TRIDENT-Z0 Diagnostic Framework: Conventional BCS Superconductivity as an Intrinsic Macroscopic Quantum Coherence System: PHYS-BCS-001.v1

This technical note applies the TRIDENT-Z0 diagnostic framework to conventional phonon-mediated BCS superconductivity as an intrinsic macroscopic quantum coherence system. Below a critical temperature Tc, certain metals exhibit zero DC resistance, Meissner flux expulsion, and long-range phase coherence arising from Cooper pairing. The superconducting state is characterized by a non-zero complex order parameter psi and an energy gap Delta. Scope boundary: This evaluation is restricted to conventional weak-coupling BCS superconductors. Excluded are Bose–Einstein condensation, unconventional high-Tc materials, strong-coupling Eliashberg corrections, non-phonon pairing mechanisms, and zero-temperature quantum critical transitions. Quantitative structure is made explicit. The weak-coupling critical temperature satisfies kB Tc = 1.13 hbar omega_D exp(-1/lambda_ep), where lambda_ep = N(0)V is the dimensionless electron–phonon coupling parameter. The zero-temperature gap ratio 2 Delta(0) / (kB Tc) approx 3.53 provides a parameter-free falsifiability invariant. The reduced temperature t = T / Tc defines the boundary scalar, with coherence collapse at t = 1.G1–G3 pass: superconductivity is reproducible; the electron–phonon pairing mechanism is microscopically grounded; and predictive relations are quantitatively testable. G4 receives a full PASS: below Tc, macroscopic phase coherence and a spectral gap are present; at T >= Tc, the gap closes and coherence vanishes. The transition is finite-temperature thermodynamic, reversible, and boundary-defined. This case functions as the quantum macroscopic coherence anchor for the PHYS node of the TRIDENT-Z0 corpus, demonstrating intrinsic equilibrium coherence under explicitly defined constraint conditions.KEYWORDS TRIDENTSHRFBCS superconductivityCooper pairingelectron-phonon couplingmacroscopic quantum coherenceMeissner effectorder parameterthermodynamic phase transitioncritical temperaturedimensionless invariantsfalsifiability