What is the appropriate term to use for titling a mathematical statement which will be proven false? Note that I'm focusing on the context of labeling and organizing results within a paper or similiar, e.g., Theorem 1.3.5 is a consequence of Lemma 1.2.4 and Proposition 1.2.3. The following conventions seem typical in my experience of research papers:
- a Lemma is a true statement, but is generally not of interest in its own right and is subservient to other grand results.
- a Theorem is a true statement of the utmost importance in the exposition.
- a Corollary is a true statement that is easily derived from other results.
- a Conjecture is a statement that is not proven, but is often believed to be true by those working in the field.
- a Proposition is a bit murkier, but my experience has been that it is generally used in papers to denote a true statement whose importance/interest is somewhere between that of a Lemma and a Theorem. Note that many textbooks use Proposition as a catch-all term for any statement whose truth is yet to be resolved. For example, "Problem #1. Prove or give a counter example. Proposition: ...."
However, there doesn't seem to be a good use for listing a result that is false, although the following would all achieve the result after jarring the reader to varying degrees.
Theorem: even plus even is odd
Counter-Example: 2+2 = 4 $\blacksquare$
My issue here is that the reader has to actually look at where the proof would go and not immediately run off to write a listicle titled "You won't believe these 10 counter-intuitive results in mathematics!"
This becomes even more problematic if your proof isn't simply a counter-example, but rather page upon page of complex manipulations. Even with guiding narration, it seems to ask a lot of the reader. Perhaps something like "Theorem:... Anti-proof:....QED" in this case?
Fallacy: even plus even is odd.
This seems the closest to what I'm after, but fallacy seems a bit too specific and yet quaint (depending on the field). I imagine that if I opened a paper and saw "Fallacy 1.3", I might immediately start looking for "Fable 2.4" and "Fuzzy Notion 6.2".
Proposition: even plus even is --NOT-- odd
This hammers home the point (depending on the amount of emphasis added to the negation), but feels...jarring? There is also a balance between the reader missing a single word and thinking you're an idiot vs. adding so much emphasis that it becomes an eyesore.