= Central-path Newton system
At a target barrier parameter $\mu$, let $r_p=Ax-b-s$, $r_d=A^\top y-c$, $r_c=y+\mu\nabla F(s)$. The Newton direction solves
$$
A\Delta x-\Delta s=-r_p,\quad A^\top\Delta y=-r_d,\quad\Delta y+\mu\nabla^2F(s)\Delta s=-r_c.
$$
Full column rank of $A$ and a positive-definite barrier Hessian give a positive-definite reduced matrix. Backtracking must keep both cone variables interior; solving the linear equations alone does not guarantee that a full step stays inside the cones.
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