Traversable Wormhole De-singularization: Almost $\eta$-Ricci-Yamabe Solitons in Static Spherically Symmetric Imperfect Fluid Spacetimes

In this paper, we investigate the almost $\eta$-Ricci-Yamabe soliton as a fundamental geometric regulator for a static, spherically symmetric black hole coupled to an imperfect fluid.

We have shown that the scaling parameter $\omega(r)$ is governed by thermodynamic friction along the radial vector field, and the geometric coupling with the Hawking temperature: $\alpha(r_H) S_{tt} = 2\pi T_H$ at the horizon.

We also derive the Poisson equation along the gradient vector field of the soliton and prove that the flow's kinematic expansion is explicitly dependent on the fluid's equation of state $\rho = \gamma \sigma$.

Diverging from traditional methodologies that assume a geometric shape function apriori, we analytically proved the geometric flow endogenously transitions the black hole geometry into a traversable wormhole throat by regularizing of temporal coordinate and satisfying spatial flare-out condition.

This transition occurs when fluid enters the dark energy era at $\gamma = -1$ and violates the Null Energy Condition $\rho + \sigma < 0$, with the soliton strictly dominating the local curvature gradient $\omega^{\prime}(r_H) > f^{\prime\prime}(r_H)$, to keep the throat open.

Moreover, by smoothly attenuating at spatial infinity, the soliton preserves the exact cosmological spacetime.

Finally, through tensorial perturbation analysis, we demonstrate that the geometric flow introduces a localized dissipative mechanism, that the perturbation evolution reduces to damped wave equation, imposing geometric drag on the manifold.