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Tax Strange Loop

Warning to readers: I am not a tax lawyer, but believe this account is correct as of the 2018 tax year.

Once upon a time there was Mrs. X and Mr. Y.  They were married residents of Pennsylvania who spent many years happily contributing to algebra problems.  They filed their taxes jointly.  Sadly, Mr. Y passed away in 2018.  His will directed that his assets be directed to a Miss Z.  At the end of the tax year, Widow X received 1099 DIV forms for Y, for which part of the income had arrived while he was alive, and the rest after he died.  

X can still file jointly for 2018.  Doing the taxes was a bit tricky, but after reading the instructions, she figured out they worked as shown below.  The blue arrows denote money received while Y was alive; the red arrows denote money received after he died.  Per instructions for Schedule B, she totaled red+blue, and then wrote an annotation indicating how much money went to the Federal 1041 for the estate.  On the bottom right, only money received before Y passed away is taxed on her return.  

Meanwhile, since Miss Z inherited the stocks, she also is entitled to the dividends from those assets. Hence the estate distributes the income to her via Federal Schedule K-1, so she pays the federal taxes on the dividends. After distribution, the estate has zero taxable federal income. In a similar manner, the estate distributes the income at the state level, from PA 41 through PA DD and RK-1. PA DD is essentially a gate computing what income can be distributed, and PA RK-1 is roughly like a Federal K-1 for residents of Pennsylvania.

Miss Z also pays the state taxes on the dividends, albeit the math is more circuitous, because PA 40 Schedule B starts with a line from 1040, and then subtracts the Federal K-1 income, since PA has it's own Schedule RK-1, for which the estate sent a copy.  Her PA 40 then adds it back via PA 40 Schedule J for income from estates or trusts.  The bottom right box shows that Miss Z pays state taxes on her dividends and the red dividends.   

This would be the end of the story, except that "Miss Z", who as a youth spent days dancing on the complex plane, turned out to be the maiden name of Mrs. X!  So they are the same taxpayer, and the blue and yellow boxes are for the same tax return, and hence must merge.  The merged diagram looks like this:

The cycle near the top, that starts at "add" has two strange features.  First, the red money appears to travel backwards in time, because the income from the K-1 has to be tallied on Schedule B before creating an extra line that sends the income to the estate.  Second, between the "add" and "split" steps, there is a subtotal that includes the red income twice.  As Widow X knows well, addition is commutative and associative, so the red income ends up contributing exactly once to the final total on Schedule B.   

Likewise she pays the state taxes on it exactly once.  No cycle appears at the state level, because the income sent to the estate's PA 41 comes back to X on PA Schedule J.

So there you have it: death, taxes, and tax-form ping-pong.
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