From Pointwise to Two-Point Continuum Limits in Signed Relational Networks

This work develops a two-point (Green-function–based) continuum limit for signed relational networks, extending and refining earlier pointwise Laplacian-based approaches. The paper shows that a two-point perspective provides a more robust bridge between discrete relational structures and emergent continuum geometry, yielding clearer mathematical footing for interpreting consistency densities as effective gravitational potentials. Through analytic arguments and numerical diagnostics, it demonstrates how discrete Green’s functions converge to their continuum counterparts and how this framework resolves limitations of pointwise operator convergence observed in prior work.