Ten thousand fools proclaim themselves into obscurity, while one wise man forgets himself into immortality. (Martin Luther King Jr.)
A paper should neither understate nor overstate its main results.
If the main result is very surprising or a substantial breakthrough compared with the previous literature, these facts should be noted (and justified in detail, for instance by explicit comparison with prior results, examples, and conjectures).
Conversely, if there are unsatisfactory aspects to the result (e.g. hypotheses too strong, or conclusions a little weaker than expected) these should also be stated honestly and openly, e.g. “We do not know if hypothesis H is actually necessary”. Similarly, it is worth noting down any interesting open questions remaining after your result.
If you are using a famous unsolved conjecture to motivate your own work, one should give a candid evaluation of the extent to which your work truly represents progress towards that conjecture, so as to avoid the impression of “false advertising” or “name-dropping”.
If for some reason you need to assert a non-trivial statement without proof or citation, it should be made clear that you are doing so (e.g. “It can be shown that…” or “Although we will not need or prove this fact here…”), so that the reader does not then hunt through the rest of your paper for the non-existent justification of that statement.
Titles of sections should be descriptive (e.g. “Proof of the decomposition lemma” or “An orthogonality argument”), as opposed to uninformative (e.g. “Step 2” or “Some technicalities”).
See also “Give appropriate amounts of detail“.