double negation to statements contain several quantifiers

I would like to use double negation to equivalently re-express sentences contain several quantifiers.

And after reading related materials Generalized quantifiers and relations, Classical vs. modern squares of opposition, and beyond∗, I have the following association and don't know if it is correct.

E.g. Assuming there are many balls which have two sizes (small and large) and two colors (red and blue). This means that it only red balls or only blue balls or both have two sizes.

Is the sentence “At most 5 red balls are larger than exactly 3 blue balls” equals to “All red balls but at most 5 are not larger than exactly 3 blue balls. ”

I think the statement “At most 5 red balls are larger than exactly 3 blue balls” is True when

$$ (R \cap L \le 5) \& (B = 3 \& B \subseteq S)$$

where R denotes the red balls, L denotes large size, B denotes blue balls and S denotes small size. And when this statement is True, it means there are at most 5 large red balls and exactly 3 small blue balls.

I'm kind of confusing whether this sentence have an equal meaning as “All red balls but at most 5 are not larger than exactly 3 blue balls. ” or “All red balls but at most 5 are larger than not exactly 3 blue balls. ”

Is the sentence “At most 5 red balls are larger than exactly 3 blue balls” equal to “All red balls but at most 5 are not larger than exactly 3 blue balls. ”

No. Assuming that the speaker isn't referring to three particular blue balls, then the former is equivalent to, for example, “It's not the case that at least 6 red balls are larger than fewer or more than 3 blue balls”, which can be symbolised as

$$\lnot\,∀x_1,x_2,x_3,x_4,x_5\: ∃p\: ∀i{\in}\{1,2,3,4,5\}\\ ∀y_1∀y_∀y_3\: ∀m,n{\in}\{1,2,3\}\: ∃q\: ∃j{\in}\{1,2,3\}\: ∀k{\in}\{1,2,3\}\\ (p\ne x_i\land Rp\land\bigg((m\ne n\to y_m\ne y_n) \to\\ \Big((q=y_j \land\lnot\big(Bq\land Lpq\big)) \lor \big(q\ne y_k \land Bq\land Lpq\big)\Big)\bigg)).$$