Actually, instead of letting you keep ignoring it, let's put it front and centre.
https://en.wikipedia.org/wiki/Founder_effectIn population genetics, the founder effect is
the loss of genetic variation that occurs when a
new population is established by a
very small number of individuals from
a larger population.
...
As a result of the loss of genetic variation, the new population may be
distinctively different, both
genotypically and phenotypically, from the parent population from which it is derived.

Every paper and article you've cited contain ample references to discrete population bottlenecks preceding the Erfurt remains, and they all talk of the founder effect with respect to the Erfurt population.
Without detailed genomic evidence of the population from which the Erfurt group descended, we can't tell how genetically similar or distinct they were. They could have been entirely representative of the wider population, or, as is vastly more likely, their DNA would not have been representative of the wider population from which they originated.
We can't tell going backwards because we don't have any genetic data to let us do that (a rather specific point to the discussion, eh?) - the original paper explains this perfectly clearly in standard English, but you appear to have ignored it, not read it, or just don't want to acknowledge it.
Instead for these models they used modern populations, which is why the authors expressly state that the data derived from their analyses cannot be used to identify origins, but rather can only establish what happened consequent to the time of the Erfurt people.
It's written in simple English.
I am not confused.
Back to that paper, the one you cited in order to make your claim:
https://www.sciencedirect.com/science/a ... 7422013782Ashkenazi Jews (AJ) emerged as a distinctive ethno-religious cultural group in the Rhineland in the 10th century (Frishman, 2008; Gladstein and Hammer, 2016). Since then, the AJ population expanded substantially, both geographically, first to Eastern Europe and recently beyond Europe, and in number, reaching about 10 million today (DellaPergola, 2015; Motulsky, 1995). Starting from the early days of human genetics, dozens of pathogenic recessive variants were identified in AJ (Charrow, 2004; Goodman, 1979; Ostrer, 2012), leading to the development of successful pre-conception screening programs (Gross et al., 2008; Kaback et al., 1993). A large fraction of these variants are extremely rare outside AJ and appear on the background of long shared haplotypes (e.g., Frisch et al., 2004; Hamel et al., 2011; Laitman et al., 2013; Raskin et al., 2011), implying that AJ descend from a small set of ancestral founders (Diamond, 1994; Ostrer and Skorecki, 2013; Risch et al., 2003; Slatkin, 2004). The Ashkenazi “founder event” is also evident in four mitochondrial lineages carried by 40% of AJ (Behar et al., 2006; Costa et al., 2013). More recently, studies found high rates of identical-by-descent (IBD) sharing in AJ, that is, nearly identical long haplotypes present in unrelated individuals, a hallmark of founder populations (Atzmon et al., 2010; Carmi et al., 2014a; Gusev et al., 2012; Henn et al., 2012). Quantitative modeling suggested that AJ experienced a sharp reduction in size (a “bottleneck”) in the late Middle Ages and that the (effective) number of founders was in the hundreds (Carmi et al., 2014a; Granot-Hershkovitz et al., 2018; Palamara et al., 2012; Santiago et al., 2020; Tournebize et al., 2022).
The origins of early AJ, as well as the history of admixture events that have shaped their gene pool, are subject to debate (Data S1, section 1). Genetic evidence supports a mixed Middle Eastern (ME) and European (EU) ancestry in AJ. This is based on uniparental markers with origins in either region (Behar et al., 2006, 2017; Costa et al., 2013; Hammer et al., 2000, 2009; Nebel et al., 2001), as well as autosomal studies showing that AJ have ancestry intermediate between ME and EU populations (Atzmon et al., 2010; Behar et al., 2010, 2013; Bray et al., 2010; Carmi et al., 2014a; Granot-Hershkovitz et al., 2018; Guha et al., 2012; Kopelman et al., 2020). These and other autosomal studies also showed that individuals with AJ ancestry are genetically distinguishable from those of other ancestries. Recent modeling suggested that most of the European ancestry in AJ is consistent with Southern European-related sources, and estimated the total proportion of European ancestry in AJ as 50%–70% (Carmi et al., 2014a; Xue et al., 2017; Yardumian and Schurr, 2019). The AJ population is overall highly genetically homogeneous, with no major ancestry differences based on present-day country of residence (Guha et al., 2012; Gusev et al., 2012; Kopelman et al., 2020). However, there are subtle average differences in ancestry between AJ with origins in Eastern vs. Western Europe (Behar et al., 2013; Gladstein and Hammer, 2019; Granot-Hershkovitz et al., 2018) (Data S1, section 1).
Despite the recent progress, open questions remain, including the localization of the founder event, or events, in time and space and the sources and times of the admixture events. Studying the genomes of individuals who lived closer to the time of AJ formation has the potential to shed light on these questions. We present here a DNA study of historical Jews, focusing on AJ from 14th-century Erfurt, Germany. The medieval Erfurt Jewish community existed between the late 11th century to 1454, with a short gap following a 1349 massacre (Weigelt, 2016). We report genome-wide data from 33 individuals whose skeletons were extracted in a salvage excavation. Our results demonstrate that Erfurt Ashkenazi Jews (EAJ) are genetically similar to modern Ashkenazi Jews (MAJ), implying little gene flow into the AJ gene pool since the 14th century. Further analysis demonstrates that EAJ were more genetically heterogeneous than MAJ, with multiple lines of evidence supporting the presence of two subgroups, one of which had higher Eastern European affinity compared to MAJ. The EAJ population shows strong evidence of a recent bottleneck shared with the bottleneck that affected MAJ, as alleles that are highly enriched in MAJ—including mitochondrial lineages and pathogenic variants—are also common in EAJ.
Even laying aside this core point about founder effects (literally the title and point of the study), the text above again directly contradicts your lie.
Genetic evidence supports a mixed Middle Eastern (ME) and European (EU) ancestry in AJ.
Nothing remotely suggests or supports the claim that Ashkenazi Jews do not have ethnic origins shared with other Jews in the Levant. Rather, the study shows how, after multiple population bottlenecks and founder effects, one particular group of Jews migrated from one part of Europe into the HRE and became the population we define today as Ashkenazi Jews, that can be traced back to this point due to the founder effect distinguishing them from their parent population, and thanks to this particular group's cultural practices of endogamy, they retained those distinguishing genes.
No one at all ever suggested that, for example, the Erfurt Jews migrated directly from the Levant to the middle of Germany, similarly, no one ever suggested that Jewish populations couldn't have existed all across the Mediterranean and Middle East for generations accruing local genes from admixture. History already knows through ample evidence that Jews underwent multiple diasporas, multiple population bottlenecks, multiple forced resettlements over a thousand and more years.