Induced Gravity at the Arithmetic Interface: A Conditional Derivation of Newton's Constant

Abstract

Two questions motivate this paper: why does mass curve spacetime, and why is gravity so weak? In the dual-domain ontology, spacetime is emergent, so gravity must be induced. Applying the one-loop induced-gravity mechanism to the DDC interface yields the Einstein equations with Newton constant G=12πl_min^2/N_eff, scheme-robustly of order unity G_ind·N_eff/l_min^2=O(10). The mode count N_eff is fixed by the interface fluctuation operator, constructed in three stages. The mass matrix decomposes as M^2=(L+V)/D, where L is the graph Laplacian of the 137-channel shell and V comes from a rate equation for coherence transport. Under strictly one-way transduction—the same irreversibility that gives spacetime its arrow of time—the surface block vanishes, producing exactly 48 massless modes with a spectral gap of 17.7. The plateau N_eff=48 is exact, giving l_min≈1.13 l_Pl. The count is one of light modes, not of participating channels: all 137 transduce, and the interior does so through the surface. The equivalence principle is automatic; G is positive for curvature coupling ξ<1/6. The cosmological constant is diagnosed rather than solved: the framework reproduces the correct form but contains only the microscopic scale l_min, whereas observation requires a second, cosmological scale. Every mechanism within the one-scale ontology fails for this structural reason. Three fundamental constants are housed in the interface: c, G, and ℏ. The fourth, Λ, requires the global integral of transduction over cosmic history. The derivation is complete up to the dictionary relating Parisi-Wu fictitious time to physical time, and all quantitative results are independent of that dictionary. Keywords: induced gravity; Newton's constant; cosmological constant; minimal length; entanglement entropy; dual-domain cosmology