No, there are plenty of things the player knows. I didn't give many details about them beause we were exploring checks and uncertainty. I can point out that the players were able to identify the statue as a hobgoblin without the need for a check.
Thanks for clarifying.
Here's an example from one of Morrus' adventures, The Business of Emotion, if it helps you visualize what I mean.
A cave outside town by a meandering river (D1). Bear pawprints in the mud. DC 15 Intelligence (Investigation) reveals it dragged a large animal, a horse whose corpse can be found in the bushes. DC 18 Intelligence (Investigation / Nature) finds a single large own feather.
Just swap in the hobgoblin statue and appropriate information & skills from my earlier post.
What this tells me is that the author believes that Intelligence (Investigation) is used to resolve tasks with a goal of finding hidden objects rather than Wisdom (Perception), and that for reasons that are not clear in what you posted above, there is an uncertain outcome to the task and there's a meaningful consequence for failure of some kind for that task. Also that Intelligence (Investigation) is used to resolve tasks with a goal of following tracks rather than Wisdom (Survival).
I think I prefer the hobgoblin example, if we have to use an example at all.
I run and play in multiple groups, and have run at conventions. It is a perfectly reasonable question for a player to ask. Saying that one at your table wouldn't ask it seems like you are playing with a narrow subset of player behavior that lets you avoid some common situations.
Can we move forward without dodging the question. The player has described their action: "I examine the statue visually without approaching. Do I know anything about it or see anything besides the scepter that catches my eye?"
It's a common sort of question in my experience with other people's games, but again, it doesn't come up in my games because it would be unnecessary. I already described what caught the characters' eyes when describing the environment. Umbran is correct in saying that the framing you offered is insufficient in my games.
Further, "do I know anything?" isn't an action I can adjudicate. If a player asked such a question, I'd ask for him or her to do as I described in my last post - tell me what sort of lore he or she is seeking to recall (and for what purpose) and why he or she might know something about the matter. Then I can decide if there's an ability check. Maybe there is, maybe there isn't.
I run and play with a lot of different groups, too, sometimes people I don't know. I have to sometimes teach players to get into the habit of describing what they want to do rather than ask questions. Questions can not only make it more difficult or impossible to properly adjudicate, they are also a way for a player to step outside the game to avoid consequences. Actions can have consequences. Having a Q&A with the DM outside the context of the setting does not. It's a clever little trick.
The player has no idea what information she seeks nor how that information would be used. That's not a valid requirement to have for a knowledge skill.
"I try to recall if this hobgoblin is a significant figure from history, perhaps a builder of this place, given that I possess great historical knowledge as an archaeologist."
The information is some set up by the adventure designer to aid with overcoming challenges later in the adventure.
Look at Morrus' adventure. The player does not know the horse corpse is there, nor that there is a large owl feather nearby. They discover that through checks.
They discover that through actions which have uncertain outcomes and meaningful consequences for failure that are not well-established in what you posted. We know they must exist because there are apparently checks. We just don't know why.
Split brain isn't something I'm particularly worried about with you, more if others jumped in. It's like a DM saying "Make and INT roll" and the player inquiring "would arcana or history proficiency help?" and the DM tells them to choose one and then only gives a response for that one roll, instead of realizing that a character knows arcana and history at the same time and they might have complementary bits of information. So since the DM only called for one INT roll, the character "forgets" everything about the skill not rolled.
I think I split my brain trying to figure out what this means.