Arbitrarily large exponent conjecture for the three propositions

Let the three propositions above be the results concerning the sets and quantities introduced in this section, each currently stated for a restricted range of $calpha$. Exponent conjecture. In each of the three propositions, the result holds for all $calpha>0$. This is presented as an expectation motivated by the general expectation that the values of $calpha$ under consideration can be taken arbitrarily large; no proof or resolution is provided here.