rsField_countable

In the Primitive Recognition Calculus minimal-field module, the RS constants are a named finite set of real values (including the fine-structure weight $w_8$, $\alpha^{-1}$, and the usual transcendental seeds). The RS field is defined as the subfield of $\mathbb{R}$ obtained by closing that set under the field operations; it automatically contains $\mathbb{Q}$ as the prime field.

A general lemma already records that the subfield closure of any finite subset of $\mathbb{R}$ is countable (itself a specialization of the countable-generators case). Finiteness of the RS constant set is proved by unfolding the definition and chaining singleton/insert finiteness. Countability of the generated field is then immediate.

One-line wrapper. Apply subfield_closure_countable_of_finite to the already-proved fact that the RS constant set is finite. No further algebraic work is required: the general lemma supplies countability of the subfield closure.

This is the countability half of the "RS physics below the continuum" claim. Downstream, rsField_proper uses it with non-countability of $\mathbb{R}$ to show the field is a proper subset; rs_physics_below_continuum and rs_scaffold_below_continuum package it with membership of $\varphi$, $\pi$, $e$, $\alpha^{-1}$, the full $\varphi$-ladder, and the chain integers 8 and 3.

On the bridge side, delta_cost_feeds_rs_chain and rs_chain_all_rungs_in_field weld the forcing chain (T6 $\varphi$, T7 eight-tick, T8 dimension 3) to a countable carrier: every named rung lands in this field, so the continuum is never the home of any chain output. Carrier values in the FRS layer inherit countability by monotonicity from the same fact.