#!/usr/bin/env python3
"""
DCT master identity verification: 2 omega_0 + 3 = c_BD * P_0**2 = 100,077.

Inputs (from DCT-FND-V2):
  P_0   = 0.851           (Brans-Dicke condensate equilibrium amplitude)
  c_BD  = 138189          (BD coupling normalisation; RG fixed point per DCT_31)
  omega_0 = 50037         (BD coupling at equilibrium)
"""
P_0     = 0.851
c_BD    = 138189
omega_0 = 50037

lhs = 2 * omega_0 + 3
rhs = c_BD * P_0 ** 2

print(f"2 * omega_0 + 3 = {lhs}")
print(f"c_BD * P_0**2   = {rhs:.4f}")
print(f"Difference      = {lhs - rhs:.4f}")
print(f"Relative match  = {abs(lhs - rhs) / lhs:.3e}")