The government must prove that the defendant acted knowingly and willfully

Those taxpayers who do not fall at the ends of the bell curve on willfulness must proceed with extreme caution .
False Statement Crime :
Willful" simply means that the defendant did the forbidden act e.g., made a false, fictitious, or fraudulent statements deliberately and with knowledge.
It further means at a minimum knowing that the conduct is illegal or acted with knowledge that his or her conduct was unlawful. Now on the bright side or the quintessence of black humor — Certifying non-willfulness   - let this conundrum throw what is left of your brain into an infinite entropic loop.
I didn’t know until I knew .....I didn’t know until I knew........I didn’t know until I knew.

The DOJ has indicated and clarified in some briefings that to make the case that a defendant acted "willfully," the government must prove that he or she knew the statement was unlawful — not just that it was false. That requires a state-of-mind element that can be hard to nail down. Defendants often claim they were unaware that lying to the government was illegal, especially in casual conversations or when not under oath.
http://lawprofessors.typepad.com/whitecollarcrime_blog/2014/06/doj-narrows-basis-for-section-1001-prosecutions.html
http://www.justice.gov/tax/readingroom/2008ctm/CTM%20Chapter%2024.pdf
As to the knowingly and willfully element, the CTM says that “several courts” permit the element to be met by proof of “willful blindness” or “conscious avoidance.” 
The Supreme Court has recognized an ignorance-of-law or mistake-of-law defense, or has required affirmative proof of the defendant's knowledge that his or her conduct was unlawful. See Bryan v. United States, 524 U.S. 184 (1998); Ratzlaf v. United States, 510 U.S. 135, 141-49 (1994); Cheek v. United States, 498 U.S. 192, 199-201 (1991).