And you are again showing confusion based on your own presumptions.
Yes, genetic information is lost at each generation. But not the same genetic information. I have about half of my mother and father's DNA. My brother has about half of my mother and father's DNA. One of my half-sisters has about half of my mother's DNA. My brother, my sister, and I do not share the exact same half of our mother's DNA, so we do not share the exact same quarter of our grandparent's DNA and so on. If my brother, sister, and I compare our genetic results then each of us will have parts of our parent's DNA that would be absence in examining only one of us. This basic principle extends to cousins, aunts, and uncles for grand parents on up, so if I compare all my first cousins, aunts, siblings, parents, and myself together then we can get some percentage of our grand parent's DNA reconstructed without ever digging anyone up or even involving my living grand parents in the process.
But that's only a part of what I am talking about. Without regard to any other DNA kits besides my own, my DNA has to split in certain ways. I didn't inherit a random assortment of DNA. I inherited roughly half of my father's autsomal DNA and roughly half of my mother's autosomal DNA; the half I inherited from my father in general isn't identical to the half I inherited from my mother, and which half goes where is roughly linked with which sex-determining chromosomes I got from whom. Which portions of what chromosomes won't be known without making matches to other people's DNA, but it doesn't matter in the mathematical model. My DNA is treated as a variable or what is called an UNKNOWN in the 1800s language of mathematics; what DNA my father contributed is another different variable or another UNKNOWN. Same for my mother. Same for my siblings. Same for my cousins, aunts, uncles, grand parents, and people totally unrelated to me.
Graphs and tables can be constructed which show what abstract portion of autosomal DNA I got from whom. I am interested at the moment ONLY in the abstract relations. Once I have the parameters of the problem to plug into a fully constructed model, I can actually start doing comparisons in order to analytically link certain portions of my autosomal DNA with certain sides of my genetic genealogy starting with the maternal or paternal difference. Like my maternal grandfather potentially gave me X-chromosome DNA but gave me no mitochondrial DNA and no Y-chromosome DNA; roughly a quarter of my autosomal DNA comes from my grandfather, and the quarter isn't continuously distributed across 1-22 of my chromosomes, so I might have my maternal grandfather's DNA on my 1, 3, 4, 6, 7, 9, and 10th chromosomes. If my maternal grandfather's DNA can be put into a set like (1, 3, 4, 6, 7, 9, 10), and I compare my DNA with a random stranger that happens to match in (1, 3, 4, 6, 7, 9, 10) then I know that random stranger is related to me through my maternal grandfather's side of the family. With successive comparisons and enough genetic samples from close family members, I can use chromosome maps of that kind to automatically sort future matches to their proper place in my family tree. I might not be able to immediately place them exactly where they are in relation to me, but I will quickly be able to place them on the maternal or paternal side then place them on the paternal or maternal's grandfather or grandmother, and so on.
Not all unknowns are equally unknown though; I know I got a Y chromosome, and I know I got an X chromosome, and I know I got the Y chromosome with roughly half my autosomal DNA, and I know I got the X chromosome with roughly half my autosomal DNA, so I know that the relationship between roughly half my autosomal DNA is not entirely independent of which sex-determining chromosomes I inherited. Because the autosomal DNA is not entirely independent then I can write a functional notation representing that non-independent relationship where either my sex-determining chromosome is dependent on roughly half my autosomal DNA or roughly half my autosomal DNA is dependent on my sex-determining chromosome. With the difference between mitochondrial DNA and X chromosomes, we can actually establish more nuanced relationships between X-linked, Y-linked, and MT-linked inheritance as cross compared to each other.
So in the way that my directly measured DNA can be treated as a variable in a system so can unmeasured DNA of ancestors long since dead. You can think of my DNA as a solved system of equations which can be compared with other partially solved or unsolved but expressed systems of equations in order to examine the state of unmeasured DNA ancestors by indirect inference. Rather than thinking of the ancestor as strictly solved or unsolved, we can think of the ancestor in terms of percentages. If you only have my DNA to work with then you can only have about (1/(2^n))% of a given ancestor at a generation n solved. But if you compare me and my siblings then you can have more than (1/(2^n))% of an ancestor solved, and if you keep adding descendants to the comparison then we can tell more about the common ancestor. There's a mathematical relationship telling us what the minimum or maximum number of such comparisons will be to get 100% of the ancestor's DNA reconstructed.
"But this is a public forum and people need to comment on anything posted which they think might mislead other readers."
This is a blatant admission on your part that you think I am trying to mislead others. Your posts so far have been technically hostile to the process of free inquiry. It is great that you want to go ahead and keep doing things the way they have always been done. I am certainly not trying to stop you from doing exactly that. I don't care that there are labor intensive alternatives to solve these problems individually or by strict experimental methods.
I know there are mathematical methods which would be somewhat difficult to develop but which would be instrumental in the development of automated reasoners for genetic genealogy which makes the problem push-button for the average user who doesn't have interest in doing genetic genealogy the way it has always been done. In the meantime, the mathematics of genetic genealogy can be used individually to setup spreadsheet macros or simple programs that search and sort through data sets to make the process of identification of family members simpler and less manually intensive.