Superfluid density ρ_S/ρ=Λ/Λ_T [London] and specific heat ratio C_S/C_N in a superconductor

By the BCS framework [Bardeen, Cooper and Schrieffer, Phys. Rev., 108, 1175 (1957); Nam, Phys. Rev., 156, 470 (1967)] with a finite pairing interaction energy range Td to have a finite transition temperature Tc [Nam, Phys. Lett. A193, 111 (1994); (E) ibid A197, 458 (1995)], we present novel results: ρ_S/ρ[T=0]= tanh(G/2) =1-ρ_N/ρ, G= Σ_j (2/w_j)tan^-1(z/w_j), w_j=(π/2)j (odd integer), z=Td/2Tc, 1/G=N(0)V [the density of states/spin at a reference energy times the BCS pairing interaction energy]; C_S/C_N[T=0]=R(z) =(χ_S/χ_N) [12(z/π)² + (1+2z)/(1+tanhz)], χ_S/χ_N =1-tanhz [spin susceptibility at T=0]; R(1.4)=0.496 may account for the data 0.5 in UTe₂ [Ran et al, Science, 365, 684 (2019)]; R(BCS)=0; C_S/C_N[T=Tc]-1=d(z) =(6/π²) tanhz/[h(z)+(z-tanhz)/2z²], h(z)=Σ_j w_j^-3 tan^-1(z/w_j); d(z>>1) =12/7ζ(3) [BCS]; d(z<<1) =18/π² [Nam, APSMAR21, M47.12]; Free energy difference (F_N-F_S)[T=0]= N(0)Δ² tanhG/[1+tanhG], Δ=Td/sinhG. Superconducting state is a miscible liquid state with both ρ_S and ρ_N. Superconductors are multiconnected [Nam, Phys. Lett. A198, 447 (1995)].