Layered dark structure with a Structuring Field: A $Z_4$-symmetric Inert Doublet-Singlet realization and implications for the $S_8$ tension

We introduce the Layered Dark Sectors with a Structuring Field (LDS-SF), a structured cosmological framework where the internal architecture of a multi-component dark sector naturally generates scale-dependent growth of structure. In this framework, the characteristic scale dependence is derived from the dominant eigenvalue, $\lambda(k)$, of the dark sector's perturbation matrix. This structurally-driven mechanism modifies structure growth while preserving the standard $\Lambda$CDM background expansion and General Relativity. We provide a minimal realization of this framework within a two-component DM $Z_4$-symmetric Inert Doublet Singlet Model ($Z_4$-IDSM). By integrating out the heavy inert doublet mediator, we derive a contact-interaction Effective Field Theory (EFT) for a 60~GeV singlet dark matter candidate. This interaction manifests macroscopically as an effective sound speed $c_s^2$, which we map to the LDS-SF eigenmode evolution. We implement this system into the CLASS Boltzmann code, employing a late-time activation function that projects virialized halo properties into the linear perturbation framework. We also compute the relic density using micrOMEGAs to further stress-test the relic abundance predictions of viable parameters. Our numerical analysis demonstrates that while the model remains indistinguishable from $\Lambda$CDM at the era of recombination, it introduces a targeted suppression of the matter power spectrum at late times ($z < 10$) and small scales ($k > 0.1~h/\text{Mpc}$). Confronting the model with Planck CMB, BAO, and growth-rate measurements, we find three instances of couplings that successfully alleviate the $S_8$ tension, bringing the predicted amplitude into $1\sigma$ agreement with weak-lensing data from KiDS-1000 and DES. This work establishes LDS-SF as a mathematically consistent and observationally viable extension of standard cosmology.