Cavity theory forecasts voter conviction crossover at 3% median error on sparse graphs up to N=720

The study maps trimodal survey fields against a tunable fraction alpha of antagonistic +-J bonds on graphs with mean degree 3-28. Exact ground-state optimizations up to N=180 and validated heuristics to N=720 show r_h* tracks the zero-field coupling energy e_J(0) within a rigorous factor-of-two bound per instance, yielding the scaling r_h* ~ <k>^gamma with gamma near 1.0 at low frustration and 0.5 at high frustration. At intermediate alpha=0.2 the replica-symmetric solution becomes unstable and the theory systematically underestimates the density where cooperative bonds saturate. The model therefore supplies a parameter-free route from network density and antagonism level to the point at which private convictions begin to override social pressure, directly relevant to adoption thresholds in polarized electorates or technology markets. Limitations include the zero-temperature assumption and the absence of dynamic rewiring; both are addressable by finite-temperature Monte Carlo and temporal network extensions now under way.