Late-time acceleration without a vacuum term in ${f(R,L_m)}$ gravity: scaling deSitter dynamics and parameter constraints

We investigate late-time cosmic acceleration in $f(R,L_m)$ gravity driven by nonlinear matter contributions, focusing on the class $f(R,L_m)=R/2+c_1 L_m+c_n L_m^{n}+c_0$ with the explicit choice $L_m=\rho_m$ and an uncoupled radiation sector. We analyze two realizations: (i) Case A: $f(R,L_m)=R/2+\beta \rho_m^{n}+\gamma$, where $\gamma$ acts as a vacuum term, and (ii) Case B: $f(R,L_m)=R/2+\beta \rho_m+\gamma \rho_m^{n}$, where the nonlinear sector can mimic dark energy without an explicit cosmological constant. For each case, we construct a bounded autonomous system, classify all critical points and their stability, and compute cosmographic diagnostics. The phase-space analysis shows that Case A reproduces the standard radiation$\to$matter$\to$de~Sitter sequence only for $n\gtrsim 4/5$, with acceleration essentially enforced by the vacuum term. In contrast, Case~B admits a qualitatively distinct and phenomenologically appealing branch: for $0<n<1/2$ the system possesses a physical \emph{scaling} de~Sitter future attractor inside the bounded simplex, yielding radiation$\to$matter$\to$acceleration with $q=-1$ and $\omega_{\rm eff}=-1$ and without introducing $c_0$. We confront both models with background data (CC, Union3, DESI BAO, plus a BBN prior on $\Omega_b h^2$) using nested sampling and perform model comparison via Bayesian evidence and AIC/BIC. The full data combination constrains $n=1.08\pm0.05$ in Case A and $n=0.05\pm0.10$ in Case B (68\% CL), the latter lying within the accelerating window while remaining statistically consistent with $\Lambda$CDM kinematics at the background level. We also record minimal consistency conditions for stability (tensor no-ghost and luminal propagation) and motivate a dedicated perturbation-level analysis as the next step to test growth and lensing observables.