  Chiral exact flat bands (FBs) at charge neutrality have attracted much recent
interest, presenting an intriguing condensed-matter system to realize exact
many-body phenomena, as specifically shown in "magic angle" twisted bilayer
graphene for superconductivity and triangulene-based superatomic graphene for
excitonic condensation. Yet, no generic physical model to realize such FBs has
been developed. Here we present a new mathematical theorem, called bipartite
double cover (BDC) theorem, and prove that the BDC of line-graph (LG) lattices
hosts at least two chiral exact FBs of opposite chirality, i.e., yin-yang FBs,
centered-around/at charge neutrality (E = 0) akin to the "chiral limit" of
twisted bilayer graphene. We illustrate this theorem by mapping it exactly onto
tight-binding lattice models of the BDC of LGs of hexagonal lattice for strong
topological and of triangular lattice for fragile topological FBs,
respectively. Moreover, we use orbital design principle to realize such exotic
yin-yang FBs in non-BDC lattices to instigate their real material discovery.
This work not only enables the search for exact chiral FBs at zero energy
beyond moir\'e heterostructures, but also opens the door to discovering quantum
semiconductor features with FB-enabled strongly correlated carriers.
