + \Lambda\text{-term} + g_Q + g_L + g_{\text{fluid}} + F_{\text{res}}(\omega_{\text{HI}}) + g_{\text{DM}} + g_{\text{expansion}} + a_{\text{dust}} \right] \times \kappa_{\text{approach}}$$

**Term magnitudes at t=0 (current epoch):**

| Term | Value (m/s) | Paper |
|---|---|---|
| g_grav = $\mu$_s$\nabla$(M_s/r) | 1.227$\times$10? | baseline |
| Ug_sum (26 layers) | 6.380$\times$10?? | 26-layer Triadic |
| ?-term | 2.998$\times$10?6 | cosmological |
| g_quantum | ~2$\times$10?6 | HUP |
| g_Lorentz | ~2$\times$10?6 | IGM EM |
| g_fluid | ~8$\times$10? | IGM buoyancy |
| F_res(?_HI) at t=0 | 1$\times$10? | PAPER_274 |
| g_DM | 1.356$\times$10? | PAPER_275 |
| g_expansion at t=0 | 0 | PAPER_276 |
| a_dust | 4.29$\times$10?? | PAPER_276 |
| ?_approach | 1.001001 | PAPER_273 |

Note: g_expansion grows linearly with t, dominating over g_grav at t = t_H.

---

## 7. exportState Update (25 ? 32 parameters)

Seven new parameters added to `exportState()`:
1. `H0` = 70.0 km/s/Mpc
2. `Omega_m` = 0.3
3. `Omega_Lam` = 0.7
4. `Mpc_to_m` = 3.086$\times$10
5. `rho_dust` = 1$\times$10? kg/m
6. `M_visible` = (1f_DM)M
7. `M_DM_mass` = f_DMM

---

## 8. Conclusions

**Three new UQFF enhancements for Andromeda (PAPER_276):**

1. **Friedmann-UQFF Expansion Coupling**: g_expansion = g_base – H(z)  t. At t = t_H, adds 98.7% of
g_base  near-gravitational doubling over cosmological time.

2. **H_UQFF = 0.987**: New UQFF constant (Friedmann-UQFF Near-Unity Resonance Coefficient).
Near-unity reflects the fundamental relationship H0  t_H  1 in flat ?CDM. Andromeda's blueshift (z =
-0.001) makes H_UQFF 0.15% below flat-universe value  a blueshift Friedmann suppression.

3. **M_visible/M_DM_mass cascade + dust drag**: Explicit DM split bookkeeping (updateCache) and ISM
dust ram-pressure (a_dust  4$\times$10?? m/s).

---

**Watermark:** 2025-2026 Daniel T. Murphy, daniel.murphy00@gmail.com – All Rights Reserved

  
## Abstract

This paper introduces a new UQFF gravity term for the Andromeda Galaxy (M31): the **Friedmann-UQFF
Expansion Coupling**, defined as g_expansion = ($\mu$_s$\nabla$(M_s/r))  H(z)  t. For Andromeda's blueshift redshift z
= -0.001, the Friedmann H(z) equals 69.969 km/s/Mpc = 2.269$\times$10?8 s-1. Evaluated at the Hubble
timescale t = t_H = 4.352$\times$10-7 s, the dimensionless Friedmann-UQFF Resonance Coefficient H_UQFF =
H(z)$\cdot$t_H  **0.987**  a near-unity value representing gravitational doubling over cosmological time.
This paper also introduces the M_visible/M_DM_mass cascade (explicit DM split tracking) and a minor
ISM dust drag acceleration term. Together these constitute three additive UQFF 2.0 enhancements
derived from the Andromeda module upgrade.

---

## 1. Background

The UQFF framework computes total gravitational acceleration g_total(r,t) as a sum of DPM-seeded,
Triadic (26-layer), cosmological, quantum, electromagnetic, fluid, resonant, and dark matter terms.
PAPER_273275 established the Andromeda-specific physics: blueshift approach amplifier ?, HI 21-cm
resonance, and DM 80/20 NFW partition.

The original Andromeda implementation (pre-Session 76) lacked a time-dependent expansion coupling
that accounts for the Friedmann-Lemaitre cosmological evolution of Hubble parameter H(z). This is
significant because over a Hubble timescale, H(z)t  1  implying that expansion-mediated
gravitational coupling is of the same order as the Step 10 Newton observational projection term.

