Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice, and the divergence of its one-loop graviton limit

We study a real-time path sum over squared edge lengths on the Kuhn triangulation of the periodic hypercubic lattice, with the Lorentzian Regge action and the flat metric I − κ11ᵀ, at κ = 1, where the edges that advance one coordinate are null. The value κ = 1 is an identification, motivated by an order on those edges. Classically, linearized Einstein gravity is recovered as a long-wavelength limit of lattice solutions. On every torus of odd side not divisible by 105 the flat Hessian has no zero modes beyond constant strains and vertex moves. For band-limited conserved sources off the light cone, the long-wave limit of the linear lattice field is the de Donder solution, with an explicit stability constant. With a resting body and a model medium, the unconstrained Regge equations on all edges have exact solutions for small coupling; their weak-field long-wave limit is that field and gives Newton's potential with G = 1/(8πs), where s is the action scale. These solutions are related to the path sum on its null stratum at linear order. At fixed source strength the long-wave equations are controlled uniformly in the torus size, while a lattice resonance rules out uniform stability at all wavelengths. At tree level the lattice has no observable signature for spacings up to 10⁶ Planck lengths. At one loop, in a stated diagram model, the quantum limit fails. Four exact massless doublers at the corners of the Brillouin zone, whose squared mass is linear in the background strain, make the zero-momentum anisotropy of the graviton self-energy diverge like 5/(44√335 π²) log(1/ε) as the regulator ε is removed after the volume, because every other degenerate zero of the operator is non-pinched; the proof's computations are exact. The long-wave self-energy also has a remainder that no gauge choice, ghost or covariant factor removes; what it means physically depends on whether the real-time path sum has a unitary evolution, which is not established. The one-loop two-mass amplitude assembled from the long-wave Feynman rules, which are the Einstein–Hilbert rules in lattice coordinates, has the coefficient −(41/(10π)) G²ħ m₁m₂/r³ in an infrared-subtracted Born reading. The lattice path sum at κ = 1 is not shown to inherit it: the doublers couple to the static field of a mass without derivatives, and in an exact local model their pair loop carries a logarithm that would add a log q²/q⁴ term to the static amplitude, more singular than the one that carries 41/(10π), unless the rest of the Brillouin zone cancels it. The path-sum measure is chosen, not derived.