"The 'hyperstatic scope' that Pauan keeps mentioning is another way of saying, 'there is no scope', only variables."
Actually, there is still scope. After all, functions still create a new scope. It's more correct to say that, with hyper-static scope, every time you create a variable, it creates a new scope:
box foo = 1 # set foo to 1
def bar -> foo # a function that returns foo
box foo = 2 # set foo to 2
bar; # call the function
In Arc, the call to "bar" would return 2. In Nulan, it returns 1. The reason is because the function "bar" is still referring to the old variable "foo". The new variable "foo" shadowed the old variable, rather than replacing it like it would in Arc. That's what hyper-static scope means.
From the compiler's perspective, every time you call "box", it creates a new box. Thus, even though both variables are called "foo", they're separate boxes. In Arc, the two variables "foo" would be the same box.
"Thus, there is no real way to make variations on the scoping scheme using hyperstatic scope, because that's all a read/compile time feature. Dynamic scope is impossible without tracking environments, which are otherwise unnecessary."
If by "dynamic scope" you mean like Arc where globals are overwritten, then that's really easy to do. As an example of that, check out Arc/Nu, which also uses boxes:
In the Arc/Nu compiler, it literally only takes a single line of code to switch between Arc's dynamic scope and hyper-static scope.
If by "dynamic scope" you mean like Emacs Lisp, then... actually that should be really easy as well. You would just replace the same symbol with the same box, and then use box mutation at run-time. Of course, at that point I don't think there'd be any benefit over run-time environments... In any case, I like lexical scope, and boxes work well for lexical scope.
Also, although Nulan has hyper-static scope, it does have dynamic variables. The way that it works in Nulan is, you create a box like normal...
box foo = 1
...and then you can dynamically change that variable within a code block:
w/box! foo = 2
Within the "w/box!" block, the variable "foo" is 2, but outside, it is 1. You can do this with any variable in Nulan.
"Now what I'm wondering is whether there is any useful equivalence to be found between a box and an environment."
No. And that's a good thing. I've found one of the major benefits of boxes is that they're a single stand-alone entity: each variable corresponds to a single box. In the environment model, you have a hash table which maps symbols to values, so you have a single data structure representing many variables.
The reason I prefer each variable being represented by a separate box is that it makes namespaces really really easy to design and implement. For instance, in Arc/Nu you can selectively import only certain variables:
(w/include (foo bar qux)
This is really easy with boxes: you simply grab the "foo", "bar", and "qux" boxes and import them. But with environments, it's a lot harder, because the environment for the file "some-file" may contain all kinds of variables that you don't want to import.
You could take the values from the environment and copy them into the current environment, but now any changes made won't show up. With boxes, the changes show up. I don't think environments work well for namespace systems. But boxes work wonderfully well.
"It can replace cons cells and regular hash tables too :P"
I've thought about ways to replace hash tables with boxes, but I don't think it'd be especially helpful. Hash tables serve a different purpose from boxes, so it makes sense for a language to have both data structures. If you want a box, just use a box. If you want a hash table, just use a hash table.
Interestingly enough, both Nulan and Arc/Nu have a giant hash table at compile-time that maps symbols to boxes. This is basically the compile-time equivalent of the run-time environments that languages like Kernel have. Creating a new scope or changing the existing scope is as easy as changing this hash table, which makes it really easy to play around with different namespace systems.