Revealing mistake: In an early shot shown from the neighbor's house, it is established that the Spencers' house is quite a ways uphill. Some of the later shots where Claire is spying on her neighbors and looking at their home, the angle of view does not appear to be correct. The neighbors' house looks like it's a lot closer and more level with the Spencers' home.
Revealing mistake: After Claire and Norman drop their daughter off at school, that night, Norman is in bed with his laptop, typing, but not once do his fingers touch the space bar.
Chosen answer: Spencer's theorem was Norman's father's theorem. Norman wants to be greater than his father was, so to be accidentally complemented on something that his father did would be bad enough. To say that the guy knows that his father is dead is to imply that the guy probably insulted him intentionally.
Myridon