The Scretching–Planck Action and Superconductivity: Pair Density, Phase Circulation and the Critical Fields of Real Superconductors

We write the Cooper-pair order parameter in Scretching–Planck (SP) variables as (ψ=√P,eiS/), with charge magnitude (2e) and effective pair mass (m^*). Expressing the Ginzburg–Landau free energy in these variables gives the stationary balance (QSP+12m^*v_s^2+α+β P=0) and the corresponding supercurrent equation. At constant condensate density, the phase equation for the current recovers the London response and penetration depth. Phase single-valuedness yields fluxoid quantization with (Φ_0=h/(2e)=2.0678×10⁻¹⁵) Wb and accounts for Little–Parks periodicity. The gauge-invariant phase difference between two coupled condensates gives the Josephson relations, with (K_J=2e/h=483597.848) GHz/V. The SP amplitude-gradient term determines the Ginzburg–Landau coherence length. The computed normal–superconducting interface energy changes sign at (κ=0.707107), and the single-vortex calculation gives (Hc1/H_c=1.000000) at (κ=1/√2) and recovers the large-(κ) limit. For a disordered niobium model with (κ=1.61), the calculation gives (μ_0Hc1=112) mT. In (d)-wave rings, a net (π) phase shift permits half-integer fluxoid quantization. The momentum-space BCS calculation gives the weak-coupling ratios (2Δ_0/(k_BT_c)=3.5279) and (Δ C/(γ T_c)=1.426). Using the volumetric electronic specific-heat coefficient (γ), the condensation-energy relation (B_c(0)=(Δ_0/k_B)√3μ_0γ/(2π^2)) reproduces the measured critical fields of Al, In, Sn, and Ta within (2.4%). Deviations of (7%) to (15%) for Hg, Pb, and Nb quantify the limits of the weak-coupling estimate for those materials. The free-electron estimates place all eight elements on the type-I side in the clean-limit calculation, including Nb, illustrating the sensitivity of the classification to the penetration depth and coherence length used. Nine solved problems are included.