"The main difference to note in the box formulation is that there is no such thing as an environment."
Doesn't step 1 need to use environment(s)? How else would it turn text into a structure that contains references to previously defined values?
"The 'hyperstatic scope' that Pauan keeps mentioning is another way of saying, 'there is no scope', only variables."
The hyper-static global environment (http://c2.com/cgi/wiki?HyperStaticGlobalEnvironment) is essentially a chain of local scopes, each one starting at a variable declaration and continuing for the rest of the commands in the program (or just the file). I think the statement "there is no scope" does a better job of describing languages like Arc, where all global variable references using the same name refer to the same variable.
"Flexible scoping becomes possible by specifically referencing the variable in the environment instead of using the default boxed values. They would still be boxes, just referenced by symbol in the environment table."
I don't understand. If we're generating an s-expression and we insert a symbol, that's because we want that symbol to be looked up in the evaluation environment. If we insert a box, that's because we want to look up the box's element during evaluation. Are you suggesting a third thing we could insert here?
Perhaps if the goal is to make this as dynamic as possible, the inserted value should be an object that takes the evaluation environment as a parameter, so that it can do either of the other behaviors as special cases. I did something as generalized as this during Penknife's compilation phase (involving the compilation environment), and this kind of environment-passing is also used by Kernel-like fexprs (vau-calculus) and Christiansen grammars.
"Now what I'm wondering is whether there is any useful equivalence to be found between a box and an environment."
I would say yes, but I don't think this is going to be as profound as you're expecting, and it depends on what we mean by this terminology.
I call something an environment when it's commonly used with operations that look vaguely like this:
String -> VariableName
(Environment, VariableName) -> Value
(Environment, Ast) -> Program
Meanwhile, I call something a box when it's primarily used with an operation that looks vaguely like this:
Box -> Value
This might look meaningless, but it provides a clean slate so we can isolate some impurity in the notion of "operation" itself. When a mutable box is unboxed, it may return different values at different times, depending on the most recent value assigned to that box. Other kinds of boxes include Racket parameters, Racket continuation marks, Racket promises, JVM ThreadLocals, and JVM WeakReferences, each with its own impure interface.
When each entry of an environment needs to have its own self-contained impure behavior, I typically model the environment as a table which maps variable names to individual boxes. The table's get operation is pure (or at least impure in a simpler way), and the boxes' get operation is is impure.
You were wondering about equivalences, and I have two in mind: For any environment, (Environment, VariableName) is a box. For any box, we can consider that box to be a degenerate environment where the notion of "variable name" includes only a single name.
I think he's talking about run-time environments a la Kernel, Emacs Lisp, etc.
Nulan and Arc/Nu use a compile-time environment to replace symbols with boxes at compile-time. But that feels quite a bit different in practice from run-time environments (it's faster too).
"Are you suggesting a third thing we could insert here?"
Once again, I think he's referring to run-time environments. Basically, what he's saying is that you would use boxes at compile-time (like Nulan), but you would also have first-class environments with vau. The benefit of this system is that it's faster than a naive Kernel implementation ('cause of boxes), but you still have the full dynamicism of first-class run-time environments. I suspect there'll be all kinds of strange interactions and corner cases though.
Slightly off-topic, but... I would like to point out that if the run-time environments are immutable hash tables of boxes, you effectively create hyper-static scope, even if everything runs at run-time (no compile-time).
On the other hand, if you create the boxes at compile-time, then you can create hyper-static scope even if the hash table is mutable (the hash table in Nulan is mutable, for instance).