From a record-keeping perspective, the biggest find was that Ben-Zion Werdesheim, the patriarch of what I had labeled "Branch 6", came to New York with his second wife, took the Americanized version of his name "Benjamin Wertheimer", and had five more children here. (We had not previously known anything about him except that he had 5 children in Mielec, 2 of whom moved to New York.) Not only did that help us make contact with his descendants, it also gave us our first break in the "1840 wall."
Here's what I mean by that: For each of the 6 branches, we have pretty good documentation of a man with the family name Werdesheim and his wife having some number of children during some date range between 1850 and 1880. (Branch 5 is the exception: in this case we have two women with the maiden name Werdesheim whose children got married to each other.) So we've assumed that the Werdesheim patriarchs and matriarchs of each branch were born in 1840 +/- 10 years, and we've taken to calling them "The 1840 generation."
Our assumption is that the members of the 1840 generation were siblings or first cousins, and that if we can determine the names of their parents, we'll know how the various branches are related. (I won't go into too much depth here, but we base this assumption on the two facts that the spelling Werdesheim appears to be unknown outside of Mielec, the fact that at one point at least one child of each of the branches lived in a consecutively-numbered house in Mielec, and the naming patterns of the various families -- for example, they almost all have a son named Moshe Y* (Yaakov, Yitzchak, Yosef) within a few years of each other, which implies that a well-loved common grandfather with that name passed away just before.)
So getting back to Ben-Zion....
I recently joined the Jewish Genealogical Society of New York, in part because that membership gets me access to a website called JewishData.com, which has an enormous collection of indexed photos of Jewish gravestones. It was there that I found the stone for Benjamin Wertheimer which started me on the examination of whether he could be the same man as Ben-Zion Werdesheimer. Once we determined that he is definitely the same person, the exciting thing is that on his gravestone, his Hebrew name is given as "Ben-Zion ben Tevel".
Which means we now have one name for the generation BEFORE the 1840 generation. (Interestingly, my great-grandfather's eldest brother had the Hebrew name "Tevel Hirsch" according to one record, although he seems to have gone by Herman in America.)
I've prepared a document that summarizes what we know about the 1840 generation and their children, as an aid in conducting our research. It doesn't follow the Genealogical Proof Standard because that's not its purpose; I have my database to keep track of my source citations, etc, and I'll write up specific documents for each family when we're ready. The purpose of this document is to give us a clear easy-to-use matrix of what our team has accepted as "fact" to help us fit new pieces together and to identify where we need to be looking.
That research dossier is at https://www.dropbox.com/s/k4eia1d9mdz3j