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 $L_m=ρ_m$ and an uncoupled radiation sector. We analyze two realizations: (i) Case A: $f(R,L_m)=R/2+βρ_m^{n}+γ$, where $γ$ acts as a vacuum term, and (ii) Case B: $f(R,L_m)=R/2+βρ_m+γρ_m^{n}$, 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