the question is:
If a borrower that lives in NY, receives a Loan Mod offer, and has told the lender that it objects certain paragraphs, and the lender keeps sending the same loan mod within the same paragraphs unchanged, then can the borrower sign the loan mod WITH SOME SORT OF OBJECTION refreed to specific paragraphs?
Is there ANY supposrting law that allows thw borrower to do that
The question suggests a fundamental misunderstanding about the essence of contracts.
I would suggest that you read some of the basics on contract law. You might start with this exposition at Wikipedia:
Generally speaking, a contract requires offer and acceptance.
If A offers to sell his Buick to B for $1,000, B can either accept the offer, in which case a contract comes into being, or reject the offer. If B does not agree and accede to the terms proposed by A in A's offer, B can always decline and make a counteroffer. But B cannot simply unilaterally alter the proposed terms of A's offer and accept based upon altered terms.
This is not merely a matter of law, but is also based upon common sense and fairness.
Suppose that A offers to purchase B's Palomino mare, a saddle and bridle for $5,000. Can B accept A's offer, while omitting the mare and saddle from the agreement binding A to the purchase of the bridle only for $5,000? Such a proposition is preposterous and intuitively we know that this cannot be the basis for a contract. To contract means to agree.
So if a servicer presents a modification agreement to a borrower, even an offensive and oppressive agreement, the borrower is presented with the choice of accepting the modification offer, rejecting the offer, or making a counter-offer.
As a practical matter, the servicer will almost never agree to a modification on terms other than those appearing in the standard oppressive offer. The only time that an agreement might be reached on other than standard oppressive terms is when the Lender is truly over a barrel, which is quite rare.
It should also be noted that while a modification agreement is often presented as if it is an offer, it almost never IS an offer. This is because the modification agreement itself will usually call for the the signature of both the borrower and the Lender/mortgage investor. But the servicer will almost NEVER sign the agreement in advance.
Presenting an unsigned agreement to the borrower is not actually an offer at all, but rather the solicitation by the servicer of an offer by the borrower on the terms set forth within the agreement. That is, since the agreement is not signed by the servicer, no agreement comes into being when the agreement is executed by the borrower. Instead, the borrower is making an offer to the lender. The lender then puts the unsigned agreement (executed by the borrower) in the loan file. This may not be a valid acceptance of the agreement.
There is much contract case law showing that an offer can be accepted by the counterparty's actions, rather than merely executing the agreement. That is, when the servicer begins accepting payments based upon the written modification agreement, the borrower has at least some good argument that the servicer accepted the offer. But the written provisions of the modification agreement itself may say otherwise and court may be required to respect the intentions of the parties as expressly shown in the written modification agreement.
The entire discussion about the possibility of agreeing to the modification agreement while rejecting or repudiating some of the terms is nonsensical. That dog wont hunt. But even more basic, the modification agreement often commits the mortgage investor/servicer to NOTHING AT ALL, because most borrowers are too foolish to realize that without the agreement being signed by the servicer, there IS NO AGREEMENT.