kappa_fibonacci_structure

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 mediating particle is a category error.

The native coupling is defined in Zero-Parameter Gravity by $\kappa_{\mathrm{RS}} := 8\varphi^5$. Because $\varphi$ obeys $\varphi^2 = \varphi + 1$, powers of $\varphi$ collapse to linear Fibonacci combinations: the identity $\varphi^5 = 5\varphi + 3$ (with Fibonacci coefficients $F_5 = 5$, $F_4 = 3$) rewrites the fifth power without residual exponents. The same Fibonacci form appears in the Quantum-Gravity Octave Duality development as $\kappa = F_6\cdot(F_5\varphi + F_4) = 8(5\varphi + 3)$.

Locally this sits beside the claims that $\kappa$ comes from $\varphi$ alone, that GW polarizations equal two in $D=3$, and that the BMV entanglement rate is the numerical value of $\kappa_{\mathrm{RS}}$.

One-line term proof: the goal is definitionally identical to the statement of the sibling lemma that $\kappa_{\mathrm{RS}} = 8(5\varphi + 3)$, which itself rewrites $\kappa_{\mathrm{RS}} = 8\varphi^5$ by the Fibonacci identity $\varphi^5 = 5\varphi + 3$. No extra algebra is performed here.

This is the clean Fibonacci certificate that $\kappa$ is algebraic in $\varphi$ alone. In the G-004 no-graviton package that fact is the first concrete claim: no gauge-group generator enters the coupling; $\kappa = 8\varphi^5$ (equivalently $8(5\varphi + 3)$) is a number-theoretic consequence of the cost function and the forced self-similar fixed point $\varphi$ (forcing chain T6). Downstream siblings use the same $\kappa$ for positivity, the two GW polarizations forced by $D=3$ (T8), and the BMV rate $\kappa_{\mathrm{RS}}\approx 88.7$ as a falsifiable emergent-gravity prediction. The declaration itself has no further dependents yet; it freezes the algebraic normal form used throughout the certificate.