The Scretching–Planck Action and the Scretching Superconductor Chain

We derive the Scretching superconductor framework from the Scretching–Planck (SP) action for the Cooper-pair condensate, (ψ=√ P,eiS/). The condensate phase stiffness defines the local inverse-square London penetration depth, (X( r)=μ_0e²P( r)/m^). With (e^=2e), (m^=2m), and (n_s=2P), this becomes (X=4π r_e n_s). The phase term (X/2μ_0( A-∇ S/e^*)^2) gives the gauge-invariant Stückelberg form of the Scretching–Proca–Fermi term. The paper uses this coefficient to organize the Maxwell–London closure, the four-tier hierarchy, and the dissipation diagnostics. The reference value (X_0=α/λ_C^2) corresponds to (n_s=1/(4πλ_C^3)). Using the single-spin density of states, the condensation-energy calculation reproduces the measured thermodynamic critical fields of Al, In, Sn, and Ta within (2.4%). Doubling the condensation-energy coefficient raises the predicted field by about (41%). When the Eliashberg parameter (T_c/ωₗₙ) is calibrated to the measured gaps of Hg, Pb, and Nb, all seven critical-field estimates agree with measurement within (4.0%). The nonlocal penetration-depth estimates are (42.3) nm for Al and (31.2) nm for Nb, compared with reference values of (50) and (29.1) nm, respectively. The 10 GHz Maxwell–London closure check agrees within (0.12%). The Mattis–Bardeen calculation gives a normalized dissipation ratio in terms of (T/T_c) and (ω/Δ_0) and the dirty-limit relation (X=πμ_0σ_nΔ[Δ/(2k_BT)]/). In the stated Homes-law convention, the weak-coupling estimate is (C=69.6), compared with (35±7) for cuprates. The phase-stiffness estimate gives (T_θ/T_c=3.5) for YBCO. Applying the same weak-coupling and stiffness criteria at (300) K gives (Δ_0≥45.6) meV and (λ≤144,nm√a/(1,nm)). The vortex-core correction increases the calculated niobium vortex line energy by a factor of (2.97) relative to the logarithmic term alone. The formulation also recovers London electrodynamics, flux quantization, the Josephson and Little–Parks effects, the interface-energy sign change at (κ=1/√2), and weak-coupling BCS thermodynamics.