Lawful espials, huh? This language does have the air of the lawyer about it. It has the sort of formal tone, the equivocation, of someone attempting to sound very reasonable. Lawful espials (espials, instead of spys, somehow makes it more lawful?)
Frankly judge, gather. . .are all more words than are strictly necessary to say, “We’re going to watch him with Ophelia.”
Seems to me that Gertrude wasn’t particularly sold on this explanation for Hamlet’s behavior. She’s thinking it’s more a death and remarriage of his parents situation than a “my girlfriend returned my letters” situation. So it does seem very possible that Claudius is working hard to convince her. What was Claudius’ job before King? Was he perhaps the royal legal counsel?