---
paper_id: PAPER_1089
title: "Inflation Buoyancy Sector Lagrangian with Stationarity Constraint"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: ['Lagrangian', 'inflation', 'buoyancy', 'stationarity', 'action-principle', 'Euler-Lagrange']
crosslinks: [PAPER_1088, PAPER_1090, PAPER_1084, PAPER_892]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER_1089: Inflation Buoyancy Sector Lagrangian with Stationarity Constraint

## Abstract

We construct the Lagrangian for the inflation buoyancy sector
of the UQFF framework:

$$\mathcal{L}_{\text{infl}} = -\beta_i \cdot U_g \cdot \Omega \cdot \frac{M}{d} \cdot [\text{UA}] + F_n \cdot \Phi$$

The action $S = \int \mathcal{L}_{\text{infl}}\, dt$ is varied with
respect to the inflaton field $\phi_{\text{infl}}$ to obtain the
stationarity condition $\delta S / \delta \phi_{\text{infl}} = 0$.
The Lagrangian decomposes into a gravitational potential term
and a neutron-phonon driving term, whose balance determines
the inflationary slow-roll dynamics.

## §1 Lagrangian Construction

### §1.1 Gravity Term

$$\mathcal{L}_{\text{grav}} = -\beta_i \cdot U_g \cdot \Omega \cdot \frac{M}{d} \cdot [\text{UA}]$$

with:

- $\beta_i = 0.603$ (buoyancy coupling)
- $U_g = \mu_s\cdot\nabla(M_s/r)$ (gravitational potential energy)
- $\Omega$ = angular rotation frequency
- $M/d$ = mass-to-distance linear density
- $[\text{UA}] = 10^{-4}$ (universal abundance parameter)

### §1.2 Neutron-Phonon Term

$$\mathcal{L}_{\text{neutron}} = F_n \cdot \Phi$$

with $F_n = 10^{-10}$ N (neutron exchange force) and
$\Phi = \Phi_0 \cdot S_{26}$ (phonon amplitude).

### §1.3 Total Lagrangian

$$\mathcal{L}_{\text{infl}} = \mathcal{L}_{\text{grav}} + \mathcal{L}_{\text{neutron}}$$

## §2 Stationarity and Euler-Lagrange Equation

The action principle requires:

$$\frac{\delta S}{\delta \phi_{\text{infl}}} = \frac{\partial \mathcal{L}}{\partial \phi} - \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{\phi}} = 0$$

Since $\mathcal{L}_{\text{infl}}$ depends on $\phi$ through $U_g(\phi)$
and $\Phi(\phi)$:

$$-\beta_i \Omega \frac{M}{d} [\text{UA}] \frac{\partial U_g}{\partial \phi} + F_n \frac{\partial \Phi}{\partial \phi} = 0$$

## §3 Gravity vs Neutron Balance

At stationarity, the ratio:

$$\left|\frac{\mathcal{L}_{\text{grav}}}{\mathcal{L}_{\text{neutron}}}\right| = \frac{\beta_i \cdot U_g \cdot \Omega \cdot M \cdot [\text{UA}]}{d \cdot F_n \cdot \Phi}$$

determines whether inflation is gravity-dominated ($>1$) or phonon-driven ($<1$).

## §4 Benchmark: Solar System

| Parameter | Value | Source |
|-----------|-------|--------|
| $M$ | $1.989 \times 10^{30}$ kg | Solar mass |
| $r = d$ | $1.496 \times 10^{11}$ m | 1 AU |
| $\Omega$ | $2.87 \times 10^{-6}$ rad/s | Solar rotation |
| $U_g$ | $8.87 \times 10^{29}$ J | $GM_\odot/r$ |
| $\mathcal{L}_{\text{grav}}$ | $-2.06 \times 10^{8}$ J | Gravity sector |
| $\mathcal{L}_{\text{neutron}}$ | $1.86 \times 10^{11}$ J | Neutron-phonon |
| Ratio | $1.11 \times 10^{-3}$ | Phonon-dominated |

## §5 Physical Interpretation

The inflation buoyancy sector is phonon-dominated under solar conditions,
meaning the SCm vacuum lattice oscillation provides the dominant
energy injection for inflationary expansion. The gravitational
restoring term acts as a regulator preventing runaway inflation.

## References

- PAPER_1088: $F_{U,Bi,i}$ Seven-Component Decomposition
- PAPER_1090: Dark Energy Buoyancy Sector Lagrangian
- PAPER_1084: SCm Phonon Inflationary Scale Factor
- PAPER_892: FUBiUnifiedFieldBuoyancyForceCalc


### Key References with arXiv/DOI Identifiers

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