To clarify: An amended return is required if you discover an error in the income, dependents, deductions or credits, whether you intended to or it was a mistake.
No an amended return is never "required." A taxpayer might want to file an amended return (to get a refund, stop interest from accruing, etc.) but there is no requirement to do so.
For my authority for this see page 228 (pdf page 7) this law review article and the statutes, regulations, and cases it cites.
https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1505&context=nlr
which says:
Treasury regulations permit the filing of amended returns for purposes of correcting errors and making or changing various tax elections. The regulations do not, however, impose a duty on taxpayers
to file amended returns. Treasury regulations provide that if a taxpayer "ascertains" that income was improperly omitted from a prior year's return, the taxpayer "should, if within the period of limitation,"
file a corrected amended return and pay any additional tax due. If the taxpayer "ascertains" that an item was improperly included in income on a prior year's return, the taxpayer "should, if within the period of limitation," file a claim for credit or refund of any resulting tax overpayment. Similarly, Treasury regulations provide that if a taxpayer "ascertains" that a liability was improperly omitted from a prior
year's return, the taxpayer "should, if within the period of limitation," file a claim for credit or refund of any resulting tax overpayment. If the taxpayer "ascertains" that a liability was improperly claimed on a
prior year's return, the taxpayer "should, if within the period of limitation," file a corrected amended return and pay any additional tax due. Thus, the Treasury regulations encourage, but do not require,
the correction of tax return errors. Id. at 229 (citations omitted) (emphasis added).
The law review article goes on to review Tax Court and Supreme Court cases coming to the conclusion that amended returns are never required (even in the context of fraud!) but the author suggests that Congress should create such a duty for significant errors.
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