Trace anomaly and isospin splitting in inverse-mapped relativistic mean-field theory

Trace anomaly and sound speed provide EOS-level probes of dense-matter nonconformality, but do not by themselves identify the microscopic channels responsible for the response. We study this question with a uniform-matter inverse-mapped relativistic mean-field ensemble constrained by chiral effective field theory, heavy-ion flow information, and neutron-star mass-radius data. The ensemble reproduces the flow-based trace trend in symmetric nuclear matter, while beta-equilibrated matter approaches the neutron-star trace bands more slowly. The resulting splitting, ${\mathrm{\Delta }}_{\text{SNM}}-{\mathrm{\Delta }}_{\text{betaeq}}$, remains positive over $2--5\text{nzero}$ and is most strongly correlated with the density derivative of the isovector-vector coupling, with bootstrap-stable Spearman coefficients $r_s\simeq 0.91--0.92$ at $2--3\text{nzero}$. Its correlation with the beta-equilibrium proton fraction is much weaker. The sound-speed splitting changes sign near $3.38\text{nzero}$, and the derivative term $-\text{dd}\mathrm{\Delta }\mathrm{/}\text{dd}\ln \epsilon$ becomes sensitive to both scalar-vector and isovector responses above $4\text{nzero}$. Data-combination and controlled-isovector tests show that this channel separation is resolved only when laboratory and astrophysical projections are combined. Thus, within the present inverse-mapped RMF space, the SNM–beta trace splitting acts as a thermodynamic probe of the high-density symmetry sector rather than as a unique signal of exotic degrees of freedom. A finite-nucleus-calibrated extension will be needed to test how much of this channel diagnostic survives in predictive covariant density functionals.