Variational Shadows of Superconductivity: Zeta-Minimizer Theorem Heuristics for Pressure-Tuned Phase Diagrams

The Zeta-Minimizer Theorem (ZMT) provides a variational framework deriving num-ber-theoretic structures (primes, zeta) as shadows of optimization in measure spaces with helical symmetries. Here, we apply ZMT heuristics to superconductivity phase diagrams under pressure, modeling Tc(P) behaviors as projections of Gibbs-like free energy landscapes. Ideal mixtures (convex weighted exponential decays) capture monotonic suppression or activation, while non-ideal excess quadratic terms (from Hessian distortions) generate domes via non-convexity and phase coexistence shadows. Across seven diverse case studies—iron pnictides, cuprates, hydrides (H₃S/Y-H), nickelates, borides (ZrB₁₂), and chalcogenides (PdSSe)—ZMT-inspired fits (e.g., Margules-like excess) achieve R² > 0.95, unifying classical thermodynamics (Gibbs-Duhem equilibria) with quantum scales (frequency embedding via spectral minima). This heuristic bridge complements Cooper pair theory, offering predictive insights for material design without overfitting. ZMT demotes microscopic mechanisms to derived artifacts of deeper variational principles, suggesting new avenues for room-temperature superconductivity.