The Two-Gate Law: A Complete Characterization of 8-Cycle-Free Cubic Bipartite Arc-Transitive Graphs of Girth Six

We establish a complete structural characterization of cubic bipartite arc-transitive
graphs of girth six that contain no 8-cycles. We prove that in such graphs, the absence of
8-cycles is equivalent to the simultaneous closure of two independent structural gates: a local
gate controlling Hamilton-local 8-cycles (those using at most two chord edges relative to a
Hamilton scaffold), and a global gate controlling 8-cycles arising from tight K3,3-subdivisions
with rectangle sum eight. Within the Foster Census of cubic arc-transitive graphs up to
60 vertices, exactly five graphs satisfy this double-lock condition: F038A, F042A, F050A,
F054A, and F056A. We provide a complete taxonomy distinguishing homogeneous locks
(single chord type, lock-level λ = 7) from heterogeneous interference locks (multiple chord
types, λ = 9). For the homogeneous family LCF[k,−k]m, we derive an explicit congruence
criterion characterizing 8-cycle existence via modular arithmetic in the underlying cyclic
Haar graph structure. All five double-lock graphs satisfy the Erdos-Gyárfás conjecture via
16-cycles, revealing a stratification phenomenon where generic girth-six graphs use 8-cycles
while double-locks are forced to employ longer witnesses.