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Buoyancy Lagrangian Euler-Lagrange EOM Calculator
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── Sgr A* System (astrophysical scale) ──
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  BUOYANCY LAGRANGIAN EULER-LAGRANGE DERIVATION
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  §1. Buoyancy Sector Lagrangian Density:
    L_buoy = ½(∂_μ φ_buoy)² − V(φ_buoy) + J_buoy · φ_buoy
  
  §2. Potential (from 4-layer Ug summation):
    V(φ) = ½ m_eff² φ²
    m_eff² = β_i·Σ|Ug_i|·Ω_g·M/(d_g·c²·ħ²)·[UA]
           = 0.603·4.0000e-10·1.0·8.5527e+36/(2.4400e+20·c²·ħ²)·0.0001
           = 8.451281e+53  kg²
    m_eff  = 9.193085e+26  kg  (5.1578e+62 eV)
  
  §3. Source Current (vacuum displacement → buoyancy reversal):
    J_buoy = (F_{U,Bi}/F_U − 1) · ρ_SCm · c²
           = (1.1/1.0 − 1) · 7.0900e-37 · c²
           = 6.372495e-21  kg/m³·(m/s)²
    Buoyancy reversed: True
  
  §4. Euler-Lagrange Equation:
    δS/δφ_buoy = 0
    ∂L/∂φ − ∂_μ(∂L/∂(∂_μφ)) = 0
    −m_eff²·φ + J_buoy − □φ = 0
  
    ══> □φ + m_eff²·φ = J_buoy  (Klein-Gordon with source)
  
  §5. Static Solution (∂_t φ = 0):
    ∇²φ = m_eff²·φ − J_buoy
    φ_static(r) = (J_buoy/m_eff²)·[1 − exp(−m_eff·r)]
    φ_∞ = J_buoy/m_eff² = 7.540271e-75
  
  §6. Buoyancy Range (Compton wavelength):
    λ_buoy = ħ/(m_eff·c) = 3.827891e-70 m
           = 1.240405e-86 pc
    This is the distance over which SCm vacuum displacement
    produces measurable buoyancy anomalies.
  
  §7. Consistency Check (existing Ubi values from δS/δΩ_g = 0):
    Ug1 = 1.0000e-10  →  Ubi1 = -2.1136e+02
    Ug2 = 1.0000e-10  →  Ubi2 = -2.1136e+02
    Ug3 = 1.0000e-10  →  Ubi3 = -2.1136e+02
    Ug4 = 1.0000e-10  →  Ubi4 = -2.1136e+02
    F_{U,Bi} (total) = -8.4546e+02
  
  §8. Connection to Existing Variational Equations:
    ✓ δS/δA_SCm = 0  →  ∇×B_SCm = μ₀ J_SCm  (magnetic, L466)
    ✓ δS/δΩ_g  = 0  →  Ubi_i (buoyancy forces, L537)
    ✓ δS/δφ̂    = 0  →  Um (helical string magnetism, L543)
    ★ δS/δφ_buoy = 0  →  □φ + m_eff²φ = J_buoy  [THIS MODULE]
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  EOM: □φ + m_eff²·φ = J_buoy
  m_eff = 9.193085e+26 kg
  λ_buoy = 3.827891e-70 m
  φ_∞ = 7.540271e-75
  J_buoy = 6.372495e-21

── Lab-Scale System (Colman-Gillespie, 0.5 kg) ──
  EOM: □φ + m_eff²·φ = J_buoy
  m_eff = 1.813268e+21 kg
  λ_buoy = 1.940702e-64 m
  φ_∞ = 9.690708e-64
  J_buoy = 3.186247e-21
  Buoyancy reversed: True

── Complete Variational Equations (9-Sector Lagrangian) ──
    ✓   δS/δA_SCm = 0 → ∇×B_SCm = μ₀ J_SCm
    ✓   δS/δΩ_g = 0 → Ubi_i = -β_i Ug_i Ω_g M/d_g [UA]
    ✓   δS/δφ̂ = 0 → Um = Σ μ_j/r_j (1-e^{-γt}) φ̂ N P E
  ★ NEW δS/δφ_buoy = 0 → □φ + m_eff²φ = J_buoy  [NEW]

[OK] Results exported: buoyancy_lagrangian_eom_results.json

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