Λ-EG Emergent Gravitational Coherence and Galactic Dynamics Without Particle Dark Matter

Abstract

Lambda-EG v10 # RELEASE OVERVIEW — This release updates the de Sitter derivation of the critical acceleration scale and clarifies the separation between the primary scalar construction and the Dirac–Lichnerowicz consistency route. ## Key changes Replaced the previous horizon heat-kernel interpretation of the $4\pi^2$ factor with the coherent-sector phase-space measure $\operatorname{Vol}(T^2)=4\pi^2$, identified as the product of a Lorentzian orbital circle and a Euclidean thermal circle. Incorporated the $4\pi^2$ measure and the $\sqrt{3}$ spectral gap into the single de Sitter partition function $Z_{S^4}=\operatorname{Vol}(T^2)[\det'(-\Box+m_{\rm eff}^2)]^{-1/2}$, removing the previous statement that their variational unification remained open. Recast the fluctuation determinant explicitly as a minimally coupled real scalar operator, $-\Box+m_{\rm eff}^2$, with $m_{\rm eff}^2=DH^2$, rather than $-\Box+R/4$. Clarified that the effective scalar mass scale originates from the dimensional-channel invariant $\operatorname{tr}(\theta_{ij}^2)=DH^2$, while the Dirac–Lichnerowicz $R/4$ term is retained only as a secondary geometric consistency relation. Added explicit heat-kernel and determinant-statistics checks supporting the scalar interpretation: $\operatorname{tr}(\mathbb{1})=1$ and the bosonic determinant exponent $-1/2$. Reworked the mass-to-acceleration discussion so that the tidal construction $a(r)=H_0^2r$ is presented as a formal consistency check using the reduced Compton wavelength, rather than as a curvature-induced Dirac mass derivation. Clarified that the $R/4$ route does not imply that the scalar field is spinorial or that the scalar fluctuation operator is a Dirac operator. Updated the critical-scale construction to emphasize the separate structural roles of the coherent-sector phase-space measure and the spectral gap: $g_{\rm crit}\propto[\operatorname{Vol}(T^2)\sqrt{\lambda_1(S^3)}]^{-1}=1/(4\pi^2\sqrt{3})$. Updated the final summary to identify the coherent-sector $T^2$ measure and the $S^3$ spectral gap as the two structural ingredients entering the critical-scale construction, with the microscopic decoherence coupling remaining as Open Problem O1.