There are many ways in which Piketty is wrong. However, it seems nobody has noticed it is logically impossible for his primary empirical assertion to be correct. (There is no true a priori statements, but given that, I think the term can be re-purposed for things like this.)
As per the above link, GDP growth is asserted to be, rounded, 1% and return on capital is 5%. If this were true, non-capital forms of income would see decline, and hit zero quite quickly due to exponential compounding.
Say 50% of GDP is return on capital and 50% is other forms of income. Say GDP grows 1% and capital return is 5%, so capital amounts to 1000% of GDP. Let's just call it $1000 because why not. So GDP year 0 is $100.
Next year capital is $1050. GDP is $101. Capital return is still 5%, so $52.50. Call it 53.
Income is now $48. Capital is now $1103
GDP $102.01. Again I'll drop the cents from income and capital - $47 and $1158
GDP is $103.03 Income - $45 Capital - $1216
$104.06 || $43 || $1277
$105.10 || $41 || $1341
Etc. After only five years, income is now down nearly 20%. If population has grown, most would be dramatically poorer. In fact, everyone has been getting richer. Eventually income will hit zero, and capital returns will begin tanking. Though of course income = zero means everyone starves and the economy collapses entirely long before that.
Obviously this is slower if return on capital is a lower portion of total GDP, but it is logically impossible that return on ALL capital could be more than GDP growth for hundreds of years. If he hasn't fudged his numbers, all he's proven is that GDP is a worthless measure.
(Indeed it's slightly worse than this - if any of that income isn't spent, it becomes capital and reduces future income even further.)
So at 50%, non-capital income sees 3.3% decline.
Let's say capital is only 10%. $200.
$101 || $90.5 || $220.5
Income sees 0.6% growth in the first year. Capital is still, by definition, seeing 5%. Within decades non-capital income will stagnate. By year 10 it's already down to 0.3%. Imagine your final, year-65 retirement wage being less than 15% higher than your first job.
This non-capital decay only doesn't happen if capital is an utterly negligible part of the economy.