1809 CBH With "Centsalf" On Edge

Thanks for the welcome!

I've found it referenced here: https://media.stacksbowers.com/Virt...talog_LR.pdf

I'll see about uploading a photo, but it is as described.

It would be helpful if you rotated your pictures upright for easy viewing. Can't say for certain but that coin has the look of "Fresh off the Chinese press". Can you give us a weight, diameter and thickness?

The new pictures are much better. I am inclined to agree with you on authenticity. There are several edge varieties with this edge lettering misalignment being one of them. If genuine this is a high grade high dollar coin worthy of slabbing by a top TPG (PCGS or NGC). Here is a link to the PCGS site for comparison.
https://www.pcgs.com/coinfacts/coin/1809-50c/6092

I found this list of varieties sorted by date and type with photos:

https://www.ngccoin.in/variety-plus...s-1807-1836/

I have another question: What are the implications of the edge markings (|||| or xxxx) in relation to edge lettering? Were the markings in specific places relative to the letters? Or did they vary depending on where the coin ended up in the production process?

Looking at varieties on the NGC site, it seems the edge markings are usually located pretty close to the start of "FIFTY". I feel like the term "markings" is . subjective. In many of the photos I've seen, what pass for markings I would normally have considered cheery-eyed optimism. Are there any markings on this edge? Or am I being cheery-eyed and optimistic?

Hooray, a bust half! I love these things. I had an O-106 prime just like yours not too long ago, but your coin has a razor sharp strike and is fairly high grade. I can answer your questions.
Did it only happen in 1809?
Edge lettering errors like yours can be found on all dates in the series, 1807-1836. It was caused by the Castaing edge lettering machine before it was struck. An independent system from striking that had it's own problems and trial runs as the First Mint employees were figuring things out as they went. Regardless, minor edge lettering errors are still found and the system was never foolproof.
Is it particularly rare?
Overlapping edge lettering is not rare. I would place the magnitude of overlapping on your example as uncommon. You can find examples like this in the wild and pay no premium. When significant portions of the edge lettering is obliterated, that is when the collectability increases.
Is it unique to a particular Overton number?
Patterns can arise (1814 a star was added on the edge, 1809 has XXX and IIII markings, etc) but there is no intentional correlation between die marriages and edge errors/mistakes.
What are the implications of the edge markings (|||| or xxxx) in relation to edge lettering? Were the markings in specific places relative to the letters? Or did they vary depending on where the coin ended up in the production process?
Problems arose during first handful of production years of the first U.S. Mint. One was slipping of the bar dies when the planchet was rolled through the machine. The planchet blank would be placed between the thin rectangular dies that contained the lettering, and the lettering would be rolled out onto the edge one coin at a time as the Mint employee worked the machine. On occasion, the bar dies would lose grip on the planchet and the edge lettering would overlap such as demonstrated on your coin. To prevent this, crude XXX and IIII designs were placed on the beginning of each bar dies in 1809 to help the planchet quickly grip the steel dies. For some reason, this only was tested in 1809 and it wasn't until 1814 that a new design element was introduced on the edge.
Are there any markings on this edge? Or am I being cheery-eyed and optimistic?
I have the records for the bar dies used for each die marriage. Out of 7 bar die marriages recorded for 1809, only marriages 3 and 4 represent the infamous XXX and IIII elements. But marriage 6 has what is described as "pseudo-engraved, light."
The 1809 O-106, your coin, is most commonly found with a plain edge. But, only in the earliest of die states, marriage 6 was used and something on the edge is noted. I do believe your coin shows the pseudo-engraved design. The planchet may have slipped off the edge when it was in the lettering machine, because you can see the design trail off to the side, making it more difficult to distinguish.

Here is my 1809 O-108a, showcasing bar die marriage 4 with the XXX design, which mistakenly overlaps the word DOLLAR:

Great post, jacrispies!

The coin is well struck (look at those stars!!), early die state, and has the most crisp detail you can get on that date. Unfortunately the coin is bright with hairlines, so it would earn the cleaned designation. AU Cleaned. A super cool coin nonetheless.