weak_field_coupling
plain-language theorem explainer
The Einstein gravitational coupling in Recognition Science equals exactly 8φ⁵, with no free parameter. Continuum and weak-field gravity arguments cite this identity when matching the discrete ledger to the Einstein–Hilbert side. The proof is a one-line appeal to the closed-form definition of the RS coupling.
Claim. The Recognition Science Einstein coupling equals $8\varphi^{5}$: $\kappa_{\mathrm{RS}} = 8\varphi^{5}$, where $\varphi$ is the golden ratio fixed by the self-similar cost fixed point.
background
The module builds the zero-parameter bridge from discrete RS ledger sites to a Lorentzian continuum: J-cost lattice to quadratic form to Laplacian to Minkowski interval, then weak-field curvature and Einstein equations. Module architecture lists the coupling $\kappa = 8\varphi^{5}$ as derived (not fitted), alongside forced Lorentzian signature, $c = 1$ voxel per tick, and $D = 3$.
Upstream, kappa_rs is the named RS prediction for the Einstein coupling, defined as $8\varphi^{5}$. The companion closed-form theorem states that this definition is literally that factor. In RS-native units $c = 1$ and $G = \varphi^{5}/\pi$, so the standard factor $\kappa = 8\pi G$ collapses exactly to $8\varphi^{5}$.
Local siblings establish the Minkowski form, causal trichotomy, and light-cone speed limit that the weak-field story sits on top of.
proof idea
One-line term proof: apply the upstream closed-form identity that the RS coupling equals $8\varphi^{5}$. That identity is reflexivity on the definition of the coupling, so no further algebraic work occurs here.
why it matters
This declaration records the derived coupling inside the continuum-emergence module so the master continuum-limit certificate can treat $\kappa$ as fixed rather than phenomenological. Downstream, the continuum limit certificate packages signature, causality, and related fields as a single proved bundle; the coupling identity is the gravity-side constant that certificate-level arguments rely on when matching defect perturbations to Einstein form.
Framework-wise it sits on the zero-parameter gravity path: $\varphi$ is forced (T6), $G = \varphi^{5}/\pi$ in RS units, and $\kappa = 8\pi G$ becomes $8\varphi^{5}$. The module contrasts this with the ILG time-kernel $w_t$, which is phenomenological; here the coupling is not fit to galaxy data. It does not by itself close the full Einstein-equation derivation, but it pins the constant that any such derivation must carry.
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