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STEP 1.  Postulated Mexican-hat potential
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V(UA) = K*rho*(UA**2/v**2 - 1)**2
     = K*UA**4*rho/v**4 - 2*K*UA**2*rho/v**2 + K*rho

a0 = coeff(UA^0) = K*rho
a2 = coeff(UA^2) = -2*K*rho/v**2
a4 = coeff(UA^4) = K*rho/v**4

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STEP 2.  Stationarity dV/dUA = 0  (Euler-Lagrange for static UA)
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dV/dUA = 4*K*UA*rho*(UA**2 - v**2)/v**4
Solutions: UA = [v]

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STEP 3.  Mass-squared at the minimum UA = v
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V''(UA)     = 4*K*rho*(3*UA**2 - v**2)/v**4
V''(v)      = m^2 = 8*K*rho/v**2

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STEP 4.  PAPER_1166 'consistency checks' -- do they fix K?
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  Check (a):  a2^2/(4 a4) = K*rho
              a0          = K*rho
              difference  = 0
              VERDICT: holds IDENTICALLY for ANY value of K.
                       Does NOT constrain K.

  Check (b):  V(v) = 0
              VERDICT: V(v) = 0 holds for ANY K (geometric property
                       of the Mexican-hat shape).  Does NOT fix K.

  Check (c):  |a2|/a4 = 2*v**2
              VERDICT: ratio = 2 v^2 holds for ANY K.
                       Does NOT fix K.

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STEP 5.  What additional condition would actually fix K?
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Solve V''(v) = m^2 for K, given an INDEPENDENTLY known m^2:
  K = m_sq*v**2/(8*rho)

So K is fixed ONLY IF m_UA^2 (the physical mass of the aether scalar)
is known from outside the potential -- e.g. from a measurement, an RG
matching condition, or a one-loop Coleman-Weinberg calculation.

PAPER_1166 plugs m^2 = (50/3) rho/v^2 INTO this equation and recovers
K = 25/12.  But m^2 = (50/3) rho/v^2 is NOT independently derived in
the paper -- it is obtained by ALREADY ASSUMING K = 25/12.  This is
circular.

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STEP 6.  Does Mexican-hat shape force K = Phi_res*|SO(5)|/D_phys?
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  Claimed:  K = Phi_res * |SO(5)| / D_phys = (5/6)*(10)/4
          = 25/12  =  25/12  =  25/12

  Question: is there any symbolic property of the polynomial
            V(UA) = K rho [(UA/v)^2 - 1]^2 that requires K = 25/12?

  Answer (from SymPy variations above):  NO.
  The Mexican-hat shape fixes:
    - minimum location:  UA_min = +/- v
    - minimum value:     V_min  = 0
    - degree-4 polynomial structure
  The Mexican-hat shape does NOT fix:
    - the prefactor K
    - the scalar mass m_UA^2 = 8 K rho / v^2  (depends on K)

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VERDICT  (SymPy, not narration)
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CLAIM (PAPER_1166):  K_Mex = 25/12 is derived from first principles.

REALITY  (this script):
  The Mexican-hat scalar potential
       V(UA) = K rho_SCm [(UA/v_UA)^2 - 1]^2
  has a one-parameter family of physically distinct realizations
  indexed by K > 0.  No internal consistency condition of this
  potential fixes K.

  All three 'consistency checks' in PAPER_1166 are tautologies of the
  Mexican-hat polynomial shape:
     a2^2/(4 a4) = a0     holds for every K
     V(v) = 0             holds for every K
     |a2|/a4 = 2 v^2      holds for every K

  The claim K = 25/12 is therefore an IDENTIFICATION with a group-
  theoretic combination, NOT a derivation from the action.

  To make K = 25/12 a REAL derivation requires ONE of:
    (i)   an independent measurement of m_UA (none in the repo)
    (ii)  a Coleman-Weinberg one-loop calculation that produces 25/12
          from the matter content circulating in the loop
    (iii) an RG matching condition at a UV scale where K is fixed
          by a higher-symmetry theory
    (iv)  a dimensional-reduction argument from 26D -> 4D where K
          drops out of integrating over the compactified manifold

  None of (i)-(iv) is present in PAPER_1166, PAPER_1167, or any
  Lagrangian script reviewed in this thread.

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