---

## 2. Friedmann H(z) Expansion Coupling

### 2.1 Friedmann Equation

The Hubble parameter as a function of redshift in the standard ?CDM cosmology:

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

For Andromeda parameters: H0 = 70.0 km/s/Mpc, O_m = 0.3, O_? = 0.7, z = -0.001.

$$H(z=-0.001) = 70.0 \times \sqrt{0.3 \times (0.999)^3 + 0.7}$$

$$= 70.0 \times \sqrt{0.3 \times 0.997003 + 0.7} = 70.0 \times \sqrt{0.9991}$$

$$= 70.0 \times 0.99955 = 69.969 \text{ km/s/Mpc}$$

In SI units: H_SI = 69.969 $\times$ 10 / 3.086$\times$10 = **2.269$\times$10?8 s-1**

### 2.2 UQFF Expansion Term

The new UQFF gravity coupling term is:

$$g_{\text{expansion}}(r, t) = \frac{G \cdot M}{r^2} \cdot H(z) \cdot t$$

This represents the gravitational effect of cosmological expansion flow coupling into UQFF buoyancy
at timescale t.

**Numerical values for Andromeda:**
- g_base = $\mu$_s$\nabla$(M_s/r) = 6.674$\times$10?  1.989$\times$104 / (1.04$\times$10) = **1.227$\times$10? m/s**
- At t = 1 Gyr (3.156$\times$10-6 s): g_expansion = 1.227$\times$10?  2.269$\times$10?8 $\times$ 3.156$\times$10-6 = **8.79$\times$10? m/s**
- At t = t_Hubble (4.352$\times$10-7 s): g_expansion = 1.227$\times$10? $\approx$ 0.987 = **1.211$\times$10? m/s**

---

## 3. H_UQFF Near-Unity Resonance Coefficient

### 3.1 Definition

$$H_{\text{UQFF}} = H(z) \times t_H$$

where t_H = 4.352$\times$10-7 s (Hubble time  13.8 Gyr).

**For Andromeda (z = -0.001):**

$$H_{\text{UQFF}} = 2.269 \times 10^{-18} \times 4.352 \times 10^{17} = \mathbf{0.987}$$

### 3.2 Near-Unity Discovery

H_UQFF $\approx$ 0.987  **1** is a remarkable result. It means:

> Over a Hubble timescale, the Friedmann expansion coupling adds an amount of gravitational acceleration equal to **98.7% of the Step 10 Newton observational projection term**  nearly doubling g_base.

This near-unity value is not coincidental. In a flat ?CDM universe (O_m + O_? = 1):

$$H_{\text{UQFF}} = H_0 \times t_H \approx 1.0$$

because H0  t_H is the dimensionless Hubble number $\approx$ 0.96§1.0 for observationally consistent H0.

**New UQFF Constant:** H_UQFF = H(z)$\cdot$t_H  **Friedmann-UQFF Near-Unity Resonance Coefficient**

| Parameter | Value |
|---|---|
| H0 | 70.0 km/s/Mpc |
| O_m | 0.3 |
| O_? | 0.7 |
| z (Andromeda) | -0.001 |
| H(z) | 2.269$\times$10?8 s-1 |
| t_H | 4.352$\times$10-7 s |
| **H_UQFF** | **0.987** (~1) |
| g_expansion(t_H) | 1.211$\times$10? m/s |
| g_base | 1.227$\times$10? m/s |
| Ratio g_exp/g_base | 98.7% |

### 3.3 Blueshift Sensitivity

For blueshift z < 0: H(z) is slightly **lower** than H0 (matter term O_m(1+z) is suppressed below
O_m for z < 0), meaning:

$$H(z < 0) < H_0 \Rightarrow H_{\text{UQFF}}(z < 0) < H_{\text{UQFF}}(z=0)$$

For Andromeda (z = -0.001): H_UQFF = 0.987 vs flat-universe H_UQFF0 = 0.9985. The blueshift
suppresses H_UQFF by 0.15%.

This is the inverse of the redshift behaviour (z > 0 ? H(z) > H0 for moderate z), making Andromeda's
approaching trajectory a unique laboratory for probing blueshift Friedmann-UQFF coupling.

---

## 4. ISM Dust Drag Acceleration

A second minor additive term captures ISM dust ram-pressure drag:

$$a_{\text{dust}} = \frac{\rho_{\text{dust}} \cdot v_{\text{orbit}}^2}{c^2 \cdot \rho_{\text{mean}}} \cdot g_{\text{base}}$$

where ?_mean = M/V_fluid = 1.989$\times$104 / 106 = 1.989$\times$10?8 kg/m.

