What you continue to ignore is that you can set explicit conditions for accessing your property. It's not complicated; movie theaters do it everyday. Your welcome to come in and watch a movie...you just have to pay for a ticket first. Different clubs and restaurants have dress codes: no tie, no jacket, no entrance. When a property owner says, or posts, "No Firearms Permitted on the Premises," they are setting a condition for permission to the property. Persons not complying with that condition are trespassing by virtue of the fact that they are entering the property without permission.
Again - I'll refer you to the Q&A compiled by the gun rights attorney from your state:
Q:"Can I carry my loaded and concealed firearm (assuming I have a license to do so) in a New Hampshire business (public place) such as Costco that has a NO FIREARMS sign posted?"
A:"NO, private property rights prevail."
Attorney Dean does state that the means of notice are not statutorily defined, and there is some degree of ambiguity as to what constitutes due notice, but her answer to the question is a direct, unequivocal, 'NO'.
It really doesn't get more simple than that. Note also that the bold is Attorney Dean's, not mine. Parse all you want. Either Attorney Dean is wrong or you are. You can't both be right, despite your protests to the contrary. Now you get with her, and when you're ready to tell me which of you two is right, and which of you two is wrong, let me know.
No I'm not ignoring it, what I'm saying is that those conditions are voluntarily obeyed. At no time does the rights of the property owner allow for anything other than willful compliance of said arbitrary rules, and barring a theft of goods and/or services, the property owners only legal recourse to someone not complying with said rules, is to ask the person to leave. If, and only if, the person refuses to leave does it become criminal.
Either Attorney Dean is wrong or you are. You can't both be right, despite your protests to the contrary. Now you get with her, and when you're ready to tell me which of you two is right, and which of you two is wrong, let me know.
This is called the "false dilemma" logical fallacy and I've already answered it previously. There is a difference between the words "prevail" and "supersede".
Regardless it's a fact that a person can walk into any establishment in NH, excluding USSC recognized "sensitive areas" such as courts and Post offices, regardless of signs being posted, out of ignorance or intentionally, and the only thing the business/owner could do, if they realized one were armed, would be to ask said person to leave.
Why one would put up such signs is a better question.