Gravity and fermions on a bond-centred lattice

Can a lattice support the weak gravitational field and relativistic matter without introducing unstable or unwanted light modes? We examine two constructions on a lattice of cells centred on the bonds of a cubic grid. The first is a linearised gravitational action with continuous time. Its curvature and kinetic terms are taken from Regge calculus and the canonical formulation of general relativity. A condition on the lapse constraints removes an instability found in a predecessor action. For the resulting law, PL-1N, an exact finite certificate establishes leading long-wavelength constraint closure in every direction, conditional on the stated exact coefficients. Spectral tests find two massless transverse-traceless modes, one mode at the lattice scale, and no physical instability in the sampled region. These tests do not prove stability throughout the Brillouin zone. The Newtonian and stationary gravitomagnetic responses check imposed normalisations; the time-dependent source sector still differs from general relativity at finite spacing. The second construction is a Wilson-type fermion action, PP+, on a separately assumed four-dimensional extension. Products of spinor projectors give one free light Dirac field and time-cell reflection-positivity evidence, with a conditional extension to U(1) and SU(3) gauge links. Hypercubic symmetry protects a common leading kinetic cone in the symmetric continuum description, while allowing species-dependent higher-order dispersion. The fermion's leading anisotropy coefficient is exactly +1/24. We illustrate how existing high-energy constraints would restrict a physical spacing if the free PP+ branch and its dispersion coefficient describe electrons. The two constructions establish properties of specified actions. Their combination into a consistent theory, and their derivation from quantum information, remain open. Release. P1, revision 5, 18 October 2026. This deposit publishes the existing 28-page manuscript as a preprint. Its internal draft wording is retained; no external peer review is claimed. Evidence availability. This is a PDF-only deposit. The manuscript source, calculation scripts, witnesses, saved outputs and internal review records are retained by the author and are not deposited with this release. A complete immutable reproducibility archive remains outstanding. Paths printed in the manuscript do not establish public availability. Scope. The results concern the specified PL-1N gravity and PP+ fermion actions. Whole-zone stability, nonlinear completion, a consistent combined matter–gauge–geometry action, universal gravitational response and derivation from quantum information remain open. The manuscript discloses its AI-assisted construction, checking and exposition.