You are conflating the quantity A, which is a label for the (unknown in the original formulation) amount in the chosen envelope, with the "smaller amount" - labelled x in the wikipedia article.
A=60 is the amount in our chosen envelope - we are given this information in the variant in this thread. What we still don't know is if that is the larger amount (and thus the other envelope contains 30, and therefore x=30), or the smaller amount (and the other envelope contains 120, and therefore x=60).
The error is (in step 7) to calculate the arithmetic expectation of those absolute values, because they do not exist "together". The correct arithmetic mean can be obtained by considering the different conditions in which those values do exist, as described in [0]. However, the ratios do exist "together" - the other envelope contains either double or half of 60 - so we could instead calculate the geometric mean, of either the ratios or the corresponding absolute values, and obtain the correct result:
(2 * 0.5) ** 0.5 = 1
(120 * 30) ** 0.5 = 60
[0]
https://en.wikipedia.org/wiki/Two_envelopes_problem#Other_si...edit: changed “first envelope” to “chosen envelope” for clarity.
A=60 is the amount in our chosen envelope - we are given this information in the variant in this thread. What we still don't know is if that is the larger amount (and thus the other envelope contains 30, and therefore x=30), or the smaller amount (and the other envelope contains 120, and therefore x=60).
The error is (in step 7) to calculate the arithmetic expectation of those absolute values, because they do not exist "together". The correct arithmetic mean can be obtained by considering the different conditions in which those values do exist, as described in [0]. However, the ratios do exist "together" - the other envelope contains either double or half of 60 - so we could instead calculate the geometric mean, of either the ratios or the corresponding absolute values, and obtain the correct result:
[0] https://en.wikipedia.org/wiki/Two_envelopes_problem#Other_si...edit: changed “first envelope” to “chosen envelope” for clarity.