A theorem stating that the interaction picture does not exist in an interacting, relativistic quantum field theory.

(physics) A no-go theorem stating that, during measurement of an entangled quantum state, it is not possible for one observer, making a measurement of a subsystem of the total state, to communicate information to another observer.

(geometry) A result in plane geometry which cannot be derived from Euclid's postulates. It states that, given points a, b, c, and d on a line, if it is known that the points are ordered as (a, b, c) and (b, c, d), then it is also true that (a, b, d).

(geometry) Pythagorean theorem

(mathematics) A fundamental result in Diophantine approximation to algebraic numbers, stating that these numbers cannot have many rational number approximations that are 'very good' (variously defined through history).

(mathematics) One of the earlier theorems leading to Roth's theorem (a Diophantine approximation to algebraic numbers), establishing an exponent d/2+1+ε.

(computing theory) A theorem stating that it is impossible for a distributed data store to simultaneously provide more than two of three guarantees: consistency (every read receives the most recent write or an error), availability (every request receives a non-error response, without the guarantee that it contains the most recent write), and partition tolerance (the system continues to operate despite an arbitrary number of messages being dropped or delayed between nodes).

(logic) The following theorem of Boolean algebra: XY+X'Z+YZ=XY+X'Z where YZ, the algebraically redundant term, is called the "consensus term", or its dual form (X+Y)(X'+Z)(Y+Z)=(X+Y)(X'+Z), in which case Y+Z is the consensus term. (Note: X+Y,X'+Z⊢Y+Z is an example of the resolution inference rule (replacing the + with ∨ and the prime with prefix ¬ might make this more evident).)

(logic) A procedure for "discharging" assumptions from an inference, causing them to become antecedents of the conclusion; or vice versa. Symbolically, the conversion of an inference of the form P,A⊢C to an inference of the form P⊢A→C or vice versa, where ⊢ is the turnstile symbol. The validity of the procedure is a metatheorem of the given logical theory.

(mathematics) A theorem stating that every Goodstein sequence eventually terminates at zero.