...isn't so much to compare what the high earner pays in taxes to what the low earner pays; but to consider impact that each additional dollar of tax has at various income levels, given the generally known costs of living.

If you have a family of four and AGI of $500K and have spendable income of $475K after taxes, you have covered the base costs of living far more comfortably than the person who has $47,500 left after taxes or about $4,000 a month. If you kept to the ratio of 30% of gross for rent, that would leave $1,200 a month for rent for the low earner and $12,000 a month for the high earner. A family of four in DuPage is going to have a very hard time finding housing for $1,200 a month.

Other everyday taxes are regressive, anyway, such as the sales tax.

To you last point, I believe that there is no such thing as objective fairness, any more than there is objective value for real estate (a taxpayer's appraiser I'm thinking of liked to talk about the "intrinsic" dollar value of the real estate, which is a fiction). Fairness in public policy is what the greatest number would agree to be fair.

...I suppose it'd be simpler to say that a flat tax has a disproportionate impact on lower incomes.

I guess there are two issues: (a) the definition of progressive/regressive and (b) the justification for progressive taxes.

If the taxes paid by an income group as a percentage of that income rises with the level of income the tax is progressive. If this falls as income rises the tax is regressive. Of course the categorization depends on a calculation of who actually pays a tax and economists have their own way of doing this.

The justification for progressive taxes is not usually based on fairness. Economists find this to be a slippery concept. The argument is utilitarian: Progressive taxes produce less pain for a given amount of revenue raised. The idea is that the reduction of utility (pain) caused by depriving someone whose income is $200,000 of $50,000 worth of consumption is less than the reduction of utility caused by depriving someone whose income is $40,000 of $5,000 worth of consumption.

Said in reverse, going from 50K to 45K has a larger negative impact than going from 250K to 245K. Maybe 50-48 utility lost equals 250-245 utility lost (numbers I picked at random) so it maximizes utility to tax the higher earner more on the margin, no?

Is a VAT considered regressive, progressive, or neutral? I honestly don't know, but my gut says "neutral" since it's based on consumption, not income. What's the expert view?

spend a larger percentage of their incomes on taxable goods

Better concepts include a refund or "pre-fund" of the tax value of a given standard of living, offsetting the basket of food, fuel, etc.

People with low incomes save less. That's the basis for the regressivity.

Yes. I guess you could wed a negative income tax to a VAT and claim that the combination is less regressive or not regressive but the VAT part is still the VAT part.

the concept from the mid-90s under the "Fair Tax" label did not have a 'non-taxable goods' category. If you bought something, the transaction was taxed. If you paid for a service, the transaction was taxed.

Monthly Universal Basic income voucher pre-pays the cost of basic goods and services, so low income has a net zero tax burden at worst, net tax income possible.

Prompted me to go online (because the answer surprised me), and I see your answer stated verbatim in a Tax Policy Center piece on this question. I hadn't though it through and was curious about the data that lay behind the answer, but the TPC answer made it quite obvious without the data.

service as we can possibly afford, with someone else's money.