Crown Interlacement and Diagonal Closure in Genus Ranges of Rigid 4-Regular Graphs

This paper studies orientable genus ranges of transversely connected rigid 4-regular graphs. It constructs an explicit crown-interlacement family of double-occurrence words W_h on 2h vertices and proves that its genus range is {h} for odd h and {h-1,h} for even h. For odd h, the construction realizes singleton genus h at the extremal order 2h. Combined with cross sums, loop nesting, and the known classification of nonsingleton ranges, this proves that the realizable singleton genus ranges form an initial segment at every order, establishing the no-gap assertion in Conjecture 33. In particular, the paper realizes both [8,8] and [9,9] in GR_18. The exact top-singleton threshold remains open at orders 2n when n >= 6 is even.