Cosmological scenarios in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity with a dy

Abstract The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> theory, which extends symmetric teleparallel gravity by including the boundary term C in addition to the non-metricity scalar Q , provides a unifying framework that encompasses both f ( Q ) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>R</mml:mi> <mml:mo>˚</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> gravities. In this work, we develop a comprehensive dynamical system analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> cosmology formulated within the non-coincident affine connection branches. Unlike the coincident case, these branches introduce a dynamical degree of freedom that significantly enriches the cosmological phase space. We show that even for simple power-law forms of the Lagrangian, the theory accommodates a broad spectrum of cosmic scenarios, including successive pressureless matter eras, stiff-matter phases between early inflation and dark matter domination, and late-time acceleration. Our analysis demonstrates that the Universe can naturally evolve from an initial de Sitter (dS) phase to a matter-dominated epoch and subsequently to a final dS attractor, consistent with the observed thermal history. These results highlight the role of the dynamical connection in shaping cosmic evolution and underline the potential of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> gravity as a viable alternative framework for addressing outstanding issues in modern cosmology.