What exactly constitutes a criminal case?

As used most commonly, a "criminal case" means that a criminal charge has been filed by a prosecutor. A prosecutor can file a criminal charge without the defendant having been previously arrested, and a defendant can be arrested without a criminal charge resulting. Of course, a crime can be committed and nobody arrested or charged--there is no "criminal case," just an "unsolved crime." The usual term for what happens after a report is made to the police is a "criminal investigation."

But none of these are legal terms of art. Anybody can use the word "criminal case" to mean anything--I have provided only the most usual use of the term.

a criminal case belongs to an act(the criminal act itself -not a person) . When police receive a report of a crime 1) they investigate to see if a crime has indeed been committed. 2) If they believe it has been then they open a "case file" 3) They investigate further and if they find sufficient evidence to believe a certain person (or persons) committed the crime then those persons are charged. If they find the allegation was frivolous or "ridiculous" as you say then no, there never would have been a case file opened,however if they investigated and found a crime had been committed BUT because of a lack of evidence no one was charged then the file would remain "open".

1) complaint filed

2) police determine if it is criminal or civil matter.

3) if criminal they investigate

4) it they find evidence of crime they turn investigation file over to District (city) Attorney

5) DA decides to prosecute... or not.

6) if s)he decides to prosecute, a criminal case file is opened.

There might have been a criminal investigation. It is not exactly a case until someone is charged or a crime has been committed.

reporting something means the police will look into it, when its a criminal case it means it is actually being handled by all the departments: FBI, CIA, etc