New York Consolidated Laws, Civil Practice Law and Rules - CVP § 3033. Contracts to submit;  enforcement of submission

1. Any written contract, otherwise valid under the substantive law, to submit any existing or future controversy to the court pursuant to section 3031 is valid and enforceable and shall be construed as an implied consent of the parties to the jurisdiction of the supreme court of this state to enforce it pursuant to the procedures of rule 3036, and to enter judgment thereon, and shall constitute a waiver by the parties of the right to trial by jury.

2. If the parties to a dispute arising under a contract to submit a controversy to the court under section 3031 are unable to agree on a statement of claims and defenses and relief sought pursuant to that section, the court on motion shall settle the terms of the statement.  In deciding the motion the court shall consider and determine any questions as to the existence of the contract or its validity or the failure of any party to perform it.  If a substantial issue of fact be raised as to the making of the contract or submission or the failure to comply therewith, the court or judge shall proceed to trial of such issue without a jury, unless either party should demand a jury trial.