Interesting perspective. I think, though, that Southern Song always had a chance to recover the north, but it was fµcked forever as soon as Mongols arrived. Allying with Mongols to take down Jurchens was the worst blunder ever.
This matches with my theory of historical continuity:
Human societies or civilizations when they reach a certain size constantly adapt and reinvent themselves, due to internal reorganization and external influence. The longer a society is left intact without too much external interference, the more they can achieve a high degree of internal cohesion and as a result are able to adapt to changing circumstances more quickly and efficiently, as compared to a society that has been the victim of external aggression and forced transformation. In other words, relatively untouched societies retain some competitive edge over societies that were victims of invasion and brutal subjugation. There could be many reasons for this, but one of the most important one seems to be that societies take a certain amount of time to heal themselves of the damages from external forces to come to a steady state where the hierarchical pyramid structure of societies become well defined and whole society starts functioning as a organic whole with all parts playing their respective specialized roles. A good analogy is the body of higher animals whose body has organs for specialized functions. Although all cells start out as stem cells, eventually their functions and forms change to become part of a specialized organ, such as the central nervous system, the pancreas, the liver, the heart etc. Just as it takes time to evolve from single celled blue-green algae to complex reptiles, mammals, plants etc., societies also take time to evolve into a more efficient and complex unit, that has a competitive edge over other societies that did not have time or freedom from external influence to evolve.
According to the Historical Continuity hypothesis, since the established powers of the time were crushed under the Mongol hammer and only Western Europe and Japan was spared and remained standing, it is no surprise then that West Europeans soon were ascendant with Renaissance, Maritime supremacy and eventual colonization of the planet. Japan was isolationist through out this period. It was opened up by force by Admiral Perry around 1860's. It adapted Western technology in a few decades and became a world power. Soon it had the honor to beat Tzarist Russia, the first White nation ever to be beaten by non-whites and went on to colonize Korea, Manchuria and parts of Asia.
The most important event took place in the twin city of Buda and Pest (today's Budapest). Subutai and Batu Khan's army had just finished off Polish army and Hungarian Teutonic Knights. A large part of the Hungarian population were decimated, except for the ones who were lucky enough to hide inside forts. So General Subutai and the Khan camps out with their army of 200,000-300,000 horsemen, with 2-3 times number of horses. Finding grassland to graze these huge number of horses were a major tactical concern, so Mongols are sending out scouts to all parts of Western Europe to come up with a plan of attack to subjugate all of remaining Europe. It takes a lot of planning and almost a year has past in Buda-Pest and preparations are almost complete to initiate the attack. At this fateful moment in 1241, comes the news that the great Khan Ogedei have passed away. All leaders of some consequence, must now go back to Kharakhorum, the seat of the great Khan and the capital of the unified Mongol empire for a Khurultai (meeting or conference) to choose the new great Khan, according to the Mongol custom/code Yasak, established by Chingis Khan. So the army marches back, never to return with the same momentum and force to complete the job they had to abandon, the job of making the borders of Mongol empire from Pacific coast in Siberia and China to the Atlantic coast of Western Europe.
Similar is the story of Japan, it was attacked 2 times by Yuan Mongol forces with Chinese and Korean maritime armada, but every time there was bad luck of typhoons that diminished their chances of victory. These divine winds are called Kamikaze (it does not mean suicide fighter pilots, although the word was used to mean that these suicide pilots in WW II would also save Japan, just as the divine Typhoons did) in Japanese, that saved Japan from Mongol subjugation.
