Strong local variational approach for superconductivity theory, and the principles of coherent interaction and action-counteraction

For the two-mode electron pairing, we propose a local stacking force pairing mechanism driven by strong local fluctuations, with two straight pairing orbits where the tying Cooper pairing $C_{-k\downarrow}C_{k\uparrow}e^{ik\cdot r}$ replaces the itinerant pairing. Based on coherent interaction and action-counteraction principles, the strong local variational theory is constructed, with the energy extremum and gap equations forming self-consistent pairs, involving the local variational parameter $\lambda$, energy gap $\Delta$, and the energy cut-off $\hbar \omega_0$. As $\hbar \omega_0(j)$ approaches its cut-off, $\lambda$ and $\Delta$ converge to fixed values. The theory predicts that the coupling strength $Vg(0)$ reduces to $\tilde{V}g(0)=e^{-\left(1-\alpha_{1}\right)^{2} k^{2} / 4 \lambda^{2}} Vg(0)$, and the Cooper pair reduces similarly. For weak coupling, $\alpha_1=1$, and when $Vg(0)=0.1$, $\Delta_{\mathrm{A \cdot C}}=108 \Delta_{\text{BCS}}$, but $\Delta_{\mathrm{A \cdot C}}$ decreases to $28 \Delta_{\text{BCS}}$ at $Vg(0)=0.2$. For strong coupling, $\alpha_1=0$, if $Vg(0)=1.4$, $\tilde{V} g(0)$ reduces to 0.2, and the smaller Cooper pair $\widetilde{C_{k \uparrow} C_{-k \downarrow}}$ reduces to $0.14 C_{k \uparrow} C_{-k \downarrow}$. Additionally, $\Delta_{\mathrm{A \cdot C}} = 0.5676~\text{eV} \gg \hbar \omega_{\text{D}}$, and the local stacking force is $\widetilde{V}_{\text{st}}=0.264 ~\text{eV}$. With $k^2/\lambda^2 =$ const, the local strength increases, causing the stacking force to grow significantly. Thus, $\hbar \omega_0$ and $\Delta$ yield a unique solution.