What does dict == null mean vs dict is null, when dict is a dictionary?

Is this table accurate?

 dict == null    dict is null    # elements   memory allocated
      true           true            0             no
      true           false           0             yes
      false          false           1+            yes
      false          true     (Is this a possibility, and if so, how to create it?)

source/app.d

import std.stdio;

void main()
{
    // value[key]
    int[string] dayNumbers =
         //   key     : value
        [
            "Monday": 0, "Tuesday": 1, "Wednesday": 2, "Thursday": 3, "Friday": 4,
            "Saturday": 5, "Sunday": 6
        ];

    dayNumbers.remove("Tuesday");
    // writeln(dayNumbers["Tuesday"]);		// ← run-time ERROR

    dayNumbers.clear;                       // Remove all elements

    // Best to check length to see if associative array is empty, rather than comparing to null.
    // Checking to see if an associative array is null only tells you if it has been allocated memory.

    // dayNumbers has no elements
    writeln("dayNumbers == null? ", dayNumbers == null);    // true

    // dayNumbers has allocated memory, although no elements
    writeln("dayNumbers is null? ", dayNumbers is null);    // false

    writeln("dayNumbers.length: ", dayNumbers.length); // 0
    writeln;

    int[string] aNonInitializedDictionary;
    writeln("aNonInitializedDictionary is null? ", aNonInitializedDictionary is null); // true
    writeln("aNonInitializedDictionary == null? ", aNonInitializedDictionary == null); // true
    writeln("aNonInitializedDictionary.length: ", aNonInitializedDictionary.length); // 0
    writeln;

    int[string] aPopulatedDictionary;
    aPopulatedDictionary["Alpha"] = 1;
    writeln("aPopulatedDictionary is null? ", aPopulatedDictionary is null); // false
    writeln("aPopulatedDictionary == null? ", aPopulatedDictionary == null); // false
    writeln("aPopulatedDictionary.length: ", aPopulatedDictionary.length); // 0
}

Console output

dayNumbers == null? true
dayNumbers is null? false
dayNumbers.length: 0

aNonInitializedDictionary is null? true
aNonInitializedDictionary == null? true
aNonInitializedDictionary.length: 0

aPopulatedDictionary is null? false
aPopulatedDictionary == null? false
aPopulatedDictionary.length: 1