Reflection algebras and conservativity spectra of theories

Turing introduced progressions of theories obtained by iterating the process of extension of a theory by its consistency assertion. Generalized Turing progressions can be used to characterize the theorems of a given arithmetical theory of quantifier complexity level $\Pi^0_n$, for any specific $n$. Such characterizations reveal a lot of information about a theory, in particular, yield consistency proofs, bounds on provable transfinite induction and provably recursive functions.
The conservativity spectrum of an arithmetical theory is a sequence of ordinals characterizing its theorems of quantifier complexity levels $\Pi_1$, $\Pi_2$, etc. by iterated reflection principles. We describe the class of all such sequences and show that it bears a natural structure of an algebraic model of a strictly positive modal logic - reflection calculus with conservativity modalities.