Gernsback
Gernsback's natural laws seem to differ from Homeline's in several
important ways. For example, on Homeline, broadcast electric power is
very difficult to achieve - on Gernsback, Tesla developed it in 1890.
However, stolen Gernsbackian technology seems to function perfectly well
on Homeline and vice versa. The fact is that we don't understand the
physical laws of Gernsback and aren't likely to in the near future,
considering the very low profile outtimers must take there.
Other than natural laws, the chief point of historical divergence occurred in the late 1890s, when Nikolai Tesla developed broadcast electrical power well in advance even of the things that needed electrical power. This, combined with several other related scientific leaps forward created worldwide prosperity and prevented the standard era of early twentieth century war.
Though Gernsback is only at local year 1969, in many ways they are far more advanced than Homeline. A thriving lunar base is home to Governor Robert Kennedy. Russia has been almost completely rebuilt since the violent and unsuccessful communist uprising of 1947. Aircars are commonplace (though dangerous), and personal jetpacks have reached the market (even more dangerous). Food pills have wiped out starvation in all but the most isolated areas of the world.
The World Science Council, a scientific and technocratic organization based in Brooklyn is one of the most influential world organizations, supported by taxes by dozens of national governments. The League of Nations is also influential. Atomic power is common. Aircars are too expensive to have completely replaced electric groundcars, but the newly-prosperous definitely have them. Lightning trains, zeppelins, superwing aircraft, and submersibles are other means of transportation used. There are even one-man helicopter/propellor-driven aircraft called "flivvers".
However, speaking socially, Gernsback suffers from parochalism and hypocrisy. Women remain in many ways unliberated, and treatment of Catholics, Jews, and racial minorities by America and Europe remains poor. Gernsback is a shining example of everything optimism can achieve and everything it can miss in a blind spot.
Gernsback is a good example of a world where the "places" and "people" of the world follow somewhat that of Homeline, with small but important diversions. You could use a map of 1969 Homeline to travel through Gernsback without much in the way of problems - a highway that was only a two-laner on Homeline might be a sixteen-lane superfreeway on Gernsback, but landmarks are quite different. (The four faces on Mount Rushmore are of Tesla, Edison, Einstein and Mendeleev, for example.)
The Big Four have tried as much as they can to keep out of Gernsback, but the lure of the place is unmistakable - it's alternate science can be of great use to Homeline, and vice versa. Further, its practically-universal "can-do" attitude would make them an enthusiastic ally in the war on Centrum, if such a thing ever became necessary.