Are all points on the Long Run Average Cost (LRAC) curve productively efficient? The definition of productive efficiency is that any given output is produced for the lowest possible cost.

In the short run, only the minimum point on the SRAC curve is productively efficient - this makes sense because at any other output, the firm could move to a different SRAC curve that would give a lower cost for that output.

However, the LRAC is, by definition, the envelope of all SRAC curve - it shows the cheapest way to produce any given output.

Agree / disagree?

If true, why is it that so many economics sources claim that "productive efficiency is achieved when the firm produces at the lowest point of its AC curve"? Is it simply that firm decisions are perpetually taken in the short run?
The statement
"In the short run, only the minimum point on the SRAC curve is productively efficient"
is false.

Suppose producing 1 unit of a good costs 5 dollars. 2 unit costs 8, x>2 units costs 5x. Clearly AC is minimal if you produce 2 units. 
But perhaps I do not want two units of the good, only one. Why spend the extra 3 dollars on the second good? Producing only 1 unit is also efficient in this case.

Note that this does not contradict your definition
"The definition of productive efficiency is that any given output is produced for the lowest possible cost."
The given output is 1 unit, and it is produced for 5 dollars, not 7 or 8, so there is no waste. Your definition does not say anything about per unit costs.

My example is also consistent with the definition of productive efficiency that I know.


  Productive efficiency is a situation in which the economy could not produce any more of one good without sacrificing production of another good. 


Sidenote:
 Perhaps you are confusing productive efficiency and minimum efficient scale of production?