**Numerical value for Andromeda:**
$$a_{\text{dust}} = \frac{10^{-20} \times (2.5 \times 10^5)^2}{(2.998 \times 10^8)^2 \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{8.988 \times 10^{16} \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{0.1788} \times 1.227 \times 10^{-10} = 4.29 \times 10^{-19} \text{ m/s}^2$$

At ~9 orders below g_base, the dust drag is physically negligible but dimensionally consistent and
additive within the UQFF sum.

---

## 5. M_visible / M_DM_mass Cascade

When M is updated via `updateVariable("M", value)`, the derived quantities are now automatically
recomputed:

$$M_{\text{visible}} = (1 - f_{\text{DM}}) \times M$$
$$M_{\text{DM,mass}} = f_{\text{DM}} \times M$$

For Andromeda defaults: M = 1.989$\times$104 kg, f_DM = 0.80:
- M_visible = 0.20 $\times$ 1.989$\times$104 = **3.978$\times$104 kg** (20% visible baryons)
- M_DM_mass = 0.80 $\times$ 1.989$\times$104 = **1.591$\times$104 kg** (80% dark matter)

These are tracked in `updateCache()` and exported via `exportState()` (params 2632), providing
explicit bookkeeping of the M31 baryonic/DM split consistent with PAPER_275.

---

## 6. Updated ANDROMEDA_UQFF_MODULE.cpp g_total

The full UQFF 2.0 equation for Andromeda with all PAPER_273276 terms:

$$g_{\text{total}}(r, t) = \left[ g_{\text{grav}} + U_{g,\text{sum}}^{(26)} + \Lambda\text{-term} + g_Q + g_L + g_{\text{fluid}} + F_{\text{res}}(\omega_{\text{HI}}) + g_{\text{DM}} + g_{\text{expansion}} + a_{\text{dust}} \right] \times \kappa_{\text{approach}}$$

**Term magnitudes at t=0 (current epoch):**

| Term | Value (m/s) | Paper |
|---|---|---|
| g_grav = $\mu$_s$\nabla$(M_s/r) | 1.227$\times$10? | baseline |
| Ug_sum (26 layers) | 6.380$\times$10?? | 26-layer Triadic |
| ?-term | 2.998$\times$10?6 | cosmological |
| g_quantum | ~2$\times$10?6 | HUP |
| g_Lorentz | ~2$\times$10?6 | IGM EM |
| g_fluid | ~8$\times$10? | IGM buoyancy |
| F_res(?_HI) at t=0 | 1$\times$10? | PAPER_274 |
| g_DM | 1.356$\times$10? | PAPER_275 |
| g_expansion at t=0 | 0 | PAPER_276 |
| a_dust | 4.29$\times$10?? | PAPER_276 |
| ?_approach | 1.001001 | PAPER_273 |

Note: g_expansion grows linearly with t, dominating over g_grav at t = t_H.

---

## 7. exportState Update (25 ? 32 parameters)

Seven new parameters added to `exportState()`:
1. `H0` = 70.0 km/s/Mpc
2. `Omega_m` = 0.3
3. `Omega_Lam` = 0.7
4. `Mpc_to_m` = 3.086$\times$10
5. `rho_dust` = 1$\times$10? kg/m
6. `M_visible` = (1f_DM)M
7. `M_DM_mass` = f_DMM

---

## 8. Conclusions

**Three new UQFF enhancements for Andromeda (PAPER_276):**

1. **Friedmann-UQFF Expansion Coupling**: g_expansion = g_base – H(z)  t. At t = t_H, adds 98.7% of
g_base  near-gravitational doubling over cosmological time.

2. **H_UQFF = 0.987**: New UQFF constant (Friedmann-UQFF Near-Unity Resonance Coefficient).
Near-unity reflects the fundamental relationship H0  t_H  1 in flat ?CDM. Andromeda's blueshift (z =
-0.001) makes H_UQFF 0.15% below flat-universe value  a blueshift Friedmann suppression.

3. **M_visible/M_DM_mass cascade + dust drag**: Explicit DM split bookkeeping (updateCache) and ISM
dust ram-pressure (a_dust  4$\times$10?? m/s).

