-Sigismund II Augustus has a male child by a Jewish mother in his late 30s, the Sejm legitimizes him in desperation for saving the Jagiellon line and Poland/Lithuania becomes ruled by a "Jewish" family starting in 1572. The young man inherits the aptitude for statecraft shown by his father and grandfather, it is his exceptional abilities along with those of his son and grandson (also born of Jewish mothers but who eventually practice Catholicism, at least on paper) who are key in solidifying the government's longevity
-Poland institutes policies of exceptional religious tolerance, allowing peoples of all faiths to settle within her borders provided they follow local laws and swear allegiance to the King. Emigration from Spain and Central Europe gives Poland/Lithuania a major headstart and allows for the establishment of several universities. Sanitation programs are also enabled along with building of sewer systems which makes the country better able to endure plagues
-Poland/Lithuania has some of the mismanagement and infighting of OTL but not to nearly the same degree, and the Polish-Lithuanian Commonwealth manages to survive the 17th century intact with some expansion at the cost of Russia, Ottomans, and the Holy Roman Empire. Her population is not wracked by wars and is supplemented with immigration (mostly Jewish but not entirely) from other areas of Europe, some of whom bring the potato with them which enables greater nutrition and more available foodstuffs.
-Commonwealth influence is able to put down rebellions in the Ukraine and bring Kiev into the orbit of Warsaw instead of Moscow. This will eventually permit the resources of the Ukraine to aid in expansion of the Commonwealth economy. As the 18th century dawns, the Commonwealth formalizes the tendencies of federation, limited monarchy, and legislation into a formal charter that becomes an influence for many governments thereafter, including the United States.
-Commonwealth industrialization results in some of the first organized labor movements in history and brings about socially progressive programs that decrease infant death and permit greater standards of living than most of Europe at the time, resulting in reliable population growth from an already burgeoning population of 22 million in 1700 to over 65 million by 1800
-Napoleonic wars and further industrialization permit Commonwealth expansion to the whole of the modern Ukraine, Byelorussia, Latvia, Lithuania, most of OTL Poland, Kaliningrad Oblast, and even some of the Baltic islands. Estonia is an area of contention while Prussia is effectively a Commonwealth satellite, especially following its neutering during the Napoleonic Wars. German (especially Prussian) infrastructure will be rebuilt with mostly Polish capital, and this causes some shift in the view of Germans about Commonwealth populations.
-By 1850 the Commonwealth government is extremely stable, and though there have been challenges with internal order and some external wars, the threat of Russia in the east with a rising Germany in the West keeps the populace even more united than their government alone would permit. With the decline of the Ottomans and the formation of a united Germany in 1848 under Austrian domination, Prussia seeks membership in the Commonwealth as a self-governing entity. This will eventually lead to a dynastic intermarriages with the Jagiellons and the Hohenzollerns, eventually causing a full merger of the two in the mid 1900s.
-By 1860 there is a population of over 30 million self-described Jews in the Commonwealth during their first official census, though in reality many more have a clandestine mixed heritage with larger amounts of intermarriage than OTL. Without the tragic events of the early and mid 20th century along with additional emigration of Russian Jews there is an even more firmly Jewish population in the different areas of the Commonwealth by 1900.