A Phenomenological Organization of BCS Superconductivity via Fermion–Boson Duality: From an Occupation-Probability Decomposition to the Pseudogap and the BCS–BEC Crossover

This work reinterprets the weak-coupling BCS gap in the language of Fermion–Boson Duality (FBD). The starting point is the Mühlschlegel formula Δ(T)/Δ₀ ≃ tanh{1.74·√(T_c/T − 1)}, which closely approximates the BCS gap. By an elementary identity, the tanh in this formula can be written exactly as the difference of two complementary logistic occupation probabilities. Within the range of the approximation, the gap is therefore read as L_B^(e) − L_F^(e): the difference between a bosonically behaving (Cooper-pair-like) B-type component and a quasiparticle-like F-type component of the electron sector.

Taking this identity as a core, the FBD phenomenology is extended to four occupation probabilities {L_B^(e), L_F^(e), L_B^(γ), L_F^(γ)} by adding a gauge sector. When the transition centers of the electron and gauge sectors coincide, ordinary single-step weak-coupling BCS is recovered. When they separate (μ_γ < μ_e), a staircase structure 1 → 1/2 → 0 appears in the observed gap, providing a phenomenological candidate for the high-temperature-superconductivity pseudogap. When the two-sector tanh arguments take opposite signs (a "competing" structure), the framework represents the BCS–BEC crossover of cold-atom systems as a mixture of BCS-like and BEC-like occupation probabilities; in this picture the unitary-point Bertsch parameter ξ_B is read as the admixture fraction of BCS character. Testable single-sample relations linking the gap, Higgs/amplitude mode, London penetration depth, and optical spectral weight are also proposed.

The rigorous element is the tanh-to-occupation-probability decomposition of the Mühlschlegel formula. The four-degree-of-freedom extension, the averaging of the observed gap, the connection to the Matsubara scale, and the BCS–BEC interpolation are phenomenological hypotheses to be tested against experiment.