kappa_integer_phi_power
plain-language theorem explainer
The RS Einstein coupling equals an integer times an integer power of φ, namely 8·φ⁵. Anyone arguing that gravitational strength is number-theoretic (not a free gauge coupling) cites this existence form. The proof is a three-line term witness: n=8, p=5, and the closed-form identity.
Claim. There exist natural numbers $n$ and $p$ such that the RS Einstein coupling satisfies $\kappa_{\mathrm{rs}} = n\,\varphi^{p}$ with the concrete values $n=8$ and $p=5$.
background
Module G-004 (NoGraviton) treats gravity as emergent ledger curvature, not a spin-2 force. Three formal claims support that stance: the coupling is algebraic in φ alone, GW polarizations equal two in D=3, and the BMV rate is a fixed positive number.
Upstream, kappa_rs is defined as $8\varphi^5$ in ZeroParameterGravity, and kappa_rs_closed_form is the reflexivity theorem that this definition equals that product. φ is the golden ratio forced as the self-similar fixed point (T6). The integer 8 tracks the eight-tick octave (T7); the exponent 5 is the same rung that appears in the RS-native constants $\hbar=\varphi^{-5}$ and $G=\varphi^5/\pi$.
The present statement packages that closed form as an existential claim: the coupling is literally an integer times an integer power of φ.
proof idea
Term-mode existential introduction. The witnesses are $n=8$ and $p=5$. The equality $\kappa_{\mathrm{rs}}=8\varphi^5$ is supplied by kappa_rs_closed_form (itself rfl on the definition). The two conjuncts $n=8$ and $p=5$ are discharged by rfl. No further algebra is required.
why it matters
Feeds no_graviton_cert as the field coupling_integer_phi. That certificate bundles five facts: gravity is not force-mediated, the coupling is algebraic in φ, the coupling is an integer φ-power, GW polarizations equal two, and the BMV coupling is positive.
In the G-004 resolution, κ=8φ⁵ being integer-times-φ-power is the concrete evidence that no gauge-group generator enters the gravitational strength. The coupling is a cost-function consequence on the φ-ladder, not a free parameter of a mediating particle. Together with the sibling kappa_from_phi_alone, this closes the "no gauge boson" leg of the no-graviton argument and underwrites the BMV prediction κ_rs≈88.7 as a pure RS number.
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