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CLAIM 1 (PAPER_1066): Mexican-hat SCm Lagrangian
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Stationary points of V(phi): [v]
V(phi_0 = v) = 0
Claim (a) V(phi_0) = -rho_SCm  --> FAILED
  Reason: lambda*(phi^2 - v^2)^2 is non-negative, min = 0, NOT -rho_SCm.
  To get -rho_SCm requires an UNDOCUMENTED additive offset -rho_SCm
  that does not appear in PAPER_1066 Section 1.

Coefficient of eta^2 in V(v+eta) = 4*lambda*v**2
=> m_phonon^2 = 8*lambda*v**2
=> m_phonon  = 2*sqrt(2)*sqrt(lambda)*v
Claim (b) m_phonon = sqrt(8 lambda) v  --> DERIVED

Euler-Lagrange residual (should equal box-equation of motion):
   4*lambda*(v**2 - phi(t)**2)*phi(t) - Derivative(phi(t), (t, 2)) = 0
Claim (c) EL recovers KG-type EOM  --> DERIVED

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CLAIM 2 (PAPER_1065): Buoyancy EOM from L_UQFF
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Euler-Lagrange residual for r:
   m*(-M_s*mu_s + (-g_buoy - g_phonon - Derivative(r(t), (t, 2)))*r(t)**2)/r(t)**2 = 0
=> r-double-dot = -M_s*mu_s/r(t)**2 - g_buoy - g_phonon

PAPER_1065 claim: rdd = -mu_s d/dr(M_s/r) + g_buoy + g_phonon
=> claim     = M_s*mu_s/r(t)**2 + g_buoy + g_phonon
=> derived   = -M_s*mu_s/r(t)**2 - g_buoy - g_phonon
=> residual  = -2*M_s*mu_s/r(t)**2 - 2*g_buoy - 2*g_phonon

Claim 2 verdict: FAILED -- charitable Lagrangian does NOT yield the boxed EOM
Note: PAPER_1065 does NOT specify functional forms for V_buoy or
L_phonon. Without explicit forms, NO variation is computable from
the paper as written. The 'derivation' is structurally an identification.

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AUDIT SUMMARY
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  [FAILED    ] PAPER_1066 (a) V(phi_0) = -rho_SCm
               min of lambda(phi^2-v^2)^2 is 0, not -rho_SCm. Needs hidden offset.
  [DERIVED   ] PAPER_1066 (b) m_phonon = sqrt(8 lambda) v
               Standard Mexican-hat second-derivative; verified by SymPy.
  [DERIVED   ] PAPER_1066 (c) Klein-Gordon EOM via EL
               EL machinery yields phi_tt = -dV/dphi correctly.
  [IDENTIFIED] PAPER_1065 Buoyancy EOM via EL
               V_buoy, L_phonon never given explicitly in paper. Cannot vary what isn't written.

Net finding:
  PAPER_1066 partially derives (phonon mass + KG-EOM ok),
  but its V(phi_0) = -rho_SCm claim is FALSE for the written potential.
  PAPER_1065 contains NO actual variation -- it is an IDENTIFICATION.
  The boxed 'delta S / delta phi = 0' acts as a label, not a derivation.
  Every downstream paper (PAPER_001-049, PAPER_877, Millennium chain)
  that cites PAPER_1065 as variational origin inherits this gap.
