From Pointwise to Two-Point Continuum

his work revises the continuum limit of discrete, signed relational networks by shifting the focus from pointwise Laplacian convergence to a two-point (Green-function) formulation. The paper argues that the physically meaningful continuum limit is encoded in the convergence of the discrete Green’s function rather than in local operator convergence, providing a more natural bridge between relational networks and emergent geometry. A theoretical framework is developed in which geometry is primarily encoded in the spectrum and two-point structure of a signed relational Laplacian, with the discrete Poisson equation elevated to a foundational role. A scaling hypothesis for the resolution parameter ε(N)\varepsilon(N)ε(N) is formulated, and a reproducible numerical protocol is presented on random geometric graphs, demonstrating stable spectra, well-posed discrete Poisson equations, and consistent two-point structure. The work further interprets coarse-graining as a renormalization-group flow on relational structure, clarifies how temporal ordering emerges from gradient stabilization of global consistency, and argues that while the framework yields a robust gravitational analogy, it does not derive Einstein’s equations. This version (V2) consolidates the two-point perspective as the primary continuum notion for signed relational networks. This v2 records the formal closure of the original one-point framework developed in Relational Consistency as an Emergent Gravitational Principle and directs readers to the superseding two-point formulation in From Pointwise to Two-Point Continuum Limits in Signed Relational Networks (https://doi.org/10.5281/zenodo.18459163