I am grateful to the AD True Believers for providing insights into how curves and equations capture, not reality, but the imagination of some economists: 
 
I think, therefore I am, 
I derive it, therefore it is.  
 
(An example: Phillips drew a curve and presented an equation, therefore, he must have been asserting that the British economy could inflate 10 times faster than its trading partners and still have a fixed exchange rate and very low unemployment  - no need to read what Phillips wrote about inflation or inflationary expectations).  
 
With respect to the IS-LM derivation, the legitimate section of an AD curve represents points associated with observed variations in M/P (or some other 'reality' constraint).  The (pointless) points outside this zone are illegitimate or (to use Joan Robinson's word) Bastard AD points. 
 
Such musing are legitimate exercises as napkin etchings during conference dinners: The Leisure of the Economic Theory Class.  The question: "what would happen if a government instructed that all prices were to be cut in half, while the money supply was left unchanged" is interesting; the answer "the only thing that would happen is that we would obtain a bigger AD curve" is BAD training for students.
 
Robert Leeson