kappa_from_phi_alone
plain-language theorem explainer
The RS Einstein coupling equals $8\varphi^5$, a pure algebraic function of the golden ratio. Anyone arguing that gravity in Recognition Science is number-theoretic rather than gauge-mediated cites this identity. The proof is a one-line term that reuses the closed-form equality already proved for the coupling definition.
Claim. The Recognition Science Einstein gravitational coupling equals $8\varphi^5$, where $\varphi$ is the golden ratio fixed by the self-similarity condition of the cost functional.
background
Module G-004 asks whether a graviton exists in Recognition Science. The RS answer is that gravity is emergent curvature of the ledger lattice, not a force carried by a spin-2 gauge boson; asking for a graviton is treated as a category error.
The Einstein coupling in RS-native units is defined as $\kappa_{\mathrm{rs}} := 8\varphi^5$. Upstream, ZeroParameterGravity records both that definition and the trivial closed-form theorem $\kappa_{\mathrm{rs}} = 8\varphi^5$ (proved by rfl). Here $\varphi$ is the unique self-similar fixed point forced by the J-cost uniqueness chain (T5–T6): $J(x)=(x+x^{-1})/2-1$, with $\varphi$ solving the fixed-point relation of the Recognition Composition Law.
The present statement simply re-exports that closed form inside the NoGraviton development, so downstream certificates can name a single algebraic fact: the coupling involves no gauge-group generator.
proof idea
One-line term proof: apply the upstream closed-form theorem kappa_rs_closed_form, which itself is definitional (rfl against kappa_rs := 8 * phi ^ 5). No additional algebra is performed here.
why it matters
This is claim (1) of the module's three-part RS resolution of G-004: $\kappa=8\varphi^5$ is algebraic in $\varphi$ alone, so no gauge boson is required to set the gravitational strength. Downstream, kappa_fibonacci_form rewrites the same equality via the Fibonacci identity $\varphi^5=5\varphi+3$, and no_graviton_cert packages it as the coupling_algebraic field of the NoGraviton certificate (alongside emergent gravity, two GW polarizations from $D=3$, and a positive BMV coupling). Framework landmarks: T6 forces $\varphi$; the eight-tick and $D=3$ steps sit nearby in the same certificate. The numerical BMV rate $\kappa_{\mathrm{rs}}\approx 88.7$ is the concrete falsifiable readout.
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