People really should use dictionaries more.
the definition of regressive
That would be a start; but it requires a bit deeper of an understanding to avoid simple mistakes.
Let's take a "tax credit" for example. You say "I'm going to give everyone a $5k deduction off their taxable income... because it's fair and equal".
So the person at a marginal tax rate of 25% gets a check for $1,250 and the person with a 10% marginal tax gets a check for $500.
The underlying issue is that "equal", while easily defined by a dictionary, is not easily defined in reality.
If I tax everyone the same amount (say $20,000)... that's "equal" right?
What if I tax everyone the same percentage (say 15%)? That's a different number. They can't both be "equal"?
What about people who pay state taxes? Should I charge them tax on the money they are paying the state in tax? Is a gross income tax the equal one, or a net income tax the equal one? If "gross", then it's possible for taxes to exceed 100% of income. If "net", then we have to say "well: what income shouldn't we tax"?
Should we tax the $500 you made that went to SS? We are going to tax it again when you get paid from SS? If not, it's an exemption (like state taxes). Now, suddenly the tax structure starts to look a lot more complex.
There's no single answer for "equal", and so there's no single answer for "regressive".
Is 10% across the board regressive because the rich pay more dollars (same percent?) Or is $20k/person regressive because the rich pay a lower percentage.
The reason that "flat" is "regressive" is, too, mired in the issue of what is and is not "equal".
The OP's flat tax proposal increases the average tax rate with increasing income.
Asymptotically, it increases from 0% to 10%, so it's effectively "progressive".
The simplest way to describe his assertion (assuming we become sane enough to make the tax rate marginal) is it's a single-bracket (10%) tax on net income where the only deduction available is a standard $25k.
So. If you made $50k; and that $50k was taken from your paycheck for medical bills. You got $0, and you owe the government $2,500 you don't have, and likely are about to go to jail.