I've tried reverting to factory settings (ie stock and non rooted and standard recovery)
Reflashing the ROM and carefully managing the first two charge cycles (up to 100, down to 15, up to 100, then normal)
Two online tickets submitted to Samsung have provided the above "helpful" suggestions which I had already tried and told them that I tried.
Installed battery monitor widget and a screenshot is attached (couldn't figure out how, if possible, to export the history) but you can see the salient points. Plugged it in at 15% at about 1015 PM, happily charging until 10:57 PM when it inexplicably stops at 63%...to resume charging at 8:12 AM and it had almost full 99% at about 9:15 when I opened it to use. (I did not touch it during the charge time).
I also installed the Galaxy charging current app/widget. I'm not 100% certain it supports the Tab 3, but the reference current swings wildly - 800mA is the most I've seen with lows of around 160mA.
Is this a hardware or a firmware problem?
How can I narrow it down?
How can I get Samsung to actually listen to me and do a warranty repair before it runs out?
Advice? Thoughts? Exorcisms?