Is Hamlet going crazy?

There is no good reason to think that Hamlet is actually going crazy. After his meeting with the Ghost, he tells Horatio, Marcellus, and Bernardo that he plans to act mentally unbalanced and gets them to swear to keep his intended behavior, as well as his meeting with the Ghost, strictly secret. 

But come!
Here, as before, never, so help you mercy,
How strange or odd soe'er I bear myself—
As I perchance hereafter shall think meet
To put an antic disposition on—
That you, at such times seeing me, never shall,
With arms encumber'd thus, or this head-shake,
Or by pronouncing of some doubtful phrase,
As “Well, well, we know,” or “We could, an if we would,”
Or “If we list to speak” or “There be, an if they might,”
Or such ambiguous giving out, to note
That you know aught of me; this is not to do,
So grace and mercy at your most need help you,
Swear.                                                  I.5

Then in Act III, Scene 4, when he is having his violent confrontation with his mother, he tells her:

It is not madness
That I have utt'red. Bring me to the test,
And I the matter will reword; which madness
Would gambol from. 

And a little later when she asks him what he wants her to do, he says:

Not this, by no means, that I bid you do:
Let the bloat King tempt you again to bed;
Pinch wanton on your cheek; call you his mouse;
And let him, for a pair of reechy kisses,
Or paddling in your neck with his damn'd fingers,
Make you to ravel all this matter out,
That I essentially am not in madness,
But mad in craft.

Hamlet ought to know whether he is mad or not. Where is there a clear instance in which he definitely seems mad and not faking? When he runs away from Rosencrantz and Guildenstern pretending that they are playing a game like Hide and Seek, he is obviously only acting mad. The same is true when he is talking to Claudius in Act IV, Scene 3. The fact that he killed Polonius is no proof of madness. He thought he had walked into a trap and that he was in imminent danger of being captured by the palace guards and thrown into a dungeon. At that time he suspected that his mother had been involved in the death of his father and was now scheming with Claudius against her own son. He has his mother crying for help in front of him and someone in hiding, possibly Claudius himself, crying for help behind him. His mother believes he is mad and that he intends to kill her, but he doesn't understand that she fears for her life when he grabs her wrist and says:

Come, come, and sit you down. You shall not budge.
You go not till I set you up a glass
Where you may see the inmost part of you.

She takes his metaphor literally. She thinks he means to cut her open with his sword and make her look at her own intestines as she bleeds to death. His pretended madness has backfired on him. She calls for the guards. Polonius calls for the guards. Hamlet thinks he has been set up--that his mother has summoned him to her chambers in order to have him apprehended by Claudius's guards, perhaps even executed on the charge that he tried to kill her. 

Hamlet insists that he is completely sane, and there is no clear-cut evidence that he is not. His behavior at the play-within-a-play seems somewhat crazy, but he has told Horatio, Marcellus, and Bernardo that he plans to act crazy in the future. He acts crazy with Claudius, Polonius, Rosencrantz, and Guildenstern, but all of it can be interpreted as pure pretense. If Hamlet is truly mad, then it seems strange that he could also be acting mad. How would we know when he might be only acting sane?