General Relativity and Newton's Constant with Scretching–Planck Matter: Four Routes and a Geometric Identity

We formulate superconductivity in Scretching–Planck (SP) density–phase variables by writing the Cooper-pair order parameter as (ψ=√P,eiS/), with charge magnitude (2e) and effective pair mass (m^*). In these variables, the Ginzburg–Landau free energy gives the stationary balance (QSP+12m^*v_s^2+α+β P=0) and the corresponding supercurrent equation. The phase dynamics recover the London equations and penetration depth. Fluxoid quantization gives the flux quantum (h/(2e)=2.0678×10⁻¹⁵) Wb and accounts for Little–Parks periodicity. Coupling two condensates gives the Josephson relations, with (K_J=2e/h=483597.848) GHz/V. The SP amplitude-gradient term sets the Ginzburg–Landau coherence length. The calculated normal–superconducting interface energy changes sign at (κ=0.707107), and the vortex calculation gives (Hc1/H_c=1.000000) at (κ=1/√2) and recovers the large-(κ) limit. A disorder model for niobium with (κ=1.61) gives (μ_0Hc1=112) mT. In (d)-wave rings with a net (π) phase shift, the same phase condition permits half-integer fluxoid quantization. A momentum-space density–phase representation of BCS pairing recovers the weak-coupling ratios (2Δ_0/(k_BT_c)=3.5279) and (Δ C/(γ T_c)=1.426). Across eight elemental superconductors, the condensation-energy relation reproduces the measured zero-temperature thermodynamic critical fields of Al, In, Sn, and Ta within (2.4%). Deviations of (7%) to (15%) for Hg, Pb, and Nb illustrate the limits of the weak-coupling estimate. Free-electron estimates of penetration depth and coherence length place even Nb on the type-I side in the clean-limit calculation, illustrating the sensitivity of that classification to material parameters. Nine solved problems are included.