has_protocol_display
plain-language theorem explainer
Every finite F_RS carrier term evaluates to a real that is the displayed value of some Delta-real protocol. Anyone packaging the Phase 3 carrier cites this to wire finite syntax into the ℝδ interface. The argument is a one-line application of protocol value-surjectivity to the term evaluation.
Claim. For every finite $F_{RS}$ expression $e$ (built from rationals and the inventory $\varphi,\pi,e,\alpha^{-1}$ under field operations), there exists a Delta-real protocol $x$ such that the real value of $x$ equals the evaluation of $e$.
background
The module builds an explicit finite-description carrier for Recognition Science: terms of type FRSExpr are finite trees with leaves the rationals and the fixed inventory $\varphi$, $\pi$, $e$, $\alpha^{-1}$, closed under negation, addition, multiplication, and inversion. Evaluation sends each term to a real; the Phase 3 claim is that these values live in a countable proper subfield of $\mathbb{R}$ and still render in the certified analytic interface.
Delta-real protocols are the display layer of that interface: each protocol carries a real value. The present lemma only needs the surjectivity half of that display map: every real arising as a carrier evaluation is already the value of some protocol. Upstream certified-analytic registries make the same point for richer expression languages (values are ranges of finite trees, hence countable), which is the countability fact that blocks continuum smuggling into the computational carrier.
proof idea
One-line term proof. Apply the surjectivity lemma on Delta-real protocol values to the real eval ex. No case analysis on the expression tree is required: evaluation has already produced a real, and every real is the value of some protocol.
why it matters
This is one conjunct of the Phase 3 headline frs_carrier: soundness into the countable RS field, correct inventory constants, countability and properness of the value set, and protocol display into the $\mathbb{R}_\delta$ interface. Without display, the finite syntax would compute reals that the certified analytic stack could not name.
In the broader framework the point is architectural. The carrier the theory actually computes on is finite generation over a fixed inventory (rationals plus $\varphi,\pi,e,\alpha^{-1}$), not the uncountable continuum. Protocol display is the bridge that lets those finitely described values participate in the Delta-real calculus used elsewhere in Primitive Recognition Calculus. Downstream packaging of frs_carrier simply conjoins this existence statement with the soundness and inventory lemmas.
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