Re: curiously recurring template pattern and default constructor

That is not CRTP. The inheritance is upside down here.
In CRTP the template is reusable base class:

I do not know of one, such a need too rarely arises. You can make a
metaprogram that checks if a type is default-constructible:

  template<typename T>
  class is_default_constructible
  {
      // make two different types
      typedef char works;
      struct doesnt_work {char a[2];};

      // try to substitute default construction
      template<typename U>
      static decltype(U(), works()) test(int);

      // fall here if failed to substitute

      template<typename>
      static doesnt_work test(...);

  public:

      static const bool value = sizeof(test<T>(0)) == sizeof(works);
  };

Then you can use it to make partial specializations of whole class
depending if the base is default-constructible or not.