C's equality operator on converted pointers

How is this refuted? Does "pointer to the same object" refer to addresses, even if no objects are there

No, I don't think that would be a plausible reading. If you stipulate that the pointer value is not a pointer to an object (and if it is not a null pointer) then an equality comparison of that (pointer) value with itself does not satisfy the "only if" condition of 6.5.9/6, and therefore the comparison must evaluate to 0.

But not so fast. Who says that (struct A *) 1 is not a pointer to an object? Consider the Standard's definition of "object":

object
region of data storage in the execution environment, the contents of which can represent values

(C 2011, 3.15/1)

Note that the definition is not inherently limited to objects that are allocated or declared by the program. To the best of my knowledge, the standard nowhere limits the scope of the term in that way. It does define means to allocate objects, but it does not specify that objects allocated in one of those ways are the only ones that exist. Thus, implementations are free to interpret that pointer value as a pointer to an object, in which case the equality comparison may evaluate to 1.

It also might still not evaluate to 1, as despite the two pointers (presumably) having bitwise-identical representations, they are not necessarily considered pointers to the same object.

(not like the compiler could know, anyway)?

Of course the compiler could and should know. It has to know in order to evaluate expressions such as you present. The most straightforward approach -- and, not coincidentally, the most common -- is to interpret every non-null pointer value that is not a trap representation as a pointer to an object.

Should we simply understand that it is implementation-defined since the previous results were already implementation-defined?

Being implementation-defined carries a requirement for conforming implementations to document their choice. The behavior you're asking about may follow from the implementation-defined behavior of converting an integer to a pointer, but it is not implementation-defined itself.

Where does the standard specify this?

It does not specify. In principle, conforming implementations may differ on this point. In practice, however, they're pretty consistent.

Constraint violation

An integer may be converted to any pointer type. Except as previously specified, the result is implementation-defined, might not be correctly aligned, might not point to an entity of the referenced type, and might be a trap representation. C17dr §6.3.2.3 5

With (struct A *) 1 code attempts the conversion. The result is implementation-defined, may lack alignment, ... might be a trap.

Next code tries to initialize a below.

struct A * a = (struct A *) 1;

Initialization constraints include:

No initializer shall attempt to provide a value for an object not contained within the entity being initialized. §6.7.9 2

It is not defined that (struct A *) 1 meets that constraint.