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 $\mathrm{\Delta }$, and the energy cut-off $\mathrm{\hslash }{\omega }_0$. As $\mathrm{\hslash }{\omega }_0(j)$ approaches its cut-off, $\lambda$ and $\mathrm{\Delta }$ converge to fixed values. The theory predicts that the coupling strength $Vg(0)$ reduces to $\tilde{V}g(0)=e^{-{(1-{\alpha }_1)}^2{k}^2\mathrm{/}4{\lambda }^2}Vg(0)$, and the Cooper pair reduces similarly. For weak coupling, ${\alpha }_1=1$, and when $Vg(0)=0.1$, ${\mathrm{\Delta }}_{\mathrm{A}\cdot \mathrm{C}}=108{\mathrm{\Delta }}_{\text{BCS}}$, but ${\mathrm{\Delta }}_{\mathrm{A}\cdot \mathrm{C}}$ decreases to $28{\mathrm{\Delta }}_{\text{BCS}}$ 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 $0.14C_{k\uparrow }{C}_{-k\downarrow }$. Additionally, ${\mathrm{\Delta }}_{\mathrm{A}\cdot \mathrm{C}}=0.5676\text{ eV}\gg \mathrm{\hslash }{\omega }_{\text{D}}$, and the local stacking force is ${\tilde{V}}_{\text{st}}=0.264\text{ eV}$. With $k^2\mathrm{/}{\lambda }^2=$ const, the local strength increases, causing the stacking force to grow significantly. Thus, $\mathrm{\hslash }{\omega }_0$ and $\mathrm{\Delta }$ yield a unique solution.