You’re correct and was about to say the same model matersy… I believe GPT4.1-mini and maybe nano support it as well.
In fact I am having HUGE success with both 4.1-mini and 4.1-nano (and price/perf better than GPT4.0)
Op. - don’t run a full reasoning model like o.1, o.3, o.3 pro etc. unless you REALLY like burning cash. I’ve not found anything that o.whatever can do as far as HA is concerned where you need it. Mini or nano are your best bets.
Edit 2: your llm will search for answers if it does not have enough information to satisfy its request. Because the web is now a tool guess what… If you don’t want that you need to EXPLICITLY describe how you want it to act re: web search.
I have a line in my ‘directives’ part of my prompt (thou shalt nots) that describes the expectation for finding data and how to handle fallback to web search. It does it pretty reliably. If you don’t tell it and it has ambiguity… Off it searches like a happy little llm.