jcost_pi_mem_T

In Recognition Science the cost of a positive ratio is the function $J(x) = (x + x^{-1})/2 - 1$ (also written $\cosh(\log x) - 1$). It is the unique continuous solution forced by the Recognition Composition Law and the T5 uniqueness step of the forcing chain.

The ambient object here is the countable subfield $T := \bigsqcup_n S_n \subseteq \mathbb{R}$, the directed union that is closed under the field operations and under the exp/log generators of the Primitive Recognition Calculus. Upstream, $\pi \in T$ is already recorded by adjoining $\pi$ at the base of that chain.

A companion lemma shows that $J$ maps $T$ into itself: it is a field expression built from $+$, inversion, division by $2$, and subtraction of $1$, and $T$ is a subfield (inversion is total, with $0^{-1}=0$). No positivity hypothesis is required.

One-line wrapper. Instantiate the general closure theorem (if $x \in T$ then $J(x) \in T$) at $x = \pi$, using the upstream membership $\pi \in T$. No extra algebraic work is done at this site.

The module's standing claim is that the cost of every named constant is itself a $T$-element. This declaration discharges that claim for $\pi$, in parallel with the sibling facts for $\varphi$ and $\alpha^{-1}$. Together they support the broader statement that cost dynamics and the named constants live in one countable field, so the Primitive Recognition Calculus never leaves a countable, explicitly generated subfield of $\mathbb{R}$.

Framework-wise this sits under the T5 J-uniqueness landmark: once $J$ is forced, every concrete evaluation of $J$ on a generator of $T$ must remain inside $T$. No downstream consumers are wired yet; the immediate parent is the module-level countable-field package rather than a later physics theorem.