There are two distinct types of lying mentioned in the article, and each should be handled differently:

1. The non-issue "lie" that was the subject of the original question. Sometimes I get a little annoyed at people that take a form completely literally and then raise a stink about it. Every form like this has an implied "to the best of my knowledge" inserted in it. You cannot know for 100% certainty that every purchase was completely necessary. Did every piece of paper that was printed from that ream you ordered get used in a necessary function, or did maybe one or two sheets get used for someone to print their baseball tickets out at the office? Could some of those papers have been stored digitally instead of printed? Of course that ream of paper wasn't used for 100% necessary business purposes. But by signing the form you are saying, "it doesn't look like there are any employees in my branch that are stealing tons of office supplies, and that our waste seems reasonable." What you are saying is that "I'm not personally aware of any fraud going on with the supply order, and I've done some due diligence to check." What you are NOT saying is "I personally tracked every piece of paper and staple in the office to make sure nobody was cheating." That is not the expectation with these certifications, and any federal employee fired for that level of expectation can go to arbitration and would certainly win. Luckily federal employees have probably the strongest employment protections of anybody in the USA.

2. The second type of lie, mentioned in the anecdote about the Army officers, is a problem but easily fixed in a way that you don't sacrifice your integrity. This is the case in which adding the implied "to the best of my knowledge" doesn't fix the issue. To fix this, you must line out the "297 days" portion and write in (a different colored ink) the correct number of days that you trained. Then you sign in ink, and return a scanned copy to wherever it needs to go. You probably cannot change the form in Adobe Reader because most government forms are protected against changes, so you must alter the form with pen. If the people that you returned the form to raise a stink (usually they want a digital signature), you simply explain that you cannot comply because to do so would be perjury. If they still won't accept your corrected version of the form you can stall, but let your boss/commander know that you cannot sign the form as-is because to do so would be a violation of federal law- and you can't be fired for not breaking the law. Most of the time, it falls out on your side. If not, you would have your defense counsel salivating to take this one past arbitration to make some real money.

Pretty much every federal employee that cares about their integrity has already discovered the answers to these situations, and I suspect that this question is more of a straw man so that the author can talk about the pervasive culture of dishonesty that exists in the workplace. He's right in that a culture of dishonesty does exist, but he's missing the mark on where it's a problem. It's not in the resource management office that signs off on the paper/staples/toner orders or the Army officer that signs a training certification form that has the wrong number of days on it (as long as he feels his troops are mission ready). The culture of dishonesty is where a person awards a contract with a particular company so that they can get a cushy job in the private sector, or with people that accept a free trip to Disneyland to overlook an issue. Those are the dishonest incidents that matter, and they do not arise from some imagined "slippery slope" after certifying an office supply order.