pi_mem_T

The module constructs a single countable subfield of the reals that is closed under the operations used to build Recognition Science constants. A finite seed set (the generators) is closed iteratively under field operations together with exp and log, yielding a directed chain of subfields $S_n$. The object $T$ is the supremum $\bigsqcup_n S_n$, documented as the exp/log-closed RS field.

Membership in $T$ is characterized by the upstream equivalence: a real $x$ lies in $T$ if and only if $x \in S_n$ for some finite stage $n$. The base stage $S_0$ is the subfield generated by the seeds, so every inserted generator is already in $T$ at stage zero. The ambient continuum is only the host in which the standard exp and log are defined; the workspace for RS constants is $T$ itself.

One-line wrapper that applies mem_T_iff.mpr. It exhibits stage $n = 0$ and invokes Subfield.subset_closure on the fact that $\pi$ is among the inserted generators of the seed set, hence $\pi \in S_0$ and therefore $\pi \in T$.

Seed membership of $\pi$ is required for the headline claim that cost, operations, and constants share one countable carrier. Downstream, rs_operations_below_continuum packages $\pi \in T$ into existence of a countable exp/log-closed field containing the seeds and derived constants, removing the continuum from the framework's workspace. alphaInv_mem_T uses $\pi \in T$ (with $\varphi \in T$) to show $\alpha^{-1} = 44\cdot\pi\cdot\exp(-w_8\cdot\ln\varphi/(44\cdot\pi))$ is built entirely inside $T$, landing in the RS $\alpha^{-1}$ band. jcost_pi_mem_T lifts the fact to $J(\pi)\in T$, and cost_and_constants_share_one_countable_field unifies J-cost closure with constant closure on the same $T$.