  When interacting motile units self-organize into flocks, they realize one of
the most robust ordered state found in nature. However, after twenty five years
of intense research, the very mechanism controlling the ordering dynamics of
both living and artificial flocks has remained unsettled. Here, combining
active-colloid experiments, numerical simulations and analytical work, we
explain how flocking liquids heal their spontaneous flows initially plagued by
collections of topological defects to achieve long-ranged polar order even in
two dimensions. We demonstrate that the self-similar ordering of flocking
matter is ruled by a living network of domain walls linking all $\pm 1$
vortices, and guiding their annihilation dynamics. Crucially, this singular
orientational structure echoes the formation of extended density patterns in
the shape of interconnected bow ties. We establish that this double structure
emerges from the interplay between self-advection and density gradients
dressing each $-1$ topological charges with four orientation walls. We then
explain how active Magnus forces link all topological charges with extended
domain walls, while elastic interactions drive their attraction along the
resulting filamentous network of polarization singularities. Taken together our
experimental, numerical and analytical results illuminate the suppression of
all flow singularities, and the emergence of pristine unidirectional order in
flocking matter.
