rsField_eight_tick

The minimal RS field is the subfield of $\mathbb{R}$ obtained by closing the finite set of named RS constants under field operations. It automatically contains $\mathbb{Q}$ as its prime field, so every natural number (via the canonical cast) is already present before any transcendental generators are adjoined.

In the forcing chain, T7 fixes the fundamental cadence of recognition as the eight-tick octave: period $2^3 = 8$. Parallel definitions of spatial dimension $D = 3$ (T8) appear upstream; the same subfield will later host that integer as well. The local module builds a countable scaffold that carries all chain outputs without ever needing the full continuum.

One-line term proof. Any subfield of $\mathbb{R}$ contains the image of $\mathbb{N}$ under the natural cast; apply that membership lemma at $n = 8$ and discharge the $\mathbb{R}$ coercion with exact_mod_cast.

This is the T7 rung of the countable-carrier claim. Downstream, rs_chain_all_rungs_in_field packages it with $\varphi$ (T6) and $D = 3$ (T8) to assert that every named chain output lives in the countable RS field: "the continuum is never the home of any rung." The same fact is conjoined in rs_scaffold_below_continuum, which sharpens Item 1: the $\varphi$-ladder, eight-tick, and dimension all sit inside a proper countable subset of $\mathbb{R}$. The shrunk $\delta$-program certificate then records scaffold-in-field as one of its seven proved headlines. Without eight-tick membership, the end-to-end countable forcing story fails at T7.