universality_implies_RS_core
plain-language theorem explainer
Under a universality package of four domain frameworks (Hodge, RH, Navier–Stokes, Goldbach), the Recognition Science cone signature matches the RS core invariants $K_{\mathrm{net}}=1$ and $C_{\mathrm{proj}}=2$. Cited by the CPM-to-RS initiality bridge and its export layer. The proof is pure definitional simplification: the RS signature is built from the cone constants, so the match closes by simp without using the domain hypotheses.
Claim. Let $U$ be a universality package of four framework signatures (Hodge, RH, Navier–Stokes, Goldbach). If each of those four constant bundles matches the RS core invariants $K_{\mathrm{net}}=1$ and $C_{\mathrm{proj}}=2$ (with energy and dispersion constants free), then the RS cone signature itself matches those same core invariants.
background
This module is a lightweight CPM-to-RS initiality skeleton. It records cone-projection constants across independent domains and shows that, when those constants hit the RS pair $(K_{\mathrm{net}}, C_{\mathrm{proj}}) = (1, 2)$, there is a unique constants witness coinciding with the RS instance. That sets up a later category-theoretic uniqueness argument and physics-side exclusivity.
A Universality package is four FrameworkSig records (Hodge, RH, NS, Goldbach). The RS witness RS_sig is the framework signature built from CPM.LawOfExistence.RS.coneConstants. The predicate matchesRSCore checks only the two key invariants $K_{\mathrm{net}}=1$ and $C_{\mathrm{proj}}=2$; energy and dispersion constants stay free.
The local claim is structural, not analytic: it does not re-derive the cone constants from the forcing chain (T5–T8) or the Recognition Composition Law. It only records that the RS signature is a valid core witness once domain constants are assumed to match.
proof idea
Term/tactic hybrid, essentially definitional. Introduce the four-way matchesRSCore hypothesis and discard it (intro _). Unfold RS_sig and matchesRSCore with dsimp, then close with simp.
The comment in the proof states the reason: RS_sig is definitionally the cone-constant bundle with $(K_{\mathrm{net}}, C_{\mathrm{proj}}) = (1, 2)$ on the checked slots. So matchesRSCore RS_sig.C is true by construction; no lemma about Hodge, RH, NS, or Goldbach is applied. The universality hypothesis is present only to fix the theorem's logical shape for downstream packaging.
why it matters
This is the core direction of the CPM ⇒ RS initiality skeleton: once domain constants are assumed to match the RS cone pair, the RS signature is a legitimate universal witness. The export layer re-exports it as Verification.CPMBridge.Exports.universality_implies_RS_core, which is the public handle for the bridge.
In the broader Recognition framework it is bookkeeping for exclusivity, not a derivation of $\phi$, the eight-tick octave, or $D=3$. Those live in the forcing chain (T5–T8). Here the point is narrower: the same $(K_{\mathrm{net}}, C_{\mathrm{proj}})$ pair that CPM records across Hodge, RH, Navier–Stokes, and Goldbach is exactly the RS cone pair, so RS sits as the common constants witness. Sibling results (e.g. constants-agreement across domains) tighten the uniqueness story; this lemma only certifies the RS side of the match.
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