---

**Watermark:** 2025-2026 Daniel T. Murphy, daniel.murphy00@gmail.com – All Rights Reserved
 
H_UQFF=H(z)*t_H~0.987 [PAPER_276]`

---

## Abstract

This paper introduces a new UQFF gravity term for the Andromeda Galaxy (M31): the **Friedmann-UQFF
Expansion Coupling**, defined as g_expansion = (\mu_s\nabla(M_s/r))  H(z)  t. For Andromeda's blueshift redshift z
= -0.001, the Friedmann H(z) equals 69.969 km/s/Mpc = 2.269\times10?8 s-1. Evaluated at the Hubble
timescale t = t_H = 4.352\times10-7 s, the dimensionless Friedmann-UQFF Resonance Coefficient H_UQFF =
H(z)\cdott_H  **0.987**  a near-unity value representing gravitational doubling over cosmological time.
This paper also introduces the M_visible/M_DM_mass cascade (explicit DM split tracking) and a minor
ISM dust drag acceleration term. Together these constitute three additive UQFF 2.0 enhancements
derived from the Andromeda module upgrade.

---

## 1. Background

The UQFF framework computes total gravitational acceleration g_total(r,t) as a sum of DPM-seeded,
Triadic (26-layer), cosmological, quantum, electromagnetic, fluid, resonant, and dark matter terms.
PAPER_273275 established the Andromeda-specific physics: blueshift approach amplifier ?, HI 21-cm
resonance, and DM 80/20 NFW partition.

The original Andromeda implementation (pre-Session 76) lacked a time-dependent expansion coupling
that accounts for the Friedmann-Lemaitre cosmological evolution of Hubble parameter H(z). This is
significant because over a Hubble timescale, H(z)t  1  implying that expansion-mediated
gravitational coupling is of the same order as the Step 10 Newton observational projection term.

---

## 2. Friedmann H(z) Expansion Coupling

### 2.1 Friedmann Equation

The Hubble parameter as a function of redshift in the standard ?CDM cosmology:

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

For Andromeda parameters: H0 = 70.0 km/s/Mpc, O_m = 0.3, O_? = 0.7, z = -0.001.

$$H(z=-0.001) = 70.0 \times \sqrt{0.3 \times (0.999)^3 + 0.7}$$

$$= 70.0 \times \sqrt{0.3 \times 0.997003 + 0.7} = 70.0 \times \sqrt{0.9991}$$

$$= 70.0 \times 0.99955 = 69.969 \text{ km/s/Mpc}$$

In SI units: H_SI = 69.969 \times 10 / 3.086\times10 = **2.269\times10?8 s-1**

### 2.2 UQFF Expansion Term

The new UQFF gravity coupling term is:

$$g_{\text{expansion}}(r, t) = \frac{G \cdot M}{r^2} \cdot H(z) \cdot t$$

This represents the gravitational effect of cosmological expansion flow coupling into UQFF buoyancy
at timescale t.

**Numerical values for Andromeda:**
- g_base = \mu_s\nabla(M_s/r) = 6.674\times10?  1.989\times104 / (1.04\times10) = **1.227\times10? m/s**
- At t = 1 Gyr (3.156\times10-6 s): g_expansion = 1.227\times10?  2.269\times10?8 \times 3.156\times10-6 = **8.79\times10? m/s**
- At t = t_Hubble (4.352\times10-7 s): g_expansion = 1.227\times10? \approx 0.987 = **1.211\times10? m/s**

---

## 3. H_UQFF Near-Unity Resonance Coefficient

### 3.1 Definition

$$H_{\text{UQFF}} = H(z) \times t_H$$

where t_H = 4.352\times10-7 s (Hubble time  13.8 Gyr).

**For Andromeda (z = -0.001):**

$$H_{\text{UQFF}} = 2.269 \times 10^{-18} \times 4.352 \times 10^{17} = \mathbf{0.987}$$

### 3.2 Near-Unity Discovery

H_UQFF \approx 0.987  **1** is a remarkable result. It means:

> Over a Hubble timescale, the Friedmann expansion coupling adds an amount of gravitational acceleration equal to **98.7% of the Step 10 Newton observational projection term**  nearly doubling g_base.

This near-unity value is not coincidental. In a flat ?CDM universe (O_m + O_? = 1):

$$H_{\text{UQFF}} = H_0 \times t_H \approx 1.0$$

because H0  t_H is the dimensionless Hubble number \approx 0.96§1.0 for observationally consistent H0.

