Large Difference between Billed Amount and Allowed Amount

This is some screwy math by your insurance, as you suspected.

First, keep in mind that an EOB is not a bill. The EOB just tells you how much insurance covered/paid, so you can then expect the amount that the doctor will later bill you for (unless everything was totally paid by insurance of course). The hospital may bill you this amount or less (but should never bill your more that what the EOB says you owe).

Now, let's figure out what they did wrong. It looks like you met your deductible with this hospital visit, of which $5,716.15 was left over beforehand. After that point you only pay 20% of the inpatient services bill (line 2: ($318.55 coinsurance) / ($1,592.73 billed) = 0.2). Line 1 shows that $5,716.15 was applied to your deductible, as expected. However, the coinsurance column for this line is wrong. It looks like they took the plan discount, added the amount left over after the deductible, and then took 20%:

$25,599.75 + ($5,830 - $5,716.15) = $25,713.60
$25,713.60 * 0.2 = $5,142.72

What they should have done was take the amount left over after what was put toward the deductible and then multiply that by 20%:

$5,830 - $5,716.15 = $113.85
$113.85 * 0.2 = $22.77

So, the total amount you should owe off of line 1 is $5,716.15 + $22.77 = $5,738.92

You have already contacted insurance, which as you discovered is often not helpful. You can contact them again with this explanation and request a new EOB. If that doesn't work then you might have to formally appeal the claim (insurance will have a defined way to do this).

Bottom line, make sure that on the bill from the hospital you do not owe more than $5,716.15 + $22.77 + $318.55 + $40 = $6,097.47.