The Impossible Triangle: A No-Go for Symmetry-Protected Scalar Portals in Interacting Dark Energy

The $S_8$ tension motivates interacting dark energy (IDE), but embedding IDE in UV-complete physics faces severe naturalness challenges. We analyze four symmetry-protected DM–DE portals ( quartic ($\frac{1}{2}\lambda {\phi }^2{\chi }^2$), trilinear ($g\phi {\chi }^2$), derivative ($(c_6\mathrm{/}{\mathrm{\Lambda }}^2)({\mathrm{\partial }}_{\mu }\phi {)}^2{\chi }^2$), and fermionic Yukawa ($y\phi \bar{\psi }\psi$) )within a $Z_2$-symmetric Inert Doublet + Singlet Model. The trilinear portal requires $\beta \sim 0.45$ ($g\sim {10}^{-16}\mathrm{G}\mathrm{e}\mathrm{V}$), overshooting the radiative bound $g\lesssim {10}^{-42}\mathrm{G}\mathrm{e}\mathrm{V}$ by $\sim 26$ orders (tuning $\mathrm{\Delta }\sim {10}^{52}$). The quartic portal needs $\lambda \sim \mathcal{O}(1\text{–}10)$ versus $\lambda \lesssim {10}^{-86}$ ($\mathrm{\Delta }\sim {10}^{87}$). The derivative portal saturates dynamically at $\lesssim 4\mathrm{\%}$ suppression. The Yukawa portal yields $\mathrm{\Delta }\sim {10}^{52}$, persisting even with SUSY cancellation. No single-mediator model simultaneously satisfies technical naturalness and resolves the $S_8$ tension. Viable solutions require either multi-field tuned cancellations or explicit symmetry breaking with quantified fine-tuning.