**New UQFF Constant:** H_UQFF = H(z)\cdott_H  **Friedmann-UQFF Near-Unity Resonance Coefficient**

| Parameter | Value |
|---|---|
| H0 | 70.0 km/s/Mpc |
| O_m | 0.3 |
| O_? | 0.7 |
| z (Andromeda) | -0.001 |
| H(z) | 2.269\times10?8 s-1 |
| t_H | 4.352\times10-7 s |
| **H_UQFF** | **0.987** (~1) |
| g_expansion(t_H) | 1.211\times10? m/s |
| g_base | 1.227\times10? m/s |
| Ratio g_exp/g_base | 98.7% |

### 3.3 Blueshift Sensitivity

For blueshift z < 0: H(z) is slightly **lower** than H0 (matter term O_m(1+z) is suppressed below
O_m for z < 0), meaning:

$$H(z < 0) < H_0 \Rightarrow H_{\text{UQFF}}(z < 0) < H_{\text{UQFF}}(z=0)$$

For Andromeda (z = -0.001): H_UQFF = 0.987 vs flat-universe H_UQFF0 = 0.9985. The blueshift
suppresses H_UQFF by 0.15%.

This is the inverse of the redshift behaviour (z > 0 ? H(z) > H0 for moderate z), making Andromeda's
approaching trajectory a unique laboratory for probing blueshift Friedmann-UQFF coupling.

---

## 4. ISM Dust Drag Acceleration

A second minor additive term captures ISM dust ram-pressure drag:

$$a_{\text{dust}} = \frac{\rho_{\text{dust}} \cdot v_{\text{orbit}}^2}{c^2 \cdot \rho_{\text{mean}}} \cdot g_{\text{base}}$$

where ?_mean = M/V_fluid = 1.989\times104 / 106 = 1.989\times10?8 kg/m.

**Numerical value for Andromeda:**
$$a_{\text{dust}} = \frac{10^{-20} \times (2.5 \times 10^5)^2}{(2.998 \times 10^8)^2 \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{8.988 \times 10^{16} \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{0.1788} \times 1.227 \times 10^{-10} = 4.29 \times 10^{-19} \text{ m/s}^2$$

At ~9 orders below g_base, the dust drag is physically negligible but dimensionally consistent and
additive within the UQFF sum.

---

## 5. M_visible / M_DM_mass Cascade

When M is updated via `updateVariable("M", value)`, the derived quantities are now automatically
recomputed:

$$M_{\text{visible}} = (1 - f_{\text{DM}}) \times M$$
$$M_{\text{DM,mass}} = f_{\text{DM}} \times M$$

For Andromeda defaults: M = 1.989\times104 kg, f_DM = 0.80:
- M_visible = 0.20 \times 1.989\times104 = **3.978\times104 kg** (20% visible baryons)
- M_DM_mass = 0.80 \times 1.989\times104 = **1.591\times104 kg** (80% dark matter)

These are tracked in `updateCache()` and exported via `exportState()` (params 2632), providing
explicit bookkeeping of the M31 baryonic/DM split consistent with PAPER_275.

---

## 6. Updated ANDROMEDA_UQFF_MODULE.cpp g_total

The full UQFF 2.0 equation for Andromeda with all PAPER_273276 terms:

$$g_{\text{total}}(r, t) = \left[ g_{\text{grav}} + U_{g,\text{sum}}^{(26)} + \Lambda\text{-term} + g_Q + g_L + g_{\text{fluid}} + F_{\text{res}}(\omega_{\text{HI}}) + g_{\text{DM}} + g_{\text{expansion}} + a_{\text{dust}} \right] \times \kappa_{\text{approach}}$$

**Term magnitudes at t=0 (current epoch):**

| Term | Value (m/s) | Paper |
|---|---|---|
| g_grav = \mu_s\nabla(M_s/r) | 1.227\times10? | baseline |
| Ug_sum (26 layers) | 6.380\times10?? | 26-layer Triadic |
| ?-term | 2.998\times10?6 | cosmological |
| g_quantum | ~2\times10?6 | HUP |
| g_Lorentz | ~2\times10?6 | IGM EM |
| g_fluid | ~8\times10? | IGM buoyancy |
| F_res(?_HI) at t=0 | 1\times10? | PAPER_274 |
| g_DM | 1.356\times10? | PAPER_275 |
| g_expansion at t=0 | 0 | PAPER_276 |
| a_dust | 4.29\times10?? | PAPER_276 |
| ?_approach | 1.001001 | PAPER_273 |

Note: g_expansion grows linearly with t, dominating over g_grav at t = t_H.

---

## 7. exportState Update (25 ? 32 parameters)

Seven new parameters added to `exportState()`:
1. `H0` = 70.0 km/s/Mpc
2. `Omega_m` = 0.3
3. `Omega_Lam` = 0.7
4. `Mpc_to_m` = 3.086\times10
5. `rho_dust` = 1\times10? kg/m
6. `M_visible` = (1f_DM)M
7. `M_DM_mass` = f_DMM

---

## 8. Conclusions

**Three new UQFF enhancements for Andromeda (PAPER_276):**

1. **Friedmann-UQFF Expansion Coupling**: g_expansion = g_base – H(z)  t. At t = t_H, adds 98.7% of
g_base  near-gravitational doubling over cosmological time.

2. **H_UQFF = 0.987**: New UQFF constant (Friedmann-UQFF Near-Unity Resonance Coefficient).
Near-unity reflects the fundamental relationship H0  t_H  1 in flat ?CDM. Andromeda's blueshift (z =
-0.001) makes H_UQFF 0.15% below flat-universe value  a blueshift Friedmann suppression.

3. **M_visible/M_DM_mass cascade + dust drag**: Explicit DM split bookkeeping (updateCache) and ISM
dust ram-pressure (a_dust  4\times10?? m/s).

---

**Watermark:** 2025-2026 Daniel T. Murphy, daniel.murphy00@gmail.com – All Rights Reserved

  
H_UQFF=H(z)*t_H~0.987 [PAPER_276]`

---

## Abstract

This paper introduces a new UQFF gravity term for the Andromeda Galaxy (M31): the **Friedmann-UQFF
Expansion Coupling**, defined as g_expansion = ($\mu$_s$\nabla$(M_s/r))  H(z)  t. For Andromeda's blueshift redshift z
= -0.001, the Friedmann H(z) equals 69.969 km/s/Mpc = 2.269$\times$10?8 s-1. Evaluated at the Hubble
timescale t = t_H = 4.352$\times$10-7 s, the dimensionless Friedmann-UQFF Resonance Coefficient H_UQFF =
H(z)$\cdot$t_H  **0.987**  a near-unity value representing gravitational doubling over cosmological time.
This paper also introduces the M_visible/M_DM_mass cascade (explicit DM split tracking) and a minor
ISM dust drag acceleration term. Together these constitute three additive UQFF 2.0 enhancements
derived from the Andromeda module upgrade.

---

## 1. Background

The UQFF framework computes total gravitational acceleration g_total(r,t) as a sum of DPM-seeded,
Triadic (26-layer), cosmological, quantum, electromagnetic, fluid, resonant, and dark matter terms.
PAPER_273275 established the Andromeda-specific physics: blueshift approach amplifier ?, HI 21-cm
resonance, and DM 80/20 NFW partition.

The original Andromeda implementation (pre-Session 76) lacked a time-dependent expansion coupling
that accounts for the Friedmann-Lemaitre cosmological evolution of Hubble parameter H(z). This is
significant because over a Hubble timescale, H(z)t  1  implying that expansion-mediated
gravitational coupling is of the same order as the Step 10 Newton observational projection term.

---

## 2. Friedmann H(z) Expansion Coupling

### 2.1 Friedmann Equation

The Hubble parameter as a function of redshift in the standard ?CDM cosmology:

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

For Andromeda parameters: H0 = 70.0 km/s/Mpc, O_m = 0.3, O_? = 0.7, z = -0.001.

$$H(z=-0.001) = 70.0 \times \sqrt{0.3 \times (0.999)^3 + 0.7}$$

$$= 70.0 \times \sqrt{0.3 \times 0.997003 + 0.7} = 70.0 \times \sqrt{0.9991}$$

$$= 70.0 \times 0.99955 = 69.969 \text{ km/s/Mpc}$$

In SI units: H_SI = 69.969 $\times$ 10 / 3.086$\times$10 = **2.269$\times$10?8 s-1**

### 2.2 UQFF Expansion Term

The new UQFF gravity coupling term is:

$$g_{\text{expansion}}(r, t) = \frac{G \cdot M}{r^2} \cdot H(z) \cdot t$$

This represents the gravitational effect of cosmological expansion flow coupling into UQFF buoyancy
at timescale t.

**Numerical values for Andromeda:**
- g_base = $\mu$_s$\nabla$(M_s/r) = 6.674$\times$10?  1.989$\times$104 / (1.04$\times$10) = **1.227$\times$10? m/s**
- At t = 1 Gyr (3.156$\times$10-6 s): g_expansion = 1.227$\times$10?  2.269$\times$10?8 $\times$ 3.156$\times$10-6 = **8.79$\times$10? m/s**
- At t = t_Hubble (4.352$\times$10-7 s): g_expansion = 1.227$\times$10? $\approx$ 0.987 = **1.211$\times$10? m/s**

---

## 3. H_UQFF Near-Unity Resonance Coefficient

### 3.1 Definition

$$H_{\text{UQFF}} = H(z) \times t_H$$

where t_H = 4.352$\times$10-7 s (Hubble time  13.8 Gyr).

**For Andromeda (z = -0.001):**

$$H_{\text{UQFF}} = 2.269 \times 10^{-18} \times 4.352 \times 10^{17} = \mathbf{0.987}$$

### 3.2 Near-Unity Discovery

H_UQFF $\approx$ 0.987  **1** is a remarkable result. It means:

> Over a Hubble timescale, the Friedmann expansion coupling adds an amount of gravitational acceleration equal to **98.7% of the Step 10 Newton observational projection term**  nearly doubling g_base.

This near-unity value is not coincidental. In a flat ?CDM universe (O_m + O_? = 1):

$$H_{\text{UQFF}} = H_0 \times t_H \approx 1.0$$

because H0  t_H is the dimensionless Hubble number $\approx$ 0.96§1.0 for observationally consistent H0.

**New UQFF Constant:** H_UQFF = H(z)$\cdot$t_H  **Friedmann-UQFF Near-Unity Resonance Coefficient**

| Parameter | Value |
|---|---|
| H0 | 70.0 km/s/Mpc |
| O_m | 0.3 |
| O_? | 0.7 |
| z (Andromeda) | -0.001 |
| H(z) | 2.269$\times$10?8 s-1 |
| t_H | 4.352$\times$10-7 s |
| **H_UQFF** | **0.987** (~1) |
| g_expansion(t_H) | 1.211$\times$10? m/s |
| g_base | 1.227$\times$10? m/s |
| Ratio g_exp/g_base | 98.7% |

### 3.3 Blueshift Sensitivity

For blueshift z < 0: H(z) is slightly **lower** than H0 (matter term O_m(1+z) is suppressed below
O_m for z < 0), meaning:

$$H(z < 0) < H_0 \Rightarrow H_{\text{UQFF}}(z < 0) < H_{\text{UQFF}}(z=0)$$

For Andromeda (z = -0.001): H_UQFF = 0.987 vs flat-universe H_UQFF0 = 0.9985. The blueshift
suppresses H_UQFF by 0.15%.

This is the inverse of the redshift behaviour (z > 0 ? H(z) > H0 for moderate z), making Andromeda's
approaching trajectory a unique laboratory for probing blueshift Friedmann-UQFF coupling.

---

## 4. ISM Dust Drag Acceleration

A second minor additive term captures ISM dust ram-pressure drag:

$$a_{\text{dust}} = \frac{\rho_{\text{dust}} \cdot v_{\text{orbit}}^2}{c^2 \cdot \rho_{\text{mean}}} \cdot g_{\text{base}}$$

where ?_mean = M/V_fluid = 1.989$\times$104 / 106 = 1.989$\times$10?8 kg/m.

**Numerical value for Andromeda:**
$$a_{\text{dust}} = \frac{10^{-20} \times (2.5 \times 10^5)^2}{(2.998 \times 10^8)^2 \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{8.988 \times 10^{16} \times 1.989 \times 10^{-18}} \times 1.227 \times 10^{-10}$$

$$= \frac{6.25 \times 10^{-10}}{0.1788} \times 1.227 \times 10^{-10} = 4.29 \times 10^{-19} \text{ m/s}^2$$

At ~9 orders below g_base, the dust drag is physically negligible but dimensionally consistent and
additive within the UQFF sum.

---

## 5. M_visible / M_DM_mass Cascade

When M is updated via `updateVariable("M", value)`, the derived quantities are now automatically
recomputed:

$$M_{\text{visible}} = (1 - f_{\text{DM}}) \times M$$
$$M_{\text{DM,mass}} = f_{\text{DM}} \times M$$

For Andromeda defaults: M = 1.989$\times$104 kg, f_DM = 0.80:
- M_visible = 0.20 $\times$ 1.989$\times$104 = **3.978$\times$104 kg** (20% visible baryons)
- M_DM_mass = 0.80 $\times$ 1.989$\times$104 = **1.591$\times$104 kg** (80% dark matter)

These are tracked in `updateCache()` and exported via `exportState()` (params 2632), providing
explicit bookkeeping of the M31 baryonic/DM split consistent with PAPER_275.

---

## 6. Updated ANDROMEDA_UQFF_MODULE.cpp g_total

The full UQFF 2.0 equation for Andromeda with all PAPER_273276 terms:

$$g_{\text{total}}(r, t) = \left[ g_{\text{grav}} + U_{g,\text{sum}}^{(26)} + \Lambda\text{-term} + g_Q + g_L + g_{\text{fluid}} + F_{\text{res}}(\omega_{\text{HI}}) + g_{\text{DM}} + g_{\text{expansion}} + a_{\text{dust}} \right] \times \kappa_{\text{approach}}$$

**Term magnitudes at t=0 (current epoch):**

| Term | Value (m/s) | Paper |
|---|---|---|
| g_grav = $\mu$_s$\nabla$(M_s/r) | 1.227$\times$10? | baseline |
| Ug_sum (26 layers) | 6.380$\times$10?? | 26-layer Triadic |
| ?-term | 2.998$\times$10?6 | cosmological |
| g_quantum | ~2$\times$10?6 | HUP |
| g_Lorentz | ~2$\times$10?6 | IGM EM |
| g_fluid | ~8$\times$10? | IGM buoyancy |
| F_res(?_HI) at t=0 | 1$\times$10? | PAPER_274 |
| g_DM | 1.356$\times$10? | PAPER_275 |
| g_expansion at t=0 | 0 | PAPER_276 |
| a_dust | 4.29$\times$10?? | PAPER_276 |
| ?_approach | 1.001001 | PAPER_273 |

Note: g_expansion grows linearly with t, dominating over g_grav at t = t_H.

---

## 7. exportState Update (25 ? 32 parameters)

Seven new parameters added to `exportState()`:
1. `H0` = 70.0 km/s/Mpc
2. `Omega_m` = 0.3
3. `Omega_Lam` = 0.7
4. `Mpc_to_m` = 3.086$\times$10
5. `rho_dust` = 1$\times$10? kg/m
6. `M_visible` = (1f_DM)M
7. `M_DM_mass` = f_DMM

---

## 8. Conclusions

**Three new UQFF enhancements for Andromeda (PAPER_276):**

1. **Friedmann-UQFF Expansion Coupling**: g_expansion = g_base – H(z)  t. At t = t_H, adds 98.7% of
g_base  near-gravitational doubling over cosmological time.

2. **H_UQFF = 0.987**: New UQFF constant (Friedmann-UQFF Near-Unity Resonance Coefficient).
Near-unity reflects the fundamental relationship H0  t_H  1 in flat ?CDM. Andromeda's blueshift (z =
-0.001) makes H_UQFF 0.15% below flat-universe value  a blueshift Friedmann suppression.

3. **M_visible/M_DM_mass cascade + dust drag**: Explicit DM split bookkeeping (updateCache) and ISM
dust ram-pressure (a_dust  4$\times$10?? m/s).

---

**Watermark:** 2025-2026 Daniel T. Murphy, daniel.murphy00@gmail.com – All Rights Reserved

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

<!-- 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.





## §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 $0.064$ (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:

$$p_{\mathrm{DVP}} = 17, \quad n_{\mathrm{channel}} = 17/26$$

Since $p_{\mathrm{DVP}} = 17$ 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 = 0.064 | PASS Threshold-consistent |
| DVP prime | $p_k \in$ {2,3,...,113} | $p_{\mathrm{DVP}} = 17$ | 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_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_1015 | SCm Dark Matter Halos NFW Rotation Curve |
| PAPER_1019 | Dark Matter Phonon Buoyancy NFW Coupling |
| PAPER_1020 | Cosmic Ray Phonon Acceleration DSA Spectrum |
| PAPER_1033 | Galactic Bar Resonance SCm Pattern Speed |

*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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