jcost_iterate_mem_T

Recognition cost is the unique symmetric functional forced by the Recognition Composition Law: $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$). It measures the cost of a positive ratio relative to the identity ratio $1$.

The ambient module builds a single countable subfield $T\subseteq\mathbb{R}$ closed under the field operations needed by the primitive recognition calculus (exponentials/logs in the companion module, and cost). The sibling lemma jcost_mem_T already records that one application of $J$ sends $T$ into itself.

The present statement lifts that one-step fact to the full forward orbit: iterating cost never forces one out of the countable base field that will later host $\phi$, $\pi$, and $\alpha^{-1}$.

Induction on the iterate count $n$.

• Zero: $J^{\circ 0}(x)=x$, so the hypothesis $x\in T$ finishes the case (via simpa).
• Successor: rewrite $J^{\circ(k+1)}(x)=J(J^{\circ k}(x))$ by the standard iterate identity, then apply the one-step closure lemma (any $T$-element maps under $J$ into $T$) to the inductive hypothesis $J^{\circ k}(x)\in T$.

No analytic estimates appear; the argument is pure closure under a unary operation.

This is the dynamical half of the claim that cost and the named RS constants live in one countable field. Sibling results place $\pi$, $\phi$, and $\alpha^{-1}$ in $T$; together with single-step and iterated closure under $J$, the module can assert that the entire cost orbit of those constants stays countable.

In the forcing chain, $J$ is the unique cost fixed at T5. Keeping every iterate inside a countable subfield is bookkeeping needed before one treats numerical special values (golden ratio fixed point, fine-structure band) as field elements rather than transcendental accidents. No downstream consumers are wired yet; the natural parent is the module-level theorem that cost and constants share one countable field.