kappa_positive
plain-language theorem explainer
The RS Einstein coupling κ_RS is strictly positive. Gravity and cosmology certificates cite this to discharge the κ>0 field (conservation and well-posed Newtonian/EFE limits). The proof is a one-line reexport of the ZeroParameterGravity positivity lemma, which reduces to 8φ⁵>0.
Claim. The Recognition Science Einstein coupling satisfies $0 < \kappa_{\mathrm{RS}}$, where $\kappa_{\mathrm{RS}} = 8\varphi^5$ and $\varphi$ is the golden ratio fixed by the self-similarity step of the forcing chain.
background
In RS-native units the Einstein coupling is not free: ZeroParameterGravity sets $\kappa_{\mathrm{RS}} = 8\varphi^5$. The factor $\varphi^5$ is the same rung that appears in the RS projection of Newton's constant ($G = \varphi^5/\pi$ in the primer units; the continuum bridge writes $G = \lambda_{\mathrm{rec}}^2 c^3/(\pi\hbar)$). The factor 8 is the discrete-to-continuum normalization that matches the linearized Einstein equation $\nabla^2\Phi = 4\pi G\rho$ once the lattice Laplacian is restored.
This module (Cubic Lattice Regge Convergence) replaces the general Cheeger–Müller–Schrader axiom by a direct argument for J-cost on the cubic lattice $\mathbb{Z}^D$. Positivity of $\kappa$ is a structural prerequisite for the Newtonian limit stated in the nearby comment: positive source implies positive potential curvature, and for the FullGR certificate fields kappa_derived / kappa_positive.
Upstream, ZeroParameterGravity.kappa_pos already proves $0 < \kappa_{\mathrm{RS}}$ by unfolding the definition and using $\varphi > 0$.
proof idea
One-line term proof: apply ZeroParameterGravity.kappa_pos directly. That lemma unfolds $\kappa_{\mathrm{RS}} = 8\varphi^5$, then uses mul_pos on the literal $8 > 0$ and pow_pos on $\varphi > 0$ (from the constants layer). No new algebra is done in this file.
why it matters
Closes the positivity obligation that every RS→GR certificate carries. Downstream, FullGRCertificate / full_gr_certificate (and the v2 variants) require kappa_positive : 0 < dimension.kappa alongside kappa_derived : kappa = 8\varphi^5 and discrete Bianchi conservation; the structure doc lists “kappa > 0 (conservation)” among the unconditional proved steps. The same fact is wired into CubicReggeConvergenceCert / cubic_regge_convergence_cert, which replaces the three NonlinearConvergence axioms for weak-field cubic Regge theory. Cosmology certificates (e.g. sphaleron-rate provenance) reuse the positivity pattern for their own κ fields. Framework landmarks: T6 forces $\varphi$; the mass/gravity ladder uses $\varphi^5$ as the Planck/recognition bridge rung; without $\kappa > 0$ the continuum variational limit would not yield a well-posed attractive Newtonian Poisson equation.
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