Where are you getting the idea of a "hidden menu" from? That isn't part of how I've specified the spectrum.
And yes, implicit in the idea of being able to make an open-ended choice is that you may be doing something the GM hasn't prepped yet. (Depending on GM style, the GM might wing it or might pause to be able to prep first.)
How open-ended a "good" sandbox is will of course be a question of taste. But a more-frequently open-ended sandbox is (as I've defined the spectrum anyway) more sandboxy than a less-frequently open-ended sandbox. If someone wants to suggest a different measure (or a more precise measure) of sandboxiness to define the spectrum, cool. But if we're getting to the point of discussing which definition for the spectrum is most useful, we're well past the point of agreeing that the spectrum exists.
Or you could just let them go anywhere they want in the setting and (try to) do anything they want. Then you're definitely "letting them do anything". If instead the expectation is that the players pick from a list of pre-defined elements to "engage with", the decisions with that expectation would tend to pull a campaign towards the non-sandbox end of the spectrum, unless you're using a much broader definition of "engage with" than you appear to be.
I'm going to address all of this at once, because it's interconnected. You seem to lump "make it up on the spot" with "already has detailed prep" with "has to stop the game to go create detailed prep" together. These are either the same thing or aren't the same thing, and it's going to depend entirely on how you define how and what the GM's allowed prep/improv space is. Because, according to this, the players can declare their going over that hill to the north, and the GM can respond with their notes -- nothing there. Then the players keep going north, but the GM has notes, a detailed setting even, and everywhere nothing is pretty much nothing. So the GM indulges the players by letting them just keep going north and describing nothing (I mean, except probably some terrain features). Alternatively, the GM can make up this same thing on the spot without notes. According to your construct, both of these are equal. This, unfortunately, means that a GM can, on the spot, decide to effectively negate actions by just making up nothing, and this is as valid a strong sandbox as a detailed, prepped setting to explore!
Now, I'm sure you don't intend this, but this is the issue with the vagueness of your metrics -- they don't actually define anything very well and encompass quite a lot of play space (before we even get to alternative fiction creation, we're still in GM-sole-creator-at-GM's-desire territory). I mean, the example above is unlikely to actually occur (although there are stories of "sandboxes" where the players can do whatever they want, it's just not going to get anything from the GM until they get back to the prepped bits) but the construction you've created allows it.
Also, I'm not agreeing a spectrum exists and we're haggling price, I'm looking at your proposed spectrum and seeing if it has any useful description to it. So far, you have to bring in so many additional yet unspoken assumptions about what play is that I can't use it at all to categorize games in a way that actually tells me something useful.
I'm saying the spectrum is still useful in the middle despite the messiness. I definitely wasn't agreeing with you that the spectrum only works at the edges. But I do agree with you that most campaigns will be in the middle--a campaign at either extreme would be quite unusual. Indeed, the fact that most campaigns are somewhere in the middle is one of the reasons I think it's important to look at sandboxiness on a spectrum in the first place.
Okay, can you tell me what I should expect to be different from a 60% sandbox game and a 40% sandbox game? I can't figure out what I should expect different from these two middle positions. It seems being above and below the mid point, and with a 20% spread, that there should be differences that we could decidedly pick out. What do you think defines these two points on your proposed spectrum? What should I be looking for in these games if they were offered to me?
