The list

Statements 1,2,3, and 4 contradict with one another. And statement 5, is the only unique one.

Let's start there. If Cellini wrote sentence 5, that would mean, Bellini wrote 2,3, or 4 of the sentences. But, Bellini tells the truth, he wouldn't be writing contradictory statements.

Hence, we know, Bellini wrote sentence 5. Which means, sentence 5 is true.

Which means that only one of the 5, were written by Bellini, which is sentence 5. Cellini wrote all 4.

Since there are 5 sentences and they all say different things (sentence 5 implies that exactly 4 sentences were written by Cellini, since 5-1 = 4) at most one of them can be true. But since both men contributed, at least one of them must be true. So exactly one sentence is true, and it must be sentence 5.

Therefore, the other sentences are all false. Since Cellini wrote the false sentences, he must have written the first four. So exactly four sentences were written by Cellini.

Statement (1) contradicts all the later 4 statements + the expert opinion so it has to be written by Cellini

Statement (2) suggests that only 1 sentence was written by Cellini, now we know that Stat(1) is for sure written by C; so now we have a small reason to believe it's a true statement, and if it's true that means all 3 statements follows this one including it with are written by B who always writes the truth, and this will lead to a contradiction with statement (5) and also stat(3) and (4); so this one also is was written by Cellini

Also the same with (3) and(4) , that will leave us with Statement(5) to be the only one written by Bellini, and that means 4 sentences were written by Cellini

Solution n1:

Since all the statements are opposites and there are two handwritings then 4 statements were by cellini (because bellini is a truth teller and cant give 2 opposite statements)

Solution n2:

If #1 is true , then there will be only one handwriting which is wrong = 1 is false =written by Cellini

If #2 is true , (One of these sentences was written by Cellini. )then two between 1 , 3 and 4 will be written by bellini which will be wrong because it will seem as if bellini is a liar since 1,2 and 3 are opposites =2 is false =written by cellini

if #3 is true and we have deduced that #1 and #2 are wrong , then #4 and #5 are true hence they are opposites then only one of them is true =3 is false written by cellini

If #4 is true , then 5 will be true also since we know that 1 , 2 and 3 are wrong hence 5 says that Exactly one of these sentences was written by Bellini. then it cant be that two statements are true =4 is false = written by cellini

that leaves with #5 true because there are two handwritings