If the contractor states that the work is under the Construction Contracts Act and invoices accordingly then you will be forced to pay on time even if you disagree on the amount, unless you have provided a calculated explained payment schedule promptly to the contractor.

Your invoices should be made in accordance with the Construction Contracts Act and you can enforce the right to an adjudication either by agreeing with the contractor on the appointment of adjudicator or you can approach AMINZ to appoint an adjudicator.