Friday's exchange of letters between the election campaigns of Barack Obama and Mitt Romney, in which Romney rejected Obama's offer to drop the tax return issue if Romney will produce just three more years' records, has moved the long-simmering brouhaha over Romney's tax returns back to the front media burner. Romney has only produced two tax returns so far. That's many fewer than any presidential candidate has disclosed in decades, setting up the hearsay accusation disseminated joyfully by Harry Reid (who may or may not actually believe it) that Romney is afraid to tell voters that he sometimes pays no taxes at all. (Romney has answered that, saying he has never paid less than 13% in taxes on his income.)

Meanwhile, Romney appears to have escaped relatively unsinged from the apparently unrelated revelation that he may have committed voter fraud in January 2010, when – despite not owning a house in Massachusetts and having given every appearance of having moved to California – he registered and voted in the Massachusetts special election to replace the deceased Senator Ted Kennedy. Given the GOP's ongoing use of the "voter fraud" fable to justify modern Jim Crow laws and its highly-publicized persecution of the voter registration group Acorn, an actual case of felony voter fraud committed by the Republican nominee could have been a big story – but Romney was able to tamp down the flames by claiming, not very credibly but also not disprovably, that he and Ann actually were living in their son Tagg's Belmont, Massachusetts basement in 2010. Without proof that Romney lied about where he lived, there's no felony – and no big national story.

Many (many, many, many, many, many) theories have been advanced to explain why Romney keeps refusing to produce any returns prior to 2010, ranging from "voters might learn he's wealthy" (which voters already know) to "he underpaid his church tithe" (doubtful).

None of them is really satisfactory, because none of them posits Romney concealing any facts more harmful than the blowback he is getting for not producing more returns. The problem may be that all of the prominent theories (with a couple of under-noticed exceptions) assume Romney is trying to conceal facts about his finances. Like the purloined letter pinned prominently in plain sight, what Romney's really hiding might be something more mundane: the home address written on the top of the tax form. That address that might reveal a connection between the "tax returns" brouhaha and the "voter fraud" fizzle – which may be the strongest explanation of all. Here's why.

Tax returns require taxpayers to state their residence address, and the Romney returns already produced, although partially redacted, state clearly that they lived in "Belmont, MA 02478" in 2012 (tax year 2011) (pdf) and 2011 (tax year 2010) (pdf):

The Romneys' 2010 tax return

But the Romneys, arbitrarily, refuse to disclose a copy of the returns they filed in 2010 or 2009 (for tax years 2009 and 2008) – which, perhaps not coincidentally, bracket the time period when Romney allegedly committed fraud by voting in Massachusetts when he actually resided in California. So here's the question: did Romney put his son's basement's address on the returns he filed in 2009 and 2010? Or did he truthfully use his real (non-Massachusetts) address, thus implicating himself in voter fraud?

This may seem like overmeticulous wonkishness, but the address given on tax returns is a big deal when it comes to proving voter fraud. As Hans von Spakovsky (senior legal fellow at the conservative Heritage Foundation, former Federal Election Commission member, and former DOJ voting-issue attorney, and himself an advocate of the GOP's restrictive voter ID requirements) explained to the Daily Caller:

"Election officials will also look at tax returns as crucial evidence in residency disputes. Where an individual declares himself to be a resident for tax purposes, thus subjecting himself to applicable state income taxes, is usually decisive on this issue."

With that in mind, let's run through the dates, keeping in mind that tax returns are filed the year after the tax year in question (and that Romney's returns, which are exceptionally complicated, likely are filed toward the end of that year: that is, his 2010 return was filed in October 2011):

• In April 2009, Romney sold his longtime Massachusetts home at 171 Marsh Street and appeared to move to La Jolla, California. He did not own a home in Massachusetts again until July 2010. (All of the Massachusetts addresses discussed here end with "Belmont, MA 02478", though one might amuse oneself surmising which address is underneath the Sharpie by the length of the redaction.)

• Sometime in 2009, probably late in the year, Romney filed his 2008 tax return, identifying the address where he lived at the time of filing. He has refused to disclose a copy of that return.

• Sometime in or shortly before January 2010 – that is, not long after he filed his 2008 return – Romney registered to vote in Massachusetts, stating on his voter registration form that he lived in his son Tagg's basement at 18 Greensbrook Way. In January 2010, Romney voted in Massachusetts' special election, which would be a felony if he was not a Massachusetts resident at the time.

• In July 2010, Romney once again became a Massachusetts property owner when he bought a new townhouse at 10 S Cottage Road Unit 3. However, as the owner of several other houses, he still could have resided elsewhere for voter registration purposes.

• Sometime (probably late) in 2010, Romney filed his 2009 tax return. If that return was filed before July and he really was living in Tagg's basement, it should give a Belmont address. If it was filed after July, and Romney truly considered Massachusetts his home, it should give the Cottage Road address. He has refused to disclose a copy of that return.

• On 15 October 2011, Romney's accountant filed his 2010 return, giving a redacted Belmont address (presumably the Cottage Road townhouse) as his residence.

If Romney's 2008 return (filed in 2009, shortly before the January 2010 special election) didn't give Tagg's basement as his address, then Romney clearly didn't consider Massachusetts his home in that year. If Romney's 2009 return (filed in 2010) gives a non-Massachusetts address, despite the fact that he claimed to be a Massachusetts resident earlier that year and had bought a house in Massachusetts in July, then Romney clearly didn't consider Massachusetts his home in that year either. If Democrats hit the daily double – in other words, if Romney declared La Jolla, California to be his home in both years – then Massachusetts prosecutors likely will have no choice but to take a hard second look at their ex-governor. (The Obama campaign's new focus on obtaining only three more years' returns – 2007, 2008 and 2009 – may suggest they're focusing in on this possibility as well.)

A felony voter fraud charge could expose Romney to fines and/or imprisonment, jeopardize Romney's standing with the Michigan State Bar, and – worst of all, in the political sense – would be a mortal embarrassment on the campaign trail, both to himself and to downticket Republicans (especially Republican Senator Scott Brown of Massachusetts, who won the special election in question but is locked in a tight, highly-publicized race against the popular Elizabeth Warren to retain his seat).

So far, none of the reasons advanced for Romney's refusal to produce tax returns seems good enough to justify the political heat his campaign is taking. But if those returns give a non-Massachusetts address, then Romney can't afford to produce them, no matter how much political fallout his campaign faces as a result. All of this is speculation, of course, though it seems at least as plausible as Harry Reid's suggestion that Romney paid no taxes before 2011, but there's only one way it can be resolved: by Mitt Romney releasing those tax returns.

• Editor's note: the second sentence in this article ("Romney has only produced two tax returns so far") was cut in error; it has now been restored at 8am (ET) on 18 August 2012.