rsField_natCast
plain-language theorem explainer
Every natural number, viewed as a real, lies in the minimal Recognition Science field generated by the named RS constants. Anyone working with the prime-field content of that subfield, or with integer tick counts and the eight-tick octave, can cite this. The proof is a one-line appeal to the standard fact that every subfield of ℝ contains the image of ℕ.
Claim. For every natural number $n$, the corresponding real $n \in \mathbb{R}$ belongs to the minimal RS field $K_{\mathrm{RS}} \subseteq \mathbb{R}$, i.e. the subfield generated by the named RS constants. Equivalently, $K_{\mathrm{RS}}$ contains the prime field $\mathbb{Q}$ (and in particular all of $\mathbb{N}$).
background
In the Primitive Recognition Calculus layer, the minimal RS field is the subfield of $\mathbb{R}$ obtained by closing the finite set of named RS constants under field operations. By construction it automatically contains the prime field $\mathbb{Q}$, so adjoining those constants is the same as working over $\mathbb{Q}$ with the constant values adjoined.
The named generators include the golden-ratio fixed point $\varphi$, $\pi$, $e$, and the fundamental time quantum (one tick, $\tau_0 = 1$ in RS-native units). The eight-tick octave period $2^3 = 8$ is the fundamental evolution period in the forcing chain (T7), so integer and natural values appear constantly as tick counts and as the octave length itself.
This lemma records the elementary containment $\mathbb{N} \hookrightarrow K_{\mathrm{RS}}$ that every subfield of $\mathbb{R}$ enjoys via the unique unital ring homomorphism $\mathbb{Z} \to K$.
proof idea
One-line term proof: apply the Mathlib lemma that every subfield of a characteristic-zero field (here a Subfield ℝ) contains the image of every natural under the canonical cast ℕ → ℝ. No RS-specific algebra is needed beyond the fact that rsField is already a Subfield ℝ.
why it matters
The minimal RS field is the ambient scalar domain for all subsequent PRC constructions: membership of $\varphi$, $\pi$, $e$, and the finite/countable character of the constant set and of the field itself. Natural and integer scalars appear as tick counts, rung indices on the $\varphi$-ladder, and the eight-tick octave $8 = 2^3$ forced at T7 of the Unified Forcing Chain.
Although no downstream theorem currently depends on this declaration by name, it underwrites the claim in the module that the field "automatically contains $\mathbb{Q}$ (the prime field)". Without $\mathbb{N} \subset K_{\mathrm{RS}}$, statements such as "the eight-tick output is a field element" and any arithmetic with integer rung offsets would sit outside the declared scalar domain. It is the prime-field half of the package that later adjoins the transcendental and algebraic generators.
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