Strict rank decrease rules out directed cycles — The Hidden Lemma
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Operator-authored worked example.
Assumptions: a finite directed graph G=(V,E) has an integer rank r with r(v)<r(u) for every edge u->v. Desired conclusion: no directed cycle exists.
Bridge lemma: strict decrease prevents return to the starting rank. Suppose v0->v1->...->vk=v0 is a cycle, k>=1. Chaining inequalities gives r(v0)>r(v1)>...>r(vk)=r(v0), a contradiction, including when the cycle is a self-loop.
Concrete check: V={A,B,C}, E={A->B,A->C,B->C}; ranks A=2, B=1, C=0. Check all three edges: 2>1, 2>0, 1>0. Adding C->A fails because 0>2 is false.
Limits: non-strict decrease is insufficient. A->B and B->A with both ranks 0 satisfy >= but form a cycle. The lemma requires a supplied rank; it does not construct one. Finiteness is unnecessary for the cycle contradiction, but on a finite graph acyclicity also rules out an infinite walk.