Mathematicians have pinned the exact point where a small directed network is guaranteed to fit inside a perfectly self-symmetric one - and where that guarantee runs out.
A loopless digraph is self-complementary if swapping every arc for a non-arc, and every non-arc for an arc, produces a digraph isomorphic to the original - a definition that has nothing to do with reversing arc direction. The researchers set out to find the largest number of arcs, q, such that any six-vertex digraph with at most q arcs is guaranteed to sit inside some self-complementary digraph of the same size. The paper proves that number is exactly 7, with the upper bound demonstrated by a disjoint union of a 3-vertex complete digraph and a 3-vertex directed path. Push past seven arcs and the guarantee can fail: the authors catalog all eight-arc digraphs that resist completion, finding five distinct isomorphism classes that collapse to three once digraphs are identified with their converse - the separate operation of reversing every arc's direction, unrelated to the complement used in the main definition.
Self-complementary graphs are a decades-old staple of combinatorics, useful in decomposition and design-theory arguments, but exact small-order thresholds like this one are notoriously fiddly to nail down by hand rather than by brute computer search. The result also settles a subtler point: every one of the five obstruction digraphs still packs alongside an isomorphic copy of itself, meaning ordinary graph packing is a strictly weaker property than same-order self-complementary completion - and the two ideas already diverge at six vertices.
No product roadmap points at this one - it's pure structural math - but exact answers at the smallest failing case are exactly what later general proofs get built and